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A group analysis of nonlinear differential equations Kumei, Sukeyuki 1981

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A GROUP ANALYSIS OF NONLINEAR DIFFERENTIAL EQUATIONS  by  V SUKEYUKI KUMEI B.Sc., The Tokyo University of Education, 1967 M.Sc., The University of the Pacific, 1972  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES (THE INSTITUTE OF APPLIED MATHEMATICS AND STATISTICS,:AND DEPARTMENT OF MATHEMATICS)  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA August'1981  (5) Sukeyuki Kumei, 1981  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study.  I further  agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives.  It is  understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  VAAVvVNAfr 'W LM  The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e  Pw^.  ,  ABSTRACT  A necessary and sufficient condition is established for the existence of an invertible mapping of a system of nonlinear differential equations to a system of linear differential equations based on a group analysis of differential equations.  It is shown how to construct the mapping, when it  exists, from the invariance group of the nonlinear system. It is demonstrated that the hodograph transformation, the Legendre transformation and Lie's transformation of the Monge-Ampere equation are obtained from this theorem. equation (u n K xJ  The  xx-u yy=0 is studied and it is determined  for what values of p this equation is transformable to a linear equation by an invertible mapping. Many of the known non-invertible mappings of nonlinear equations to linear equations are shown to be related to invariance  groups of equations associated with the given  nonlinear equations.  A number of such; examples are given,  including Burgers' equation u  +uu -u =0, a nonlinear XX  X  l  _2 diffusion equation (u u ) -u =0, equations of wave propagation XX u {Vy-wx=0, vy-avw-bv-cw=0}, equations of a fluid flow {w^+v x =0, w -v~"'"wP=0} u =e u . J and the Liouville equation H x xy As another application of group analysis, it is shown how conservation laws associated with the Korteweg-deVries equation, the cubic Schrodinger equation, the sine-Gordon equation and Hamilton's field equation are related to the  invariance groups of the respective equations. All relevant background information is in the thesis, including an appendix on the known algorithm for computing the invariance group of a given system of differential equations.  ACKNOWLEGEMENTS  I would like to express my sincere acknowlegement to Professor George Bluman, my thesis supervisor, for making it possible for me to come to the University of British Columbia and to carry out this research.  Without his helpful,  enlightening and often very patient discussions the completion of this thesis would have been difficult.  His encouragement at  times of difficulty is particularly appreciated.  I am grateful to other members of my thesis committee, Professors C.W. Clark, F.W. Dalby, R.M. Miura and B.R. Seymour, for their valuable time and criticism on this thesis.  I am  particularly indebted to Professor Miura".:for a number of critical comments which helped to improve the clarity of the presentation of the thesis.  I am also grateful for the award of the University Graduate Fellowship which enabled me to concentrate on the research.  I dedicate this thesis to Professor Carl E. Wulfman whose faith and love to nature deeply influenced my thought. particularly grateful to him for introducing me to the wondrous world of.symmetries..  I am  TABLE OF CONTENTS  page 1  INTRODUCTION I.  CONTACT TRANSFORMATIONS. 1.1 1.2  Contact transformations. Invertible contact transformations.  15 17  1.3 1.4 1.5 1.6  Infinitesimal contact transformations. Properties of generators. Invariance contact transformations. Invariance contact transformations of differential equations. Invariance groups of linear equations. Contact transformations of the spaces y,X.  20 23 26 29  1.7 1.8 II.  15  INVERTIBLE MAPPINGS OF NONLINEAR SYSTEMS TO  30 37 41  LINEAR SYSTEMS. 2.1  Theorems on the existence of invertible  2.2 2.3  mappings. Remarks on the use of theorems. Examples.  '. ^ 42 50 52  2  A. The equation z + %(z ) - z = 0. r2 J xx x y ^ B. Hodograph transformations, 54 C. The Legendre transformation. 56 D. Lie's theorem con the Monge-Ampere equation. 59 E. The equation (z - z =0. 61 X  XX  j7  III. NON-INVERTIBLE MAPPINGS OF NONLINEAR SYSTEMS TO  67  LINEAR SYSTEMS. 3.1  Examples of non-invertible mappings,  67  3.2  A use of potential functions.  70  A. A nonlinear wave equation. B. An equation of a fluid flow.  IV.  71 75  3.3  The Liouville equation.  79  3.4  A series of Lie-Backlund generators and the linearization.  81  A SUMMARY AND FUTURE PROBLEMS.  82  4.1  A summary of the main results.  82  4.2  A generalization of the concept of invariance.  83  A. A hierarchic structure in Lie-Backlund sequences.  84  B. On integral dependent generators.  88  BIBLIOGRAPHY.  93  APPENDICES. 1. A proof of theorem 1.  97  2. Proofs of propositions 9-12.  99  3. An example of computing invariance groups: a determination of an invariance group of the equation (z  2  ) -4z z =0. xy J x y 4. On the remarkable nonlinear diffusion equation. n  101  K  ,105  5. Invariance transformations, invariance group transformations and invariance groups of the sine-Gordon equation.  110  6. Group theoretic aspects of conservation laws of nonlinear dispersive waves: KdV type - equations and nonlinear Schroedinger equations,  118  7. On the relationship between conservation laws and invariance groups of nonlinear field equations in Hamilton's canonical form.  127  INTRODUCTION.  A variety of transformations arises in the study of differential equations.  Two important classes of transform-  ations are integral transformations, which include the Laplace and Fourier transformations, and geometrical trans-formations, which include contact and point transformations. In this work we are concerned with transformations of the latter type.  One of the important aspects of geometrical  transformations is that their formulations generally do not depend on the linearity of differential equations while those of integral transformations .depend critically on it.  For this particular reason, in connection with  recent developments in nonlinear physics, there has arisen a revived interest in various geometrical theories of differential equations [l] .  In this thesis we focus our attention on a group analysis of differential equations [2-9] and its. use for the study of relationships between differential equations. More specifically, we are interested in answering the question when a given system of nonlinear differential equations can be transformed into a system of linear differential equations. There is a good reason to believe that a group analysis of differential equations is helpful in answering this question.  Before we elaborate the motivation, we need to review some basic ideas of geometrical  transformations important for  the study of differential equations.  In order to keep  the geometrical picture simple we only consider the case involving one dependent variable u and one independent variable x.  Let w be a vector space with coordinates (x,z,z,z,...) 1 2 where xtR, z^R, ze.R and consider a mapping w+w: k x'=x'(x,z,z,...,z), 1 P  z'=z' (x,z,z,...,z), 1 q .  z ' =z ' (x,z,z,...,z) k k 1 qfc.  (0.1) where x', z' and z' are sufficiently differentiable functions k of their arguments and k=l,2,3,... . Let u(x) be a function R-*R, sufficiently differentiable in the domain of interest and u(x) = (d/dx) u(x). k z = u,  If we set  z = u, k k  k=l,2,3,...,  (0-2)  then the set w [u] consisting of points  (x,z,z z,...) = (x,u,u,u, . . .) , 12 12  x  R  defines a curve (more generally a manifold) in w (Fig.l). Under the transfbrmation (0.1) the curve %[u]is mapped into (w tul) ' whose equation is obtained by introducing (0.2)  (0.3)  into (0.1)  x'=x'(x,u,u,...,u), z '=z' (x,u,u, . . . ,u), z' = z'(x,u,u,. . . ,u) . q k k ^k (0.4)  Solving the first equation of  z  1  (0.4) for x and introducing it into the rest of (0.4) we  \w[u|]: (x,u,u)  obtain z', z 1 as functions of k x* : Z ' = V C k ; ) (x') .  Z ' = U ' (X ' ) ,  2  k  Fig.l  Obviously not all transformations (,0.1") have the property v ( k ) ( x ' ) = (d/dx')k u'(x') = u'. k  (0.5)  When the equality (0.5) is satisfied for any choice of u(x), i.e.,  z' = u',  z' = u', k k  k=l,2,3,... ,  we call transformation (0.1) a contact transformation. Namely, a contact transformation maps a curve w[u] defined by (0.3) into a curve w[u'] (Fig. 2) consisting of points  (x, z , z , z ,...) = (xj u',u',u','...) , 12 1 2  x'tR.  It is intuitively clear that in contact transformations  (0.6)  z' is related to x' and z' k because z' must behave as k  z 1  w[u] : (x,u,u)  the k r t h derivative of z' with respect to x' when (0.3) is introduced.  Obtaining an w[u'] : (x'.u'ju')  explicit form of z' in terms k of x' and z' from condition that (0.2) yields the (0.6) is not only messy but also becomes •  Fig, 2  confusing for vector x and vector z. It is very convenient to replace lq.(0.2) and Eq.(0.6) by equivalent differential forms.  dz = udx, 1  dz = udx, 1 2  Eq.(0.2) implies  dz = udx, 2 3  (0.7)  and if we use (0.2) it can be written as  dz = zdx, 1  dz = zdx, 1 2  dz = zdx, 2 3  (0.8)  If we let z = u in the first equation of (0.8), we find z=u and if we let z=u in the second we find z=u and so on, 11 11 2 2 recovering (0.2). Thus, we may replace (0.2) by (0.8) which we represent collectively by {dz - z dx=0}. Similarly, k k+1 we replace (0.6) by {dz' - z'dx'=0}. We redefine a • k k+1 contact transformation as a transformation (0.1) which has  the property that:  if {dz - z dx=0}, then {dz' - z'dx'=0}. k k+1 k k+1  (0.9) +  We call (0.8) a tangent (or contact) condition. The simplest contact transformation is a point ft  transformation! x' = x'(x,z),  z' = z'(x,z),  z' = z'(x,z,z,. . . , z) , k k l k  (0.10)  and the next simplest one is Lie's contact transformation  t  The first equation of (0.8) written in the form of scalar product (dz , dx)-(l, - z) = 0 implies that the vector (dz,dx) is perpendicular to the vector (1,-z) which represents a normal vector t o a curve' z-u=0 when z is replaced by u. 1 1 Consequently, the variation (dz,dx) must always be tangent to the curve z-u=0. In order to determine z' from z' and x' we introduce z',x' into dz'-z'dx'=0, i.e., 1 0=dz'-z'dx'=(z'dx+z'dz+z'dz+...) v J - z'(x'dx+x'dz+x'dz+...) J ,, 1 x z zi i x z zT 1 . . . . 1 1 where z'=8 z', x'=8 x', ... . We eliminate dz, dz, dz,... x x ' x x ' 1 2 using (0.8), and obtain 0=(z' + z'z + z'z + ...) - z'(x' + x'z + x'z + ...). x z1 z2 l x Z! z2 Solving this for z', we obtain z!. Using this z in dz'-z'dx« = 0 we find z', and so on.  tf  Usually a set consisting of the first two equations in (0.10) is called a point transformation. In the following we call (0.10) a point transformation.  t  x' = x'(x,z,z), 1  z' = z'(x,z,z), 1  z' = z ' (x, z,z,...,z) . (0.11) k k 1 k  The most general contact transformation of the form (0.1) was considered by Backlund [26],  A particularly important class of contact transformations is that of infinitesimal contact transformations. Consider a transformation  x' = x + e£(x,z,z,...,z) 1 P z ' = z + e?(x,z,z,...,z) 1 5  z' = z + er(x,z,z,...,z ),  k  where  k  k  1  qv  (0.12) k=l,2,3,...  and z are functions of their arguments and e is k  a small parameter.  We call (0.12) an infinitesimal contact  transformation if it satisfies the condition (0.9) to order 0(e).  When  (x,z), £=£(x,z) the transformation (0.12) is  called an infinitesimal point transformation and when E,, £ and c are functions of x,z,z, the transformation is called 1 1 Lie's infinitesimal contact transformation. All other cases  t  Usually the term contact transformation refers to Lie's contact transformation. Lie's contact transformation has the property of mapping two curves z=u(x) and z=v(x) which are tangent to each other at Xq into two curves z'=u'(x') and z'=v'(x') which are also tangent at the transformed point x^j. Under the definition (0.9), not all- contact transformations have this property.  will be called higher order infinitesimal contact transformations .  As in the case of finite contact transformations  (0.1), once £ and £ are given the function | are determined from the condition (0.9) as described in the footnote on page five.  A succession of infinitesimal transformations (0.12) leads to a finite transformation which is called a group transformation.  Its geometrical picture is the following.  Eq.(0.12) associates a variation vector (Ax, Az , Az , . . .) = e (£, t;, ^ with every point (x,z,z,...) and defines a flow in the space w (Fig.3).  Let the z 1  equation of a flow curve originating at a point (x,z,z,...) be V  X 1 —X 1 (x j Z j Z j  » • .  jcl}  z'=z'(x,z,z,...;a)  (0.13)  , ... ,  Z ' — z ' (x, z , z , . . . k k ' '1  z  j  Fig.3  X  where a_- is a parameter of the curve.  For simplicity, we  denote co= (x, z , z , . . . ) and write (0.13) as 1 a)' = a)' (w; a) = T(a) w,  where T(a) is generally a nonlinear operator acting on to.  (0.14)  The transformation (0.14)  forms a one-parameter group  of transformations. Namely, there exists a parametrization with parameter a such that  <L' (u>f (oj;a) ;b) = u'(oa;a+b),  (L' (oi; 0) = oj,  (0.15)  or equivalently,  T(b)T(a) = T(a+b),  T(0) = I,  where I represents the identity transformation.  (0.16)  The explicit  form of the operator T(a) is given by  T(a) = e  a£  OO J| = I * £n, n=0  (0.17)  where I is defined by £ = C8 x + ?3 Z +  1  •+ C3 Z + . . . , 2 1 2  (0.18)  The operator I is called a generator of the group T(a). If (0.12) is an infinitesimal contact transformation, then its "integrated form" (0.13) is also a contact transformation which we call a group contact transformation. Depending on the type of infinitesimal contact transformations mentioned above we call the corresponding group contact transformations a point group transformation, Lie's group contact transformation or a higher order group contact transformation.  The first two group transformations were studied extensively by Lie  [2,4]' in the last century while the higher order  group contact transformations were introduced recently by Anderson, Kumei and Wulfman [9,10,11,12].  They are also  called Lie-Backlund (L-B) group transformations [l3] .  Up to now a picture of differential equations has been absent.  To find the significance of contact transformations  in the study of differential equations we consider an equation  f(x,z,z,... ,z) = 0, 1 n where f is an analytic function R n + ^->R.  (0.19)  Eq.(0.19) defines a  hypersurface (or a manifold) in the space (x,z,z,...,z). I n We again consider (0.2).  Let w' n ^[u] be the set consisting  of points (x,z,z,...,z) = (x,u,u,...,u), I n I n The set  x£R.  (0.20)  [u] defines a curve (generally a manifold) in  the Sp£LC6 (x j Z } Z , , , , , z ) , Let us demand that the curve 1 n w fn) [u] be imbedded on the hypersurface f=0 (Fig. 4). This will be possible only if the function u  happens  to be a solution of the differential equation  f(x,u,u,...,u) = 0. 1 n  (0.21)  When u solves (0.21), we call w'-11-' [u] a solution curve (more generally a solution manifold).  The  projection of the curve onto the x-z plane defines a solution curve z=u in the usual sense (Fig.4). Fie.4  We now suppose that the  z=u  contact transformation (0.1) maps the hypersurface f=0 into a hypersurface  (0.22)  f' ( x \ z \ z \ . .,z') = 0, n 1 Since the transformation.is a contact transformation, the curve w^-11-' [u] is mapped into the curve  [u'] (Fig.2)  consisting of points  (x,z,z,,..,z) 1 n  (x',u',u',. 1  n  (0.23)  Obviously, if the curve w^ n ^[u] is on the surface f=0, then the curve w^-11-' Ju'] must be on the surface f'=0 (Fig. 5), namely, z 1 f'(x',u',u',...,u') = 0 . . 1 n  (0.24)  In other words, a contact transformation maps a solution of the Fig.5  differential equation (0.21) into a solution of the differentia equation (0.24).  When the contact transformation maps the  hypersurface f=0 into itself, the transformation is called an invariance contact transformation of f=0.  Obviously, such  a transformation maps a solution of the differential equation (0.21) into another solution of the same equation (Fig.6). When a solution happens to z be mapped into itself under such a transformation, it is called an invariant solution of the transformation. cularly  Parti-  important invariance  contact transformations are  Fig. 6  those which form groups in the sense of (0.13)- (0.15). We call, them invariance group contact transformations, or invariance groups for short.  Lie [2-6] studied invariance groups extensively and established the foundation of a group analysis of differential equations.^  More recently, Ovsjannikov  [6,7] extended Lie's  theory and applied it extensively to partial differential  From a practical point of view it is important that Lie gave the algorithm for finding the invariance groups of any given differential - equation. • We give one explicit example of the computation of such an invariance group in Appendix 3.' .  equations.  Bluman and Cole  [§]/ used invariance groups to  construct solutions to certain types of boundary value problems.  All these works just mentioned are concerned  with either point groups or Lie's group contact transformations.  It has been found that the Lie-Backlund group transformations are also useful, particularly for the study of nonlinear differential equations.  It is shown [Appendices .5-7  + that the well known infinite number of conservation laws admitted by the Korteweg-deVries equation u 2  +uu +u =0, XXX  X  L  the cubic Schroedinger equation u +u u*-iu =0 and the XX L sine-Gordon equation u.^-sinu=0 are all related to invariances 4- 4*  of the corresponding equations under L-B groups.  More  interestingly, soliton solutions admitted by these equations are shown to be invariant solutions of these invariance L-B groups [Appendix 6 ] . [Appendix  These findings, lead to a general theorem  7 ] that with any conservation law, admitted by  a Hamiltonian system,  + tt  Appropriate references are given in the appendices. A group theoretical aspect of conservation laws of the Korteweg-deVries and sine-Gordon equations was also studied by Steudel using Noether's theorem D-4] . He called groups, leading to conservation laws, Noether transformations " [15,16] .  3pn  SH  3t  6H  (0.25) .  is associated an invariance L-B group, of the equation. Further studies of invariance L-B groups have been reported [17-24] ,  Now we return to the question posed at the beginning of this introduction. .Suppose that there exists ian invertible transformation mapping a given system of-nonlinear, differential equations to a system of linear differential equation's: We should expect that the invariance groups the same structure.  of the two systems have  Since it is well known that any linear  system admits an invariance group related to the superposition principle, it is evident that the given nonlinear system must admit the corresponding  invariance group.  From such  an observation, we establish a theorem which tells one definitively:  '  1) when a given nonlinear system can be mapped into a linear system; 2) how to construct the mapping when it exists.  It should be emphasized that in applying this theorem one needs .only  to  calculate > a • . system' s invariance Vi'nf initesimal- v  contact transformations (0.12) by a known algorithm.  Moreover,  the types of infinitesimal transformations to be considered  are simple: in general one may assume £=0, £=g(x,z,z); for 1 a system, i.e., z is a vector, it turns out that c is at most linear in z (corresponding to a point group). 1 In the first chapter, we define and formulate mathematically those basic concepts which were illustrated above and lay out the basis for subsequent developments. In the second chapter we establish the theorem mentioned above. A number of examples are given.  In the third chapter, we  examine non-invertible mappings which connect nonlinear equations to linear equations.  Appendices .5,6,7; consist  - of already published works on L-B groups and conservation, laws.  CHAPTER  1.  CONTACT TRANSFORMATIONS.  In this chapter we discuss various properties of transformations which will be considered in subsequent chapters.  All transformations cprhs;idered:%n this work, are  basically of "contact" type.  The term "contact trans-  formation" will be used in a context more general than it is usually referred to.  Throughout the work, we adopt the customary summation rule for repeated indices: Roman indices are summed from 1 to M and Greek indices from 1 to N.  1.1  Contact transformations.  Let w be an infinite dimensional vector space with coordinates CO = (x , z , z , z , . . . , z , . . . ) 12 n where Z..R™  x = (x^ ,X2 , • • •  e  z  =  consists , of- coordinates zV . 12 and i, = 1,2,...,M. For instance,  >  > • • • >  e  an  d  . with v = 1,2,...,N n  1 1 N N. i_z1,z2, . . . , z M _ 1 , z m j ,  r  z  f  z  1  w'-11-' denotes the space with coordinates where  = (x,z).  1 1 N . L z -j. 1' z 1 2 ' Z 21' ' ' " ' Z M M J  = (x , z,j , . . . ,z) ,  Let y be a space o£ functions  R,  k = 1,2,3,..., analytic in a given domain D(w) of w. We consider a transformation T:w ->• w defined by  z 1 = z'(x, z , z , . . .) ,  x' = x ' (x, z , z , . . .) , 1  k'  =  ^'  ' z ' |' ' ° ° '  (1.1)  1 k  =  1> 2 > 3 , . . .  ,v v where x-' l e y, z e y, z-' i 1 i-2 . . . i.k ey.  We write (1.1) as  0)' = w' (co) = Tco,  (1.2)  and the first n+2 expressions of (1.1) as  0) (n) = 60 ' ( n ) (co) .  A set of equations  dz v - zVdx i = 0 j v . _ dz: - z-v dx 1 ^ 1 n t • ill 1 x12 . . . i k j j 12 k is called a' contact condition. simply by {dz - z-dx= 0}. 1 k k1  1  (1.3) y3y • • •  We express the set (1.3)  Definition 1.  The transformation (1,1) is a contact trans-  formation only if it preserves the contact condition, i.e.,  {dz k  - z.dx. = 0} k1 1  +  {dz' - z!dx! = 0}, k k1 1  Let u(x) be a function  +  (1.4)  We consider a space  w[u]Cw consisting of points  03 = (x,U,U,U,,,,,U,,.,), 12 k  X€ R "  where, as z, u is a vector with components k k uV . . = 3 1 1 1 2." ' , : L k  x  3 i ixi2  3 u v (x) , ' ' ' xik  The transformation (1,1) maps the space w[u] into a new space which we denote by Tw[u],  If T is a contact transformation,  then Tw [u] = w[u'] for some function u'(x).  Namely, for  any function u(x) the transformed space Tw[u] consists of points expressed as w = (x ,u' ,u' ,u' , . . , ,u' , , . ,) for some 1 2 . k function u'(x),  1.2  Invertible contact transformations.  We call the transformation oo' r11-* = aj'^Cw) the n-th extended transformation of one can determine  o)'^^. =  Given g o ' ^ ,  n>0, from the condition (1.4),  A contact transformation is said to be invertible, or 1-1,~ if there exists a- s p a c e - w ^  in which the n-th  extended transformation is a 1-1 mapping w^  , n=0,l,2,,,, .  Under the present difinition (1,4),.not all contact transformations are 1-1.  Actually only very limited classes  of contact transformations are invertible.  Two cases,  z scalar and z vector, are considered separately.  z scalar.  Backlund [26] proved for scalar z that the  most general 1-1 contact transformation is the extended Lie contact transformation.  The following theorem by Meyer  [27,28] characterizes Lie's contact transformation:  Theorem [ Meyer] .  A transformation  x' = x'(x,z,z), 1  z' = z'(x,z,z),  w^^ ,  1  z' = z'(x,z,z), 1 1 1  (1.5)  x ' r w ^ 1 ^ R M , z ' r w ^ 1 ^ R, z'-.w^ 1 ^ R M , is a Lie contact 1 transformation, i.e. dz' - z|dx| = p(x,z,z)(dz - z^dx^)  (1.6)  if and only if 1) xj, i=l,2,...,M, and z' are M+l independent functions of x,z,z and satisfy [x!,x'-] = 0, [z',x-']= 0, 1 i J i 2) z!, i=l,2,...,M, are determined from  z' + z- 3 z 1 = z!(3 x, i z 3 'i or from 3  3  z' = z!3 i  J  z  x» + z-3 x!)  (1.7)  Z"J  i!  (1.8)  ± J  and p(x,z,z) from p = 3 z z' - z!3 z x! =  z'],  k=l,2,...,M,  (1.9)  where the Lagrange bracket, [ ,] , of two functions <j>(x,z,z) and ^(x,z,z) is defined by 1 1 =  -  ^x.^  +  U-10)  The extensions of the transformation (1.5) to higher order coordinates z', n>l, are found from the contact condition (1.4). n Remark 1.  In the literature, the term "contact transformation"  usually refers to Lie's contact transformation.  z vector. For a vector z e R^, N>1, the most general 1-1 contact transformation is the extended point transformation [29] of x' = x'(x,z), R^,  R^.  are determined from (.1.4).  z' = z'(x,z),  (1.11)  The transformations for z, n>0, n  Remark 2. In the present context, a canonical transformation of classical mechanics f  = t,  q' = q'(p,q,t),  p' = p'(p,q,t),  (1.12)  t time, q generalized coordinates, p generalized momenta, corresponds to a point transformation with  t = x,  . 1 2 n^ , n+1 n+2 2n. q = (z ,z ,...z ), p = (z ,z ,.. . ,z ) .  We write the m-th extensions of (1.5) and (1.11) and their inverses as  a)' ( m ) = <I)'M(a) (m) ) = To, ( m ) ,  u) ( m ) = <L(m)(co'  Cm) (1 13)  and the infinite extensions as  "  to = co(co') = T_1co'.  u' = aj'(w) = Too,  J  (1.14)  1.3 Infinitesimal contact transformations.  We consider a contact transformation which depends analytically on some parameter a and reduces to the identity transformation I at a=0: CO' = (L' (go ; a) ,  w = w' (co; 0)  (1.15)  or co' = T (a) oo,  T(0) = 1.  (1 .16)  Expanding co' in a power series in a, we obtain «> =  n  I n=0  ( 1  •  -  1 7 )  Defining an operator £ = with ^ ^  'x. 1  T+  CV9 v + ' u z v ' ,»i0zV I  +  = (3 x!) n , C V = (3 z'V) v J a i J a = 0' s a a = 0'  = (3 £.'V) v a l Ja = 0 ,  we write (1.17) as co' = (1 + a£)w + 0(a 2 )  (1-18)  where 0(a 2 ) represents terms of order a 2 .  For (1.15) to be  a contact transformation it is necessary that  ,  , ...  in £ satisfy the recursion relation  *i...jk = V i . . . j  - zi...jmDxk^m  (1.19)  where D  xk  =  9  xk  +  z  k a z V - + z ik 3 zV  +  ••• •  Now instead of starting from a transformation of the form (1.15), we start with an operator +  A = i  ?V9zv  +  ?i9zV i  +  ••• >  CI - 20)  where E,1, £ V ey, and <;V , . , . . . are determined recursively from i 13 (1.19). We consider a group transformation  e a£  00 ft r ct w = 2, n! ^T U )  (1.21)  Transformation (1.21) always satisfies the contact condition (1.4). It was previously shown [Appendix 5] that a transformation of the type (1.15) can be represented as an infinite composition of group transformations of the form (1.21). Namely, (1.15) can be written as  (1.22) where  are operators of the form (1.20).  £ is called  a generator of a group contact transformation (1.21). Depending on the forms of E, and <;, the one-parameter groups (1.21) are classified as  = £ x (x,z), £ V = £ V (x,z)  a)  Point groups:  b)  Lie contact transformation groups: z is scalar and there exists a function C 1 = 3 Z .G, i  R such that (1.23)  C  c) Lie-Backlund groups: all other forms of E,1 and  .  The transformation  (1.21) corresponding to the first two  cases defines a mapping w^11-' -»- w'-11-' for any n>0.  For the  third case, however, the transformation (1.21) must be considered in the infinite dimensional space as w + w . In all these cases, group transformations are 1-1 in the domain where e a ^ exists.  1.4 Properties of generators.  The commutator of two generators I and  ' is  defined by  i v  b  l . . . "J J  (1.24) . .] J  Let A be the space of all generators a.  zv  1... J It is easy to  show:  Proposition 1.  If £eX and 5,'eX, then [£,£'] eA.  Thus, the form of the commutator is determined from the two leading terms using (1.19).  From now on we represent the  generators by two leading terms.as I =  X 1*  + £V3 v . z  The following property is very useful [in Appendix 6]:  Proposition 2.  Generators of the formZ = C v 3 z v commute with  the total derivative operator D i Proof. Noting that D  U,D  i  ] = U"l)3  eX, we have 1  + (lzV - D x C V ) 3 z V - (D C v - D x C v )3 z v= 0. • i i i i  We introduce into the space X an equivalence relation  Definition 2.  Two generators I and  and only if (SL - & 1 ) g |  by:  are equivalent if  q = 0 for any g e y.  The equivalence  is represented by Z = 8,'.  The symbol I g= g indicates the evaluation of the quantity for those values of  g = 0,  The equivalence  w =(x,z,z,...) which satisfy the equations  D x g = 0, i  D  D i  g = 0, ...  .  (1.25)  j  class N of OtX consists of generators  A = CXDX , i  5 l e Y-  (1.26)  + £ v 9 7 v is equivalent  Proposition 5. A generator £ = to £ = U v -  Proof.  zV?1)3zV.  + zV^ 1 3  Obviously, £ - £ =  From (1.19) we  find its full expression to be  £ - £ = £ X (3  X-  + zYa  1  1  zV.3  V+  Z  1  ZV  + ...) = J  J  ^  X-  .  •  1  As a result, elements of the quotient space XA. can be represented by generators of the form £ = 0 V ( w ) 3 z V , 6v ey.  The generators X and ^£ satisfy the same commutation  relation.  Namely,  =  Proposition 4.  Proof.  ^3  ^^  a n d  on  /\  l y if  /N  From (1.24) we have (Jl  [£ 1 ,£ 2 ] = (£ 1 5 2 " Therefore,  = {(l^  lC2 " A 2 ? 1 ) 3 Z v  - £ 2 C^) - z ^ O ^  =  - £ 2 ^J)}3 z^  On, the other hand, we have A /S  j J 3 zV  V  l  l  J  The converse is obvious.  1.5  J  •  Invariance contact transformations.  A set ,Q,f K C. functions  f V (to  ) = f V (x,z,z, . . .z) , I n  v = l,2,...,K,  (1.27)  are said to be functionally independent in the domain D(w) iff there exist K components of (z,z,...,z), denoted by 1 2 n y1 ,y 2 ,...,y K , for which the Jacobian of (f 1 ,f 2 ,...,f K ) is nonzero: '' ' D C y 1 ^ 2 , ... . ,y K )  Q in D (w) .  (1.28)  We denote by D(w;f=0) the set of points to^n-' satisfying the equations f^(to^n-^)=0, v = l,2,...,K.  f V ( t o ( n ) ) = 0,  K equations . ... ..••'..  v = 1, 2 , . ! . , K,  K<N,  (1.29)  o  are said to be independent iff the set of functions f v are functionally independent in D (w) ' aridD.(w; £=0) is nonempty. The implicit function theorem ensures that if {f V } is a set of functionally independent functions and D(w;f=0) is nonempty, then in every neighbourhood of a point toeD(w;f=0) there exists a unique set of K C^ functions ^ V ( w ( n ) ) , v = l,2,...,K, independent of y^ ,y 2 , . . . ,y K , with the property that the functions f V all vanish with the substitutions V  y  = ipV (to  ) ,  v = l, 2 , . . . ,K.  (1.30  The system (1.30) is called an explicit form of the system (1.29).  The system (1.29) is said to be a linear system  iff its explicit form is linear in z,z,...,z, namely, 1 n yv =  + 4>V (x) ,  v = l, 2 , . . . ,K,  (1.31  where Ayv is a linear operator defined by  A?z v = (a^(x) - a i x ; i (x)D Y +....+ a ^ Z - ^ f x J D , D y . . .D }z y , y y y x. y x. x. x. i i ii i 2 inn (1.32) J a^ :R (xj+R, <j> :R (x).->-Rt Defining an operator. ,matrijc tor. .matrix A  =  1^1'  w e  sometimes write (1.31) in the form y = Az + cf> (x)  (1. 33)  where y,z and (p are column vectors.  We consider a system of equations f = 0,  D  f = 0, i  D  D f = 0, ... i j  where f=0 is the independent system (1.29)'.  (1.34)  We use the  same notation D(w;f=0) to represent the set consisting of points weD(w) satisfying the system (1.34).  The set D(w;f=0)  defines a manifold in w. A contact transformation w'=co'((jo) is called an invariance contact trans formation  of the equation f = 0  iff it transforms D(w;f=0) into itself, namely,  fCS'CuO) l £ ( a ) ) = 0 - 0.  (1.35)  For a group contact transformation (1.21) to be an invariant transformation, it is necessary and sufficient that  * f(w) l £ C a ) ) = 0  =  CI-36)  Eq . (1. 36)-.. is called the determining ^equation v. Because o.f the local nature of the generators of an invariance group, we have:  Proposition 5.  A system of independent equations {f v (w)=0}  and its explicit form {yv=<j>v (co)} admit group, generators.  the same invariance  Lie gave' an algorithm [-6,7,23] to determine invariance group generators  =  x  + £V(co'-n-')3  z  i  v  satisfying the  condition (1.36) for given m and n. Clearly we have Proposition 6.  If  of the system f~0, then  and I is an invariance group generator is also an invariance group  generator of f=0.  Thus, recalling Propositions 3 and 4, we see that for the study of invariance groups, it is sufficient to consider generators of the form £ = 8 V (co  ) 3z V .  Often generators of  this form are easier to work with than generators of the form (1.20. ) and in the rest of this work we only deal with such generators.  1.6  Invariance.-contact transformations of differential equations.  In §1.1 we introduced the notation w[u] to represent a space consisting of co given by  co - (x ,u ,u ,u, . . . ) ,  M  xeR ,  M  N u:R (x)-tR .  When u is a solution of a system of differential equations,  - id. -  £ (x ,u ,u ,u, . . . ,u) = 0, 12 n  (1.37)  we call w[u] a solution surface o£ the differential equations (1.37).  It is clear that for wCu) to be a solution surface  it is necessary and sufficient that wLu] be imbedded in D(w;f=0).  Now let T:w' =w' (to) be an invariance contact trans-  formation of f(co^n-')=0.  Then from the contact property of T we  have, as mentioned in §1.1,  Tw[ u ] = w [u'] ,  (1.38)  and from the invariance property of T, we have for a solution surface wEu],  Tw [u] <=. D(w;f=0) , hence w[u'] must be a solution surface of the equations (1.37).  Therefore, any invariance contact transformation  of f(a)Cn))=o maps a solution surface of the differential equations (1.37) to another solution surface.  1.7  Invariance groups of linear equations.  Generators of invariance group contact transformations of linear equations bear special properties.  (1.39)  We consider two cases separately:  i)  ii)  SL = 0 V ( x ) 3 z V ,  0 V :R M (x)+R  £ = ev(w^)3zV,  eV:w^n^R.  To be consistent with notations to be used later.," capital letters X,Z,U,Q,... will be used in this section.  We use the term  "linear" in the sense defined in §1.5.  i) The first case.  It is known that any linear equation  admits a generator of the form L=0 V (X)3£ V .  Proposition 7.  Namely,  A system of linear equations ^ Z ^ - $V(.X) = 0,  v=l,2,...,K, K<N, $ V :R (X)->-R, with linear operator  A.  defined by = {A V (X) • + AV:1.-(X)DV y - y x^  . j4 y  + . . . + AV1l12:,-Jin(x.).Dv AD v ; • • • y in ' (1.40)  Avi.••j:R  M  ( x ) ^ R , admits a generator L=U y (X)3 z y depending  upon an arbitrary solution {Uv(X);v=l,2,...,N} of the system of linear differential equations ^U y (X)=0, i.e., A V (X)U y + A v i (X)uV + ... + A v i l i 2 - • -1 ^1( X ) ^ . • = 0, 1 y y y ii^2 • • ^ n (1.41) =  v 1, 2 , . . . , K.  }  Here and in the following, U y •  , = 9  13 • • .K  3  ...3  X-j^ Xj  U y (X) .  Xj,  Let Z=(l+cL)Z, c constant and L=U y (X) S^y def ined.-.above . It is easy to see that if 4 v Z y - $ v ( X ) =0, then the Z, a superposition of cU(X)' with Z, satisfies the equation 4^z y -§ v (X)=0. Definition 3.  An operator L ^ ^ X ) ^ ^  is said to be a super-  position generator of the equation  - ( X ) = 0 , v=l,2,...,K,  if 4 v U y (X) = 0. J U It is clear from Proposition 5 that:  Proposition 8.  A linear system F v ( f t ^ ) = 0 , v=l,2,...,K, K<N,  admits a superposition generator L=Uy(X)32y of its explicit form Y  v  ii) The second case.  = A ^ Z W + $ V (X) .  (1.42)  In this subsection, we restrict our-  selves to a linear system Fv(f2^n^)=0 whose explicit form is linear homogeneous, i.e.  4^Z y =0.  types of generators can arise:  In this case, two • "" . .  0 V is linear in Z,Z,...,Z, i.e., 0 V = B v Z y + Y V ( X ) 1 m y for some linear operator B . a)  b)  0 V is nonlinear in Z,Z,...,Z. 1 m  The generators will be called linear generators and nonlinear generators respectively.  For a linear system,  we often assume its invariance group generators to be linear generators as the computation of generators becomes much simpler.  Although this assumption has been found  to be valid for most of linear systems, there exists no theorem stating the range of validity of this assumption.^ Indeed, there exist exceptions. equation Z ^  For instance, the wave  - Zyy = 0 admits a nonlinear generator  L-0(Z^+ZyjX+Y), 0 (•,•) being an arbitrary function. We now show the completeness of linear generators, namely, that if the system admits nonlinear generators, then they all can be found by examining linear generators admitted by the system.  Theorem 1.  A linear homogeneous system  = 0, ^  by (1.40), admits a nonlinear generator ' 0 V (ft  )3  defined iff  the system admits a linear generator of the form  /  y L = {BuV K(X)Z + BuV 1v(X)ZV J J l +  + B (1.43)  t)  Only for a limited class of scalar linear equations  has this assumption been shown to be valid [6,7].  where  = {3 ZV 1  .0 v ^ (m) )>lz=u,z=u,...,z=u'  • • •J  1.1  mm  t1-44)  and U:R^(X)->-R^(Z) is an arbitrary solution.of a system of differential equations 4 v U y = 0 . y This is a new result and the proof is given in Appendix 1. According to this theorem, if a linear homogeneous system admits a nonlinear generator, then the system must admit a linear generator which depends on an arbitrary solution of the corresponding system of differential equations. By virtue of this result, in the study of a linear homogeneous system, we may restrict ourselves only to linear generators  of the form  L = (B^Zy + y V ) 8 z v  (1.45)  where B* = B ^(X) + B^ 1 (X)D y y y  + ... + B ^ 1 l i 2 - - - i m ( X ) D D ...D , A A y ii i X ± i 1 2 .J-m (1.46) M V M B V1. . • j : R ( X ) + R , and ¥ :R (X)-*R. To illustrate a use of the theorem we consider the following example.  Example.  Consider the equation ZXX  zYY  (1.47)  0.  This equation admits a linear generator  (1.48)  L ='{g(U x +U Y )-(Z x +Z Y )}3 z X Y where g is an arbitrary function of U^+Uy, U being an  arbitrary solution of the differential equation U x x - U Y Y = 0 . In this example, the equation is scalar, hence we drop all the Greek indices in the theorem.  Comparing (1.43) and  (1.48), we have  B(X.,Y)' = 0, • B X (X ,'Y.)' = g(U x +U Y ),  B Y (X,Y) = g(U x +U y ). (1.49)  Comparing (1.49) with (1.44), we look for a function 0 satisfying 0Z  ez  = g(zx+zY).  Clearly, we have 0=G(Z x +Zy), G being an arbitrary function of Z x +Zy, and consequently Eq(1.47) admits a nonlinear generator L=G(Z Y +Z Y )3 7 .  We state a few basic properties of the linear generators. Their proofs are given in Appendix 2.  (1.50)  Let B be an operator matrix B  Proposition 9.  .  If L=(£^Z y )3 z V , L=(B^Z y )3 z v and £=(B^Z y )3 z V  satisfy the commutation relation [L,L]=l!, then [B,  Proposition 10. If L=(B^Z y )9 z v and L=(B^Z y )3 z V are generators of invariance groups of a linear  homogeneous system 4^Z y =0,  then so is L= ( S V 3 K Z y ) 3 7 v •  Let A be an operator matrix A  =\|.  Proposition 11. L=(5^Z y )3 z V is an invariance group*-generator of a linear system ^Z = 0 iff [A,B~[Z =0 for any Z satisfying ^Z=0.  Proposition 12. If L=(£ V Z y )3 7 v is an invariance group y  L  generator of the system ^z = 0, and if U(X) is a solution of a system of differential equations a4U(X)=0, then (5) u(X) also solves the same differential equations, i.e.  4(B) k U(X)=0, k=l, 2 , 3,... . Invariance groups of linear equations have been  studied extensively in recent years in connection with representation theories of groups [30,31] and with quantum physics 125]  In these works, problems were formulated  in terms of the linear differential operator :B instead of  the operator L= ( s V Z y ) 3 ? v .  A systematic method of finding  the operator B was first presented by Winternitz et a;l\[32 ,33] in their study of symmetries of Schroedinger equations. The relationship between the operators L and B was first established by Anderson, Kumei and Wulfman and Propositions 10-12 have been known to them.  Proposition 9 is a. new result.  1.8 Contact transformations of the spaces y and X.  Function's f and generators I were defined in space w.  If the space w is mapped into a new space W by  some transformation T, then f and £ undergo corresponding transformations.  We assume the transformation T to be a  1-1 contact transformation, analytic in D(w) and write it as  ft = ftO) = Tto  with  Q = (X,Z,Z,Z, . . .) . 12  The inverse is written as (jo = gj(£!)eT  D(W) denotes the  image of D(w) , and r the space of functions F:W^->-R, k=l,2,3,..., analytic in D(W).  A represents the space of  generators L of group contact transformations in W.  The transformation T:y->T.  The transformation of the  function fe.y by T is written as Tf(oo'-n^) and defined by  (1.51)  TfO^)  = f(T"Wn;)).  (1.52)  The inverse transformation r^-y is defined by  T _ 1 F ( f t ( n ; ) ) = F (Tco  ) .  (1.53)  The transformation T:A->A. We write the transformation of _i I by T as T£T . Regarding (1.51) as a change of variables one finds  T £ T _ 1 = (£X i )9 x _  + (£ZV)9ZV,  where X,Z are X,Z components of (1.51).  (1-54)  Since the trans-  formation (1.51) is a contact transformation, if leX, then T£T  In view of Poposition 3, we have • (JtZV - zVj?,X )9 . T£T 1 = i zv i x- i  •  (1.55)  Similarly, T _ 1 L T = (LzV - z V L x i ) 3 z V .  We let £ = 6V(to(-m^) 9 v and L = v Z lu cases, z scalar and z vector, separately.  (1.56)  and consider two  z scalar. As mentioned in §1.2, the most general 1-1 contact transformation in this case is the extended Lie contact trans-  formation.  It takes the form  X- = X. (x , z,z) , x 1 1  Z = Z (x ,z,z) , Z. = Z. (x , z,z) , ... . x x i 1 (1.57)  Using (1.7)-(1.9) where we let x'=X, z'=Z, z'=Z,..., we obtain:from (1.55) and (1.56) the expressions  T0 (oo  ^) 9 T" T Z  1  ;= {^ertTT-^1fotC^m ^ )p ^frTT""11^o1(-11)^ } 9 7 , Li  (1.58)  and T  1  0(fi ( m ; ) ) 9 7 T = { e C T ^ ^ a ^  1  ) ) }9 ,  (1.59)  where p(T_1ftC1)) = p ( w ( 1 ) ) l u = - ( n )  and  a((. ( 1 ) ) = { p ( W ( 1 ) ) } _ 1  (1.60) pfw^ 1 ^) = 9 Z - Z-9 X.. v J z 1 z 1  with  z vector.  In this case the most general 1-1 contact trans-  formation is the extended point transformation:  = X i (x,z) ,  Z v = Z v (x, z) , . . . ,  (1.61)  yielding Te V ((0 ( m ) )9 Z  V  T-1  = {6y(T"1n(m})pV(T-1ft(1))}97V, ji Lt  (1.62  where = {9 z p Z V -  (1.63)  and T-V(^m))8zVT  = {©y(Ta)(m))a^(1))}^v> yV ^ - - ZV  U-64)  where a ^ h  = {3 z yz v - z ^ y V l ^ ^  (1.6S)  From these results follows that:  Proposition 13.  An operator L = 0 V (fl ^  ) 9z V  is an invariance  group generator of the equation F(fl^n))=0 iff T~ 1 F=f (w ^ ) =0 admits the generator (1.59) (z scalar) or the generator (1.64) (z vector) .  - 41 "  CHAPTER 2.  - INVERTIBLE MAPPINGS OF NONLINEAR SYSTEMS TO LINEAR SYSTEMS  In this chapter, we study transformations mapping nonlinear differential equations to linear differential equations in a 1-1 manner.  Based upon the group analysis  of differential equations, we obtain necessary and sufficient conditions for the existence of such transformations.  The established theorems not only allow us  to determine the existence of the transformations but also enable us to actually construct these transformations from invariance groups of the nonlinear equations.  In the following analysis, two types of transformations are considered: 1)  The invariance group transformations of differential  equations; and 2)  The mappings which transform nonlinear differential  equations to linear differential equations. Theorems will be proved based upon the following observations. Clearly if there exists a 1-1 mapping between any two differential equations it must inject properties of one equation into the other, including their invariance  +) properties.  For this reason the often ignored fact,  mentioned in §1.7, that any linear differential equation admits a superposition generator becomes significant.. As we have seen, this particular generator depends upon an arbitrary solution of the linear equation.  It follows then  that any nonlinear equation transformable to a linear equation by a 1-1 mapping must admit a generator which depends upon an arbitrary solution of some linear differential equation.  2.1  Theorems on the existence of 1-1 mappings. We consider two cases, z scalar and z vector,  separately since each admits a different type of 1-1 contact transformation.  First we consider the case z scalar.  In the following theorem, the Lie contact transformation discussed in §1.2 will be used with new notations X=x', Z = z ' , Z = z' , ... 1 1  t) The idea of comparing invariance groups of differential equations in the search of mappings connecting the equations was first used by Bluman in his study of Burgers' equation ['34 1 and it was applied to the study of mappings of one dimemsional linear parabolic equations to the heat equation [35]. ft) The results in this chapter and a part of the results in the next chapter have been reported as Technical Report 81-3 of the Institute of Applied Mathematics and Statistics, University of British Columbia.  Theorem 2.  A scalar n-th order nonlinear equation  xeR M ,  - f (x , z , z , z , . . . , z) = 0, 12 n  zeR,  (2  is transformable by a 1-1 contact transformation to a linear equation if and only if the equation f=0 admits a generator £ of the form  £  = .{a(w(1))U(X( J  1 }  ) ) }9z,  (2.2)  where 1)  M U(X') :R.->R is an arbitrary solution of some n-th  order linear differential equation  I4U-- = '"{A (X') ' + A 1 (X) D V x  i  +  ... + A 1 - 1 • • • LN (.X).I) V  •  X  i1-'-  x  ..PY  }U = . 0 ,  in  and  (2.3) 2)  X(O)( 1 ' ) )  i w ^ + R 1 ^ is a component of a Lie contact  transformation X = X(EO(1)),  Z = Z(U)(I;)),  Z = Z ^  1  ) )  (2.4)  and a(u> C 1 ) ) = {p(a) (1) )}" 1 = (B z Z - Z i 9 z X i ) ' 1 . The transformation (2.4) maps Eq.(2.1) to a linear equation which has an explicit form j4Z-$(X)=0 with  A  defined by (2.3).  (2.5)  Proof.  Suppose that Eq.(2.1) is transformable to a linear  equation by an extended Lie contact transformation.w+fl.  By  Proposition 8, this linear equation admits the superposition generator L=U(X)8 Z of the equation 4Z-$(X)=0.  Hence,  according to Proposition 13, Eq.(2.1) must admit (2.2). Conversely, suppose that Eq.(2.1) admit a generator of the form (2.2) with the properties 1) and 2).  The  transformed equation of Eq.(2.1) by (2.4) is written as Tf=F(X,Z,Z,...,Z)=0. 1 n  In view of Proposition 13, F=0 admits  the generator L=U(X)9 Z .  Thus, by the invariance condition  (1.36), we have LF = F U for any  +  Fz-Ui i  +  ...  +  Fz _ Uiii?...i 1 1 x n i 2'"" n 1 2  = 0  (2.6)  satis fying F(fi(n^)=0 and for any U(X) satisfying  the differential equation (2.3).  It is easy to show that  (2.3) and (2.6) involve the same set of  We  assume without a loss of generality that both contain U. Eliminating U between the two equations, we get  0 = (AF  i  - A 1 F z ) U i + (AF  ij  - A 1 ^F„)U.. + ... . J  Since U represents an arbitrary solution of Eq.(2.3), at any point X an arbitrary set of values may be assigned to U-.U-...... and thus all the coefficients in (2.7) must v J i Ij vanish. This is possible only if F has the form G(4Z,X),  (2.7)  G(«,*) being an arbitrary function.  Therefore, the extended  transformation of (2.4) maps Eq.(2.1) to an equation G(^Z,X)=0, which is solvable in the explicit form AZ-$(X)=0.  •  In this theorem, z , z ...... Z , Z-,... are coordinates ' ' 1' ' ' I' of the spaces w, W.  Because of the contact condition (1.4)  imposed upon T this theorem implies: Corollary 1.  A scalar nonlinear differential equation  f(x,u,u,...,u) = 0, 1 n  x e R ,M,  u:RM (x)+R  (2.8)  is transformable to a linear differential equation by a 1-1 mapping if and only if the equation f(x,z,z,...,z)=0 1 n admits a generator of the form (2.2).  The mapping is given  by the extension of (2.4) and it transforms Eq.(2.8) into a differential equation which is solvable in an explicit form 4U - . $(X) = 0.  We now turn to a system of nonlinear equations. We have:  (2.9)  Theorem 5.  A system of K independent n-th order nonlinear  equations f V ( > ( n ) ) = fV(x,z,z,z,...,z) = 0, v=l,2,...,K, n 1 2  K<N,  (2.10)  xeR M , zeR^, is transformable by a 1-1 contact transformation to a linear system if and only if the system  (2.10)  admits a generator of the form  I = {U y (X(x,z))crV(x,z,z)}3 , y — zv  (2.11)  where 1)  U V (X), v=l,2,...,N, is an arbitrary solution of  some system of n-th order linear differential equations  LJP = {A V (-'X) -+ A V 1 ( X ) D Y X.. . 11 11 1 *  + .... + A V l 1 - • • 1 n(X)D Y • y Mi  . . . D Y . • }U U .= 0 -n  (2.12)  v=l,2,...,K, and 2)  X(x, z) :  i  s  a component of a point trans-  formation X = X(x,z),  (2.13)  Z = Z(x,z)  with inverse transformation x=x(X,Z), z=z(X,Z) and  v <J*(x,z,z) =O  z  y Z  V  -  zV3  z y  xi>  X=X,Z=Z.  (2.14)  The extended point transformation of (2.13) maps Eq.(2.10) to a linear system with an explicit form  AvZy  Proof.  - 3>V(X) = 0, v = l, 2 , . . . ,K.  (2.15  We recall that the most general 1-1 contact trans-  formation involving vector z is the point transformation. Suppose that there exists a point transformation  X = X (x, z) ,  Z = Z(x,z),  Z = Z(x,z,z), ... 1 1 1  mapping Eq.(2:,l,0) to a linear system  (2.15).  (2.16)  By Proposition  8  this linear system admits the superposition generator L=UV(X)87V.  In view of Proposition  13  the system  (2.10)  Li must admit the generator (2.11) with properties 1) and 2). Conversely, suppose that Eq.(2.10) admit the generator (2.11).  Under transformation (2.13), the  generator (2.11) is transformed into L = U v (X)9 z v  (2.17)  and Eq.(2.10) into, say, Fv(X,Z,Z,...,Z)=0, v=l,2,...,K. 1 n The system {F v =0} is solvable in explicit forms for K components of the Z,,Z., . . . ,Z.. Without loss of generality, 12 n  for these we can choose Z^, v=l,2,...,K, and write the explicit forms as  Z^ + <j)V(X,Z,Z, . . . ,Z) = 0, 1 I n  v = l, 2 , . . . , K,  where cj>v are independent of Z y , y = l, 2,...,K.  (2.  According to  Proposition 13, the system {F — 0} admits the generator (2.17), and hence so does the system (2.18).  +  +  *zy u i i  +  •••  +  ^zv . u? 1-....1I 1 n  where v=l,2,...,K and cj> ^y = ^ ^l-1 ^ V ' e t C - -  Thus,  .i n  =  °>  v (2.19) J  This holds for  any U(X) satisfying the differential equations (2.12). It is easy to show that Eq.(2.19) can involve only those U y ,UV,... appearing in Eq.(2.12). for uV, y=l,2,...,K.  Eq.(2.12) is solvable  This is seen as follows.  this be not the case, i.e. rank |A^|<K, y,v<K,  Suppose .We fix X at  X=X° and assign to U V (X°), U^(X°), U ^ (X°) , ., . a set of values consistent with Eq.(2.12).  Here, the indices on  U^(X°) are restricted either to i>l or to {i=l,y>K}. . For this set of values, there exist non-unique values of U^(X^), v<K, satisfying Eq.(2.12) because of the above rank condition.  On the other hand, the introduction  of the same set of values into Eq.(2.19) uniquely determines the values of U^(X^).  This contradicts the condition that  Eq.(2.19) holds for any solution of Eq.(2.12).  Thus,  Eq.(2.12) is solvable as  U^ a V I ^+ /A. U  l  +  A ^ uUV•  y  +  ^  ... ...  +T  A/V ^ l - ' -" UW•  yi  y  = 0, v = l,2, . . . ,K.  .. .  (2.20)  Eliminating U^ from (2.19) and (2.20), we have  I n  (2.21) The equality (2.21) is possible only if  (J)V = A V Z y +  y  AvizV y i  + ... +  A^l-^nzV . x y i•••xn  +  (X)  /s 1 where A^ =0 for y<K, and consequently the explicit form (2.18)of the transformed system {F v =0} is linear.  It is  also clear that Eq.(2.18) is equivalent to Eq.(2.15). D  As in the case of scalar z, from this theorem follows:  Corollary 2.  A system of K independent nonlinear differential  equations  f v (x,u,u,...,u) = 0, 1 n  v=l,2,...,K,  K<N,  (2.22)  - so.  x 6 R M , u:RM(x)-*RN, is transformable by a 1-1 mapping to a system of linear differential equations if and only if the system fv(x,z,z,...,z)=0 admits a generator of the 1 n form (2.11).  The mapping is given by (2.13) and it trans-  forms Eq.(2.22) to a system solvable in explicit form as A VU y  2.2  - $ V (X) = 0,  v=l,2,...,K.  (2.23)  Remarks on the use of theorems.  These results just obtained ensure that if a given system-of nonlinear equations • is transformable 'to a system of linear equations•by,a 1-1 mapping,.one can always, find the mapping by examining the nature of the invariance group of the nonlinear equations. The type of groups to be considered depends on the dimension of the space z. . For a scalar equation we need only to consider a generator £ of the form  I = 6(x,z,z)8 z 1  .  (2.24)  If the equation is transformable to a linear equation, then it admits a generator of the form (2.2).  It should be  emphasized that the function X(oo'-"^), the factor a(o)^"^) and the linear differential equation (2.3) can all be found by examining the generators admitted by Eq.(2.1).  Once X is obtained, the function Z(a)^)) is determined from the condition [Z,X^]=0 which represents a system of first order partial differential equations for Z. Z still admits functional arbitrariness.  At this point,  From Z and X we  determine Z • using conditions (1.7) or (1.8).- "Next-we. use ^ (2.5) for the Known p=a -1 to limit the arbitrariness m Z. The resulting transformation X=X, Z=Z, Z=Z maps the nonlinear equation to an equation with an explicit form AZ - $(X) = 0.  The form of $(X) depends upon the remaining  arbitrariness: in Z. For a system of equations, in view of (2.11) and (2.14), we need to consider generators of the form  £ = UV(x,z) - z^i(x,z)}8zV.  (2  By Proposition 3, (2.25) is equivalent to a generator of a point group  + C V (x,z)3  £ - 5^x^)3 i  z  v.  If there exists a mapping to a linear system, we can find the functions X(x,z), a v (x,z,z) by comparing the resulting ^ 1 invariance group generators (2.25) with (2.11).  The functi  Z V (x,z) are to be determined from these functions using equations (2.14). invariance group.  Eq.(2.12) is found on determining the  Remark 1. It is possible for differential equations to admit generators whose forms are more general than those of (2.2) or (2.11) with the forms (2.2) or (2.11) as special cases.  The Monge-AmpSre equations of a special type are  such examples as we see in the following examples.  A system  of ordinary differential equations also admits s.uch generators.  2.3  Examples.  To illustrate the use of our theorems, we consider some well known equations transformable to linear equations. Since the linearization of differential equations £v('X,u,u, . . . ,u)=0 is equivalent•• to^that:;of. the equations 1 ri v f (x,z,z...,z)=0 by a contact transformation, we only deal 1 n with the latter.  In the following examples we let x^=x,  x ? = y and, where convenient, adopt the customary notations z =p, z =q, z =r, z =s, z =t. x ^' yi xx ' xy ' yy  A.  The equation z  + h{z )  2  zy  o.  We consider the equationt  f  t  z  y  o.  This is an integrated form of Burgers' equation z +zz -z =0 which will be discussed in 3.1. xx x y  (2.26)  The generator (2.24) is now i=Q(x,y,z,z^,z )3 z .  Applying  Lie's algorithm, we find that Eq.(2.26) admits 2  1  = (y z + 7yxz + %x y x  JU = z 3 , 3 y z ' 4  x  equation  +y) 3J 3 , z'  = z 3 , z '  - (yz x + x)3 z ,  where,:.in  2  £  5  = w(yz + Jgxz )3 , y • x' z'  2 = 3 , z'  (2.27) ^ ^  ij = U(x,y)e _ J s Z 3 z ,  , U(x,y) is an arbitrary solution of the heat U^  ••• IP =. 0 . This indicates that Eq.(2.26) is  equivalent to a linear equation. compare  To find the mapping, we - 3"Z  with (2.2) to get X=x, Y=y and a=e  2  .  From  the conditions [X,Zj = [Y,Z]=0, we obtain Z = Z(x,y,z). the mapping is a point transformation. From (2.5) and p=(a)  =e  2  , we find that 3 z Z = e  X = x,  Y = y,  2  Thus,  v  .  The mapping is then  Z = 2e  + h(x,y),  where h(x,y) is an arbitrary function of x and y.  (2.28)  It is  easy to check that the extended point transformation of (2.28) maps Eq.(2.26) to the linear equation  AZ - $(X,Y) = Z x x - Z y - ( h x x - h y ) = 0 .  (2.29)  Setting h=0, from Corollary 1 we see that the transformation  X = x,  Y = y,  U = 2ehu  (2.30) 2  maps the differential equation u ^ + h(u^)  - u^ = 0 to  the heat equation ,4U=U-^-Uy=0 , and moreover the inverse of (2.30), x = X,  y = Y,  u = 2£n(%U)  (2.31)  defines an implicit solution u(x,y) of this nonlinear differential equation for any solution U(X,Y).  In this case  the explicit form is  u = 2£n{%U(x ,y) } .  (2.32)  B. Hodograph transformations. 1  2  In this example we let z =w, z =v and consider a system of quasilinear equations = a l x ( w , v ) w x + a i y (w,v)wy+ b l x ( w , v ) v x + b iy (w,v)v^= 0,  i=l,2, (2.33)  where the coefficients a's and b's are functions of w and v. For the invariance group of this equation we have:  Proposition 14.  Provided J = w x v y ~ v x w y ^ ® , the system (2.33)  admits a generator of the form  I = -{U 1 (w,v)w x + U 2 (w,v)w y }3^ - {U 1 (w,v)v x + U 2 (w,v)v^}3 v 1  2  where {U (w,v),U (w,v)} is an arbitrary solution of the system of linear differential equations 2 i.r t r T _V\ lrv ll) lU 24- +na i x^(^w ,\j \j b i y (w,v)U^ - a l yf(w,v)U^ - b l xfu (w, v )jUkj =— 0,  —  i = l,2  (2. X  1  1=  where U =3 U , U w w ' V Proof.  1  3 U . V  Since Dx f x =Dy f 1 = 0', we find that - Afx = w U1fa1Xw x w^ x  + a i y w ) + v vU 2 ( b 1 X v y yv x  + b l yJ v ) y  + w U 2 v( a l x w y w x  + a l y w J ) + v U 1v ( b l x v y x v x  + b i y vJ ) y  + v U1(blxw. x w^ x  + b i y wJ ) + w U 2 ( a l x v y y v x  + a i yJv ) y  + v U2(b1Xw y w^ x  + b i y wJ ) + w U ^v ( a l x v y X' v x  + a l y vJ ) y  Using Eq.(2.33) in the first two rows of this expression, we find that „r i i iy ,l ,ix 2 , T r ,iy Tt l if .c r n = J-(b - a 7 UT - b U TT + a v I£^=£2=0 W V w  ixTT2. U )J V  which vanishes by the condition (2.35). •  To construct the mapping to a linear system,  we compare (2.34) with (2.11) which in the present case takes the form  I = {U1(X,Y)aJ + U2(X,Y)a2>3w + {U1(X,Y)a2 + U2 (X, Y) a2 }.3 . Clearly3 X=w, Y=v, af=-w,;, ai=-w , a 2 = -v , a 2 = -v . The ' 1- x' 2 y 1 x' 2 y v definition of a leads to 9ljrx=l, 9,,x=0, 9,^=0, 9„y=l. y W ' ¥ ' m3 ' V' we have a solution x=W, y=V.  Thus  Combining these together, we  find the hodograph transformation [ 36 ]  x = W,  .y = V,  w = X,  v = Y  (2.36)  which maps Eq. (2.33) to a- linear system  /Zy  = b i y ( X , Y ) W x - a l y ( X , Y ) W Y - b l x ( X , Y ) V x + a lx (X,Y)V Y .,= 0 (2.37)  where Z  1=  2  W , Z =V  and i=l,2.  C. The Legendre transformation.  We consider a second order quasilinear equation  f = a (p , q) r + 2b(p,q)s + c(p,q)t = 0,  where p,q,r,s and t denote the variables defined at the  (2.38)  beginning of this section and a,b and c are functions of p and q.  We have  Proposition 15.  Eq.(2.38) admits a generator &=U"(p,q)9  depending on an arbitrary solution U(p,q) of the linear differential equation  U = a(p,q)U q q - 2b(p,q)U pq + c(p,q)U p p = 0,  where U  =(9 ) 2 U , U  p  pp Proof.  =9 9 U, U  pq p q  qq  Introducing w=p, v=q,  (2.39)  =(9 ) 2 U .  q , v =t, w^=v x =s, we write  Eq.(2.38) as a system  a(w,v)w x + b(w,v)(w w  - v y  + v x ) + c(w,v)v y = 0  = 0. x  (2.40) (2.41)  K  J  According to Proposition 14, this system admits the generator 1 2 of the form (2.34) where {U ,U } is an arbitrary solution of the linear differential equations  c(w,v)U^ - b(w,v)(U^ + U 2 ) + a (w,v)U2 = 0 "  U  w =  Eq.(_2.43) allows us to introduce a function U(w,v) with  (2.42) (2-43)  a property U1 = 8 U E U , iir T»r 7  U  2  = 3 U E U , "\t "\r '  (2.44) v  and then Eq.(2.42) takes the form  aU  VV  - 2bU  WV  + cU  WW  =0.  (2.45) J  Introducing (2.44) into (2.34) and using (2.41), we find that i = (D x U)8 w + (D y U)3 v .  Recalling that w=p=z x and  v=q=Zy, we see that this is the first extended part of the operator  I = U(p,q)3 , and hence follows the assertion. •  In order to construct a mapping of Eq.(2.38) to a linear equation we compare the generator &=-U(p,q)3z with (2.2).  Clearly, X=p, Y=q and a=l.  To find Z, we use  [Z,Xj = [Z,Y]=0, i.e.,  Zx + rpZ z = 0,'  Zy + nqZz = 0.  The solution is Z=Z(p,q,a), a=-z+px+qy.  Z + xZ = P, p a ' with P=Z X and Q=Zy  From (1.8), we get  Z + 7yZ = Q x q a  Now from (2.5) with a=l, we get Z  =  1,  and consequently Z=-z+px+qy+h(p,q) for ah arbitrary.function h. Setting h=0, we get the Legendre transformation [3.6] :  J  X = p,  Y = q,  Z = -z + px + q'y, P  X,  Q = y.  This transformation maps Eq.(2.38) to  AZ E a(X,Y)T - 2b(X,Y)S + c(X,Y)R = 0  where T = ZyY> ^=  an<  (2.46)  (2.47)  ^  D. Lie's Theorem on the Monge-Ampere equation,  The Monge-Ampfere equation takes the form  f = A(rt - s 2 ) + Br + Cs + Dt + E = 0,  (2.48)  where the coefficients A,B,C,D and E are functions of x,y,z,p and q.  In studying this equation, the concept  of intermediate integrals plays an important role [-37= ]An equation I(a(a) (1) ),3(^ (1) )) = 0,  (2.49)  I(.,«) an arbitrary function R2->-R, is said to be a general intermediate integral of Eq.(2.48) if a and 3 satisfy the equality  (D x a)(D y 3) - (D a)(D x 3) = f.  (2.50)  Lie [ 38,37] proved a theorem which in our notation reads as  Theorem[Lie] . A Monge-Ampere equation admitting two 1 1 1 2 2 2 general intermediate integrals I (a ,3 )=0 and I (a ,3 ) = 0 is transformable to the equation Z^-y^ ^y a Lie contact transformation  = fi (to ^ ^ ) whose four components are  given by X = a1, • ' Y = a2,  P = 31,  Q = 32.  (2.51)  Two intermediate integrals in this theorem are related to an invariance group of the corresponding Monge-Ampere equation:  Proposition 16. A Monge-Amp&re equation possessing two general intermediate integrals I 1 (a 1 ,3 1 )=0, i=l,2, admits two invariance group generators  = a(co ( : i ) )I 1 (a 1 ,3 1 )3 z ,  i = l,2,  = [a2,3^]-  where a ^ = p =  Proof.  It is easy to check that the equation Z ^ Y ^ admits 1 (X,P)9 and L£ = I2 (Y,Q)9 with arbitrary generators Z Z 1 functions I  2 and I .  On the other hand, according to Lie's  theorem above, there exists a Lie contact transformation  (2.52)  mapping the Monge-Amp&re equation in question to Z X y=0. In view of (2.51) and Proposition 13, the inverse of this transformation maps L^ and L^ to (2.52). D  The generators (2.52) appear to have different-..forms', from the generator (2.2). special case.  However, they contain (2.2) as a 1 1  1  To see this we choose special forms I -I (a ),  I 2 = I 2 ( a 2 ) and let i=Z 1 +l 2 .  Then from (2.52) we obtain  I = a{I 1 (a 1 ) + I 2 ( a 2 ) } 9 z = a-U(a 1 ,a 2 )8  .  (2.53)  Observing that U(X,Y)=1 1 (X)+1 2 (Y) is the general solution of the equation U X y = 0 >  w e  s e e  that indeed the Monge-Amp&re  equation in question admits a generator of the form (2.2) - 2 (tofll v J with X=a 1 (eofl~) ) and Y=a ^ J ) . From this result it is clear that we can find Lie's linearization mapping of a Monge-Amp£re equation to Z X Y = 0 , when it exists, by examining the invariance group of the equation .  E. The equation A(z ) a z - z =0. xx YL  A special Monge-Ampere equation of the form  g(zxJ)z xx - z yy = 0 , '  g:R+R, s '  (2.54) K J  arises in a variety of physical problems such as nonlinear  vibrations (g(z )>0) [39], and irrotational transonic flows (g(z )=l+az ) [40].  In the following, we consider a class  of equations of the form  f = v(z ) a z - z = 0, a real, x^ xx yy ' '  v(2.55) J  and apply the foregoing analysis to examine a possible mapping to a linear equation other than the Legendre transformation.  The invariance group of Eq.(2.55) depends  upon the value of a.  Assuming a generator to be of the form  (2.24), i.e. £=0(x,y,z,p,q)9z, we find that the following cases occur:  (1) a f 0, -2, -4:  Jl1 = {-(a+4)xpq + azq - 2(a + l)yq 2 - 4y/ p a + 1 d p } 9 z = { (a + 4)xp + (3a + 4)yq - az}8 z , = (z + %ayq)9 z , (2) a = -2: l^,  ^3' £4>  (3) a = -4: J^ = (xp -yq)3z>.  = y9z, a s  £ 5 = U(p,q)3 z .  above and I 6 =  =  U  (2.56)  (2.57)  (P^)9Z'  £3 = {pl 1 (p" 1 -q,(p" 1 -q)y+z)}9 z ,  = {pl2(p"1+q,(p"1+q)y-z))9z.  (2.58)  (4) a = 0:  £  = z3  1  z  I  z  = h(x,y)9  £ 3 = I 1 (x+y,p+q)3 z ,  z  (2.59)  £ 4 = I 2 (x-y,p-q)3 z .  In all cases, the function U(p,q) represents an arbitrary solution of the differential equation  U - (p) a U = 0 pp qq  v  (2.60) J  and the corresponding generators are related to the Legendre transformation discussed above.  The function h(x,y) in  the last case is an arbitrary solution of the differential equation h x x  - h  = 0 and the corresponding generator  is the superposition generator of the equation z ^ - z  =0.  1 2 I and I in the last two cases are arbitrary:functions 1 2 of their arguments and the equations I =0 and I =0 are the general intermediate integrals of the corresponding equations. We examine the case a=-4 in some detail. The equation is  (z )" 4 z - z = 0. v x^ xx yy  Comparing (2.52) and the generators  (2.61) and I^ of (2.58),  and using (2.51), relations [X,Z] = [Y,Z]=0 and (1. 8) , (2 . 5) , • we find the Lie contact transformation mapping (2.61) to  to be  X = p " 1 -q,  Y = p " 1 + q,  Z = - 2 p _ 1 ( z - px -qy),  (2.62) P = - ( p - 1 -q)y -z,  Q = ( p _ 1 + q)y - z.  The inverse transformation is  y = -(X + Y ) - 1 ( P - Q) (2.63)  x = %Z - % (X + Y) (P + Q) , z = -(X+Y) _1 (YP+XQ),  p = 2(X+Y) _ 1 ,  If we introduce the general solution of  U  X  q = %(-X+Y).  y=0,  U=F(X)+G(Y),  where F and G are arbitrary functions, into (2.63) by Z=U, P=U X  and Q=U y , then we obtain a parametric representation 1  of the general solution of the differential equation  (u ) x^  u  - u xx  = 0.  (2.64)  yy  Explicitly this is , x = %{F(X)+G(Y)} - %(X+Y){F'(X)+G'(Y)} / y = -(X+Y)  {F'(X)-G' (Y) }  u = -(X+Y)" 1 {YF t (X)+XG'(Y)},  where F' and G' denote derivatives. We note in passing that Eq.(2.61) is also trans-  (2.65)  formable by the Legendre transformation (2.46) to  Z  XX " 00 " \ Y  =  (2  and in turn Eq.(2.66) can be mapped into  -66)  by a  composition of the transformations (2.46) and (2.62). Explicitly, the transformation x = 2(X+Y) _ 1 ,  maps the equation  Remark 2.  - (x)  y = %(-X+Y),  z = (X+Y)"1Z  -4 z ^ = 0 to the equation  Let Eq.(2.55) be written as  D (z ) a + 1 - D (a+1)z x\ xJ y y  = 0.  (2.67)  r\j If we introduce a potential z by (zx)a+1 = zy,  (a+1)Zy = z x ,  (2.68)  z satisfies the equation (z K  f z -z = 0, yJ yy xx '  _1 3=-a(a+l) . K J  The * transformation (2.68) may be viewed as a Backlund  (2.69) K J  +) transformation between Eq.(2.67) and Eq.(2.69). For a=-2, the transformation (2.68) becomes an auto-Backlund transformation: (z )" 2 Z -z = 0 , ( 2 - 6 8 ) , (2 ) ~ 2 z v K x^ xx yy yJ yy  =0. xx  K  (2.70) J  For a=-4, Eq. (2.69)takes the form  v  z Y  - z = 0 yy xx  (2.71)  K  J  and the transformation (2.68) together with the general solution (2.65) yields a general solution for Eq.(2.71). Seymour and Varley [41] obtained the general solution of Eq.(2.54) when g(z ) satisfies the equation X d  5  X  7  g = yg""" + vgT ,  y,v constants.  The case y=0 yields Eq.(2.61) and the case v=0 leads to Eq.(2.71)  t)  Consider a system (g  =0}. k  If this  £  has the property that for any solution of a system {Fy(x,z,...,z)=0} the corresponding z,z,z,... satisfying m 12 {g =0}solve a system (x,z,...,z)=0}, then the system v •• n u {g =0} is called a Backlund transformation between {F =0} and =0}. In particular, if F=F, then it is called an auto-Backlund transformation. For the discussion of Backlund transformations, see the article by Lamb in Ref.l.  CHAPTER  3.  NON-INVERTIBLE MAPPINGS OF NONLINEAR SYSTEMS TO LINEAR SYSTEMS.  In the preceding chapter we have considered mappings which transform a system of nonlinear equations to a system of linear equations in a 1-1 manner.  If we  require only that a mapping transform a solution of some linear system to a solution of a given nonlinear system, the class of mappings widens and includes non-invertible (non 1-1) mappings.  In the following we investigate  these types of mappings and show that such mappings are frequently related to invariance groups.  3.1  Examples of non-invertible mappings.  We first consider Burgers' equation  f = z xx  + zz  - z  x  y  = 0.  (3.1) v  J  This equation admits a five parameter point Lie group [42]. However, none of the generators is of the form (2.2), and hence by Theorem 2 there exists no 1-1 mapping to a linear equation.  It is known that the Hopf-Cole transformation [43,44]  x = X,  ~ ' -2'Zy z = ,  v - Y,  (3.2)  relates Eq.(3.1) to the heat equation Z^-Zy-O.  Introducing  the transformation C3.2) into Eq.(3.1), we find that  - 2  f ~~ 2Z  ^^^XXX ~ ^X^X^X  ^  ^  Y  ^  ^  ^^  which factorizes as  f = 2Z" 2 (ZD X - Z X ) ( Z X X - Z y ) = 0.  (3.4)  It follows from (3.4) that the transformation (3.2) maps a solution of the heat equation to a solution of Burgers' equation.  It is incorrect to say that the Hopf-Cole trans-  formation maps Burgers' equation to the heat equation. It is also clear that (3.2) is not a 1-1 mapping.  Although  this type of mapping is out of the scope of the discussion in the preceding chapter, this particular transformation is found to be related to a Lie group.  One standard  argument [ 45] to rationalize the Hopf-Cole transformation is to introduce z through z=z  and, after integrating once, A  one considers the equation  Z  xx  +  - z  = 0.  One then sa-ys'"by inspection" that the transformation  (3.5)  x = X,  y = Y,  z = 2£n(cZ),  c constant,  (3.6)  maps Eq.(3.5) to the heat equation Z ^ - Z y = 0 and from this follows the transformation (3.2).  Eq.(3.5) corresponds to  Example A in the previous chapter where we found the transformation (3.6) with c=% by applying Theorem 2.  The second example is a nonlinear diffusion equation  f = D (z~ 2 z ) - z  =0.  (3.7)  This equation admits a four parameter point Lie group [ 6] with no generator we let z=z  of the form (2.2).  As in the above  and instead of Eq.(3.7) we consider an  X integrated form f = (z )~ 2 z - z = 0. v x^ xx y  (3.8) J ^  This equation admits [ 23] seven generators of point transformations including the generator  S. = U(z,y)z x 9 z ,  involving an arbitrary solution of U z z - U =0.  (3.9)  Comparing  (3.9) with (2.2), we find a point transformation  X = z,  Y = y,  (3.10)  Z = .x  which maps Eq.(3.8) to the heat equation  AZ = Z^-Zy = 0  From (3.10) it follows that z=z =(Z V ) ^ and one can X  A  verify that the transformation  x = Z,  y = Y,  z = (Zx)-1  (3.11)  transforms Eq.(3.7) to  f = Zx3CZXDX " Z X X ) C Z X X " Z Y )  =  (3.12)  Hence the transformation (3.11) maps a solution of the heat equation to a solution of Eq.(3.7).  3.2  A use of potential functions.  The equations we have just considered are of the form D x g ( z x _ x , . . . ,z x ,z) - D y z = 0.  (3.13)  For such an equation we can always introduce a potential z by  2 = zx,  The equation governing z is  g = z .  (3.14)  ^zx...xx"--'zxx'z^  - Zy  =  C3  -15)  As the examples in §3.1 show, Eq.(3.15) can admit a larger invariance group than the original equation (.3.13).  Although  this is not always the case, it is well worth keeping in mind. The possibility of introducing a potential,.of course, is not limited only to equations of the form (3.13).  To  illustrate the importance of considering such a potential in more general circumstances, we take two examples.  A. A nonlinear wave equation.  v  - w 7  v  We consider a system  =0 X  (3.16)  = avw + bv + cw,  y where a,b and c are constants. in physical problems.  Equations of this form arise  For instance an equation governing  a fluid flow through a reacting medium I 46,45]  w  y  + v  y  + cw  x  = 0  (3.17)  Vy = k^(a - w)v - k2w(b - v)  and an equation describing a two wave interaction [47-49], vy w  i x = -avw - bv - cw  + C-.V  + c„w = avw + bv + cw y 2 x  (3.18)  t) can be put in the form (3.16) by simple changes of variables. Rescaling the variables in (3-16) as . r (x,y,v,w)  v, cv v —bw. v- (-' F {-' a a) ,  ,7 (3.19)  we obtain v  - w  y v  y  = 0 x =vw+v+w.  (3.20a) J ^ • (3.20b)  This equation admits only a trivial invariance point group generated by  A- = 3 , 1 x'  £0 = 3 , 2 y'  £ _ = x3 -y3 - (v+l)3 + (w+1)3 . 3 x 7 y K v ^ ^w  (3.21) J ^  However, if we introduce a potential z by v = z , w = z , x' y'  (3.22) y. J  then the corresponding equation for z, i.e. z - z z - z - z = 0 xy x y x y  (3.23) v J  admits a larger point group with generators  t)  For Eq.(3.17), x+cx, y->x+y and for Eq.(3.18), x-^^x-c^y,  y-*x - y.  1  = 3 , x'  £0 = 3 , 2 y'  2, = x3 - Jyd - v(x-y) 3 , 3 x y - 1J z'  (3.24)  = U(x,y)e Z 3 z ,  3 4 = 3z >  where U(x,y) is an arbitrary solution of the differential equation u  x y - U x - U y = 0.  Now applying Theorem 2 to X = x,  (3.25)  we find a transformation  Y = y,  ^ Z = e" Z ,  (3.26)  +) which maps Eq.(3.23) to Z  J  X Y " Z x - Z y = 0.  (3.27)  The transformation connecting Eq.(3.27) to Eq.(3.20) is then  x = X,  y = Y,  Z  v = - -JL t  w  z  = -  I.  (3.28)  It is known that Eq.(3.17) and Eq.(3.18) admit transformations of the form (3.28) [ 45 ,4.6 ,48 ,49 ] .  t)  When bc=0, the equation corresponding to Eq.(3.23) admits  an additional generator and the equation can be mapped into  z X Y =o.  To gain some insight as to why the introduction of the potential, (3.22), enlarges the invariance group let us express the generators (3.24) in terms of variables x,y,v and w.  The generators  extension of  *3 =  X3  a r e  and  unchanged.  The first  takes the form  x  'y3y  +  " Cx-y)3? -  Clearly, this corresponds to  (3.21).  tV  1 ) 3  z  y  C3  -  '29)  The first  extension of Z^ is  £4 = 3 Z  and hence  +  0-9 z x  + OOz , y  = 0 in the (x,y,v,w,) system.  (3.30)  For the generator  we have lrD = Ue Z 3 z + (Ux +Uzx )e Z 3z + (Uy +Uzy )e z 3Zy . x  (3.31)  Because of the appearance of z in the coefficients of 3  S3 V  and 3  =3 , the first extended part of (3.31) can  Z-y W  not be expressed in terms of 3c,y,v,w alone and consequently is not a point group generator in the (x,y,v,w) system as we should expect.  Now suppose that we consider z,v and w  as; functions of x and y and express z by a line integral as z = / z x dx + Zydy =. / vdx + wdy.  (3.3 2)  Then the first extended part of (3.31) can be written as /vdx + wdy } 3 i s = {(Ux+Uv)e x v  +  { ( u  +Uw)e/vdx+wdy}9  y  _  w  (3.33)  One can verify that Eq.(3.20) indeed admits the generator (3.33).  We note that the generator (3.33) depends not only  on x,y,v and w but also on the integral /vdx+wdy. In other words by introducing the potential, we have in effect introduced an "integral dependent" generator which is beyond the .framework . of the Lie-Backlund groups. same can be said for the preceding two examples. 2  the generator £=U-(;x ,y).e  The For instance  d • o t". I • q .(3.5) b e c o me s , via z =  an integral dependent generator,  £ = {(U -%Uz)e- i s / 2 d x }9 z  (3.34)  A  of Burgers' equation (3.1).  We will discuss some aspects  of  integral dependent generators in the following chapter.  B.  An equation of a fluid, flow.  Sukharev [ 50] investigated  the invariance point group of the equation  a,3 real wX  a  v w'  6  = 0  which describes a fluid flow through a long pipe-line.  (3.35)  The system (3.35) was found to admit an extra generator when a =-1: w w  y X  + v  x  =0  (3.36)  - v -1w "3 = n0 .  The extra generator is  A = gO,y)9x  +  {w" 3 3 w g(w,y)}3 v ,  (3.37)  where g(w,y) is an arbitrary solution of a linear differential equation  The property of the generator (3.37) was not studied.  It is  shown here that it is related with a linear equation associated with Eq.(3;36).  >  According to Proposition 3, we have  I = v(w g)3 x°J  w  + (v gs - w X  3 g)3 . 6 w -^  V  (3.39) v J  This, however, is not in the form of (2.11)"^ 'and consequently  1  2  f) With z =w, z =v, the operator (2.11) takes the form 1=(U^cr^+U^u^)3w+(U^a^+U^a^)3y, where U 1 =U 1 (X(x,y),Y(x,y)), l^CX.Y) being an arbitrary solution of some linear system of differential equations.  Clearly, (3.39)-is not of this form,  there exists no 1-1 mapping of the system (3.36) to a linear system.  Now, introducing a potential z by  w = zx,  v = -z y ,  (3.40)  we write the second equation of (3.36), as  (z v  )3z xJ  XX  + (z )_1 = 0 v Y  (3.41)  The generator (3.39) can be written in terms of x,y and z as  1  where  =  (D  xU)8w  "  ( D  y  U p  V  =  ^DxU)3zx  +  (D  yU)9zy'  (3  U=U(z ,y) is ••••def ined~ by X U(z x ,y) = / * g(s,y)ds.  (3.43)  From (3.38) and (3.43), we obtain an equation for U:  (z x )"' e O z ) 2 U - 3 U = 0. y x  (3.44)  Noticing that (3.42) is the first extended part of the generator £ = U(z x ,y)3 z ,  (3.45)  we expect that Eq.(3.41) admits the generator (3.45) subject to  the _ condition (3.44).  A direct calculation verifies  '4  this.  We can now identify (3.45) with (2.2) obtaining  X=z , Y=y.  Going through the steps illustrated in the example  of the Legendre transformation in the preceding chapter, we find a Lie contact transformation  x -  y - Y,  z - Z -XZ X ,  z x - X,  zy  Zy> (3.46)  which transforms Eq; (3 . 41) * to a : linear equation •  X  Z  XX " Z Y  =  (3.47)  Let U(X,Y) be a solution of the differential equation  x; 3 U xx - U Y  = 0.  (3.48)  Then, from the first three relations in (3.46), we obtain an implicit solution of Eq.(3.41):  x = -U X (X,Y),  y = Y,  and the rest of (3.46)  z = U(X,Y) - XU X (X,Y),  (3.49)  leads to  w = X,  v = -U y (X,Y),  (3.50)  which together with (3.49) defines an implicit solution of Eq. (3.36).  To obtain an explicit solution, we solve the  first two equations of (3.49) with respect to X and Y, say, X=f(x,y) and Y=y, and introduce these into (3.50) to get  w=f(x,y),  v =-UY(f(x,y),y).  (3.51)  So far in this chapter, we only considered those equations which admit potential functions.  The following  equation does not admit a potential, but it is related to a linear equation.  3.3  The Liouville equation.  The Liouville equation is defined by  zxy  - e Z = 0.  (3.52)  One of the generators admitted by this equation is  £ = {f(x)zx + g(y)z y + f x (x) + g y (y)}9 Z ,  where f(x) and g(y) are arbitrary functions and f  and g  x are their derivatives.  (3.53)  y  This is the only generator which  depends on arbitrary functions.  The generator (3.53) is not  of the form (2.2) and hence there exists no 1-1 mapping of Eq.(3.52) to a linear equation.  Let us consider the  invariant solution z = u(x,y) associated with the generator (3.53).  It is a solution of the system  f(x)u x + g (y)Uy + f x (x) + g y (y) = 0 (3.54) - eu = 0.  u xy  The solution is found to be  u = £n  2<j> il) O  (3.55)  + M  -1 -1 where (J) (x) = Jf dx and ip (y) = Jg solution of the Liouville equation  dy. [37].  This is the general Introducing.  U=cp+ip, we write (3.55) as  u = £nI2U 1  (3.56)  U xU y  Recognizing U as the general solution of the equation U x y = 0 , we conclude that the transformation  z = ,£n|2Z 1  maps a solution of Z r  Z Z x y  =0 to a solution of z - e xy xy One can also see this from the equality  (3.57)  = 0.  z - e Z = v(Z_1D xy x x  + Z_1D  y y  - Z_2Z x xx  - Z~ 2 Z  y yy  - 2Z_1)Z  xy'  (3.58) hence Z =0 xy  3.4  z -e =0. xy  A series of L-B generators and the linearization.  Another sign which indicates a possible connection of a nonlinear equation to a linear equation is the admission of an infinite sequence of Lie-Backlund generators by the nonlinear equation.  From Proposition 10 it is clear that  a linear homogeneous system admitting a generator of the v y form L = (Z? !fl Z ) L9 y V admits an infinite sequence of L-B  generators.  Consequently, if there exists a mapping connect-  ing this linear system to a given nonlinear system, then the mapping is likely to transform these generators into group generators of the nonlinear system.  In Appendix 4, we  investigate L-B generators of a nonlinear diffusion equation D X  { (zTz > - z. X  L  = 0 and  it is _ shown that only for  the equation admits an infinite  a=-2  sequence of L-B generators.  The analysis of these generators in turn leads to a transformation similar to (3.11).  The Hopf-Cole transformation  of Burgers' equation can be obtained in a similar manner.  ,  -  8 "2  -  CHAPTER 4.  A SUMMARY AND FUTURE PROBLEMS.  4.1  A summary of the main reuslts.  In the second chapter we proved that by examining the invariance group of a system of nonlinear differential equations one can determine definitively whether the system is transformable to a linear system by an invertible mapping. Moreover, the mapping can be constructed from a generator of the group.  In all cases, we need only to consider group  generators of the form (2.24) or (2.25) in which no higher coordinates than z appears , i . e.-. e v = 0 V (x , z , z) .  In the third chapter we investigated the question of the existence of non-invertible mappings relating linear and nonlinear equations.  It is a considerably more complex  question than that of invertible mappings.  The problem of  finding such mappings is -equivalent to.finding a condition under which a given nonlinear equation admits a transformation leading to a factorization such as (3.4), (3.12) and (3.58).  No definitive condition has been found yet.  However, as it has been demonstrated here,  the group analysis  supplemented by the introduction of a potential function and higher order Lie-Backlund generators are effective means  to discover such non-invertible mappings.  The examples investigated in this work cover all linearizable equations ^known to the author and two new equations, i.e,, Eq.C3.7) and ^ Eq.(3.36).  With our method it should be particularly  emphasized that even if one is unable to linearize given nonlinear differential equations, one is always left with their invariance groups.  In turn these can be used for the  construction of invariant solutions, conservation laws and other invariance properties of equations [6,7, 8]. Some such examples, are given in Appendices 5-7.  4.2  A generalization of the concept of invariance.  In the group analysis of differential equations, it is very important to find the largest invariance group associated with the equations.  During the course of the  present work a question concerning the possibility to enlarge an invariance grouphas arisen.  Obviously, the meaning  of "largest" changes according to the type of groups we consider.  The group can be enlarged by considering  higher order L-B groups or by introducing more general types of invariance.  In the following we discuss some  aspects of integral dependent invariance.  We use notations  u,u x ,u x x ,... in place of  z  »z x » z.xx » • • • •  a. A hierarchic structure in L-B sequences.  To present the  basic idea clearly, we take a specific example, namely, Burgers' equation,  u  + uu - u xx x t  =  0.  (4.1)  A generator of the L-B invariance group of Eq.(4.1),  I = e(x,t,u,u x ,...,u x > >  x)3u  ,  (4.2)  (Dx)2e + uDxe + uxe - Dte = o  (4.3)  must satisfy the determining equation  for any u satisfying Eq.(4.1).  Now we let u+u+ev, |e|«l, t)  in Eq.(4.1).  Then v  satisfies- the linearized equation  J  v + uv + u v - v^ = 0. xx x x t In view of Eq.(4.3) and Eq.(4.4), it is clear that 0 is  t) Here, the term "linearization" is used in a different sense than in the preceding chapters.  (4.4) y. j  a very special solution of the linearized equation (4.4): A solution expressed in.terms of a solution of the original equation (4.1).  This observation leads us to examine the  invariance group of - therlinearized equation (4.4) for.from such an invariance group we may be able to construct those particular solutions which satisfy Eq.(4.3).  So, we  consider the following problem:  Find invariance groups of the differential equation v  +uv + u v - v^ = 0 with unknown v and an arbitrary xx x x t solution u(x,t) of the equation u + uu - u = 0. XX  X  X  Since Eq.(4.4) is linear in v, it is sufficient, according to Theorem 1, to consider a linear generator i = (Bv)3  (4.5  = 63  where B is some linear operator.  Once B is found, we can  find, using Proposition 12 in §1.7, an infinite sequence of solutions of Eq.(4.4) in the form v(n)(x,t)  ( B)n v (x , t) ,  n=l, 2 , 3 , .  where v(x,t) is any solution of Eq.(4.4).  In particular,  if we choose as v one of 0 satisfying- Eq.(4.3), we obtain a sequence of functions  (4.6  0 ( n ) (x,t,u,u x ,u x x , . . .) = (B) n 6(x,t,u,u x , . . .) ,  which solve Eq.(4.4), and hence satisfy Eq.(4.3). words, once we find a generator  £ =(#v)3^  In other .  of the linearized  equation, , we can construct a sequence of L-B generators  £ = {(3)ne}.3u,  for the original generator  £=03^.  n=l, 2,3,...,  (4.8)  nonlinear equation from any known Obviously, this statement holds for any  system of nonlinear equations.  We apply this result to  analyze L-B invariance groups of Eq.(4.1). Some years ago I found that Burgers' equation admitted hierarchies of L-B generators.  The question is whether these  have the structure of the form (4.8).  A simple calculation  shows that the only point group generator, i.e.  £ = (Sv)3v = {b X (x,t)v x + b t (x,t)v t + b(x,t)v}3 y (4.9) admitted by Eq.(4.4) is  £=v3 v , i.e. 5=1.  Thus, as long  as £ is restricted to the form (4.9), there is no sequence of L-B generators of the form (4.8).  Olver found in his study  [18] of symmetries of time evolution equations that Burgers' equation admits an infinite sequence of L-B generators of the form  (4.7)  £ = {(£) n u x } 9 u ,  (4.10)  n=l, 2 , 3 , . . .  with D = D  X  + hu + hu D" 1 , X  X  (4.11)  '  where D ^ denotes an integral operator with the property -1  X  D D = D - 1 D =1. Comparing (4.10) with (4.8), we let s =D XX X X and consider an operator corresponding to (4.5):  69  v  - v(5v)3 7  v  = v(v x  + %uv + hu D _ 1 v ) 9 x x ^ v  .  (4.12)  It is easy to check that Eq.(4.4) indeed admits (4.12). The sequence (4.,10) is one of the- hierarchies- of generators I found previously.  This suggests the existence of other  linear operators B,  An important aspect of the generator  (4.12) is that e depends not only on x,t (through u), v and v  but also on the integral D ^v. X  This leads us to  X  consider a generalization of (4.9) by including a D  term:  £ = (Bv)9 v = {b X (x,t)v x + b t (x,t)v t + b(x,t)v + b" x (x,t)D x 1 v}9 v (4.13) The ' b's are functions of x and t.  For this type of  integral dependent generator , we can adapt Lie's algorithm for finding generators.  With a straightforward calculation  we find that Eq.(4.4) admits, in addition to (4.12) and the trivial  £=v9 , -: the generators with-following B's:  -1 B f = D t + (%u x + %u 2 ) + hu tXD t x 5" = tJ5' + %xD  X  + %xu + %(u+xu ) D _ 1  (4.14a-c)  -1 to • + '-iX + -2 D I X  where in 5"', D is the operator (4.11).  Using any o£ (4.14)  we can produce sequences of L-B generators of the form (4.10). More generally, recalling Proposition 10, we see that operators of the form  (4.15)  u  where f(•,•,•,•) is an arbitrary polynomial of its arguments are all invariance group generators of Burgers' equation for any 0 satisfying the determining equation (4.3), for example 0=u . They include all hierarchies mentioned.above.  b. On integral dependent generators.  The example we have  just seen and the example in §3.2 lead us to generalize the concept of invariance by allowing generators to depend not only on derivatives but also on integrated quantities. A fundamental difficulty in considering integral dependent generators is that there are too many possibilities. Consider a time evolution equation  G  (U'UX'UXX'••'•'ux...x) " u  with invariance group generator £=09^.  t  =  0  C4.16D  We want to allow  to depend on integrals such as  D _ 1 u = / u dx, X  ( D _ 1 ) 2 U = // U X  dxdx,  (4.17)  A problem is that these are not the only possible integrals involving u.  There is no a priori reason not to include  integrals such as  Dx1(u)2,  Dx1(ux(Dx1u)),  etc.  .  (4.18)  One way to get around this problem is again to study the linearized equation of Eq.(4.16):  G v + G v: + . . . + G u u x u X  Let  v x...x  X• • •X  - v t  = 0.  be an invariance group generator of (4.19).  The merit of considering Eq.(4.19) instead of Eq.(4.16) is that since Eq.(4.19) is a linear equation in v, we can expect that 0 of £ depends linearly on v, i.e. 0=£v, for some integro-differential linear operator B . of B, however, is still quite arbitrary. Bv could contain terms such as  The form  For instance,  (4.19) J  D^Cfv),  D ^ g D ' V v ) ) , ...  where £ and g are functions of x,t.  (4.20)  We can avoid considering  such complex expressions if we observe that quantities such as (4.20) can be formally represented by a series of the form  {  I b(k)(D )k x k=0  where b^-' and b^-  +  I k=l  b ( - k ) ( D " 1 ) k } v = 5V,  are functions of x,t.  (4.21)  For example,  by repeated integration by parts, we have for the first expression of (4.20),  D x 1 ( f v ) = Jfvdx = f D ^ v - ( D x f ) ( D x 1 ) 2 v + ((D x ) 2 f) ( D ; 1 ) ^ + .... (4.22) For the second expression of (4.20), we apply this procedure to each integration operator D ^. a generator  t= (Bv)3v  Thus, we should start with  with Bv of the form (4.21) rather  than quantities such as (4.20).  Once such an operator B is  found, we can obtain a sequence of invariance group generators for the original equation (4.16) using the formula (4.8). Although the approach described here is of great generality, it will be of little practical value unless a ^closed form of"the infinite sum in (4.21)-is found.  Such a closed expression  may be found by examining the first few terms in the sum. In applying this analysis to the KdV equation u  + uu  + u  = 0,  a variety of B'S of the form (4.21)  has been found.  The only B with a closed form is  B  =  (D  x)2  +  1  u  +  1 "A  1  (4  -  -23)  This particular operator arose in the study of the inverse scattering method [51] to solve nonlinear time evolution equations.  It was also obtained by Olver in connection  with invariances of the KdV equation.  We use (4.21) with  (4.23) to obtain invariance group generators of the KdV equation.  The equation admits point group generators  £=03 u with the following 9: (a)  u  ' (b) x  9  >  =  _u  _ (c) t  '  0  =  '  t u  , „ (d) ; " '  x  . 1 2 tu XU " t" 3 x" 3U' (4.24)  Using these 0's in (4.21) we obtain sequences of invariance group generators.  It turns out that there are only two  sequences instead of four. u =u n , u X  -L  With 0 = ( B ) n 0 ^ ^  and.notations  =u 9 ,..., one sequence is:  XX  Z  d(0) Q (a) D (l) 0V J _ .Q «., ./_ u (3)  =  u  5  r  +  5 uu 3T 3 7  +  Q (b)  A  J = Uj + uu^ = 0V  10 5 -=- U..U- + 3 1 2 6  2  u  u  x  ,  ....  J  ,  ,  and the other is  e  C°)  =  0  (  c  )  =  tia1-l,  0  ( 1 )  =  t(u  3  +uu  1  )  - |x  U ; L  - |u  =  0  ( d ; )  ,  (4.26) 1  1  Since the first sequence involves no integral quantity, it corresponds to L-B invariance groups and was first found in connection with conservation laws associated with the KdV equation (Appendix 5).  The second sequence is new and  involves integrals.:- -.... '  Olver was interested in the operator D with the property that £=(Z?n^6)9 is an L-B invariance group generator u of a nonlinear time evolution equation. The present analysis shows that D is related with a generalized invariance of the corresponding linearized equation.  Clearly, our formulation  is not restricted to time evolution equations and can lead to more general invariance groups than L-B groups.  There  should be further investigations of these generalized symmetries and their uses.  BIBLIOGRAPHY  1.  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Dokl. , 17_ (1976) 1242-1246.  18.  P.J. Olver, Evolution equations possessing infinitely many symmetries, J. Math. Phys. 18_ (1977) 1212-1215 .  19.  A.S. Fokas, Invariants, Lie-Backlund operators and Backlund transformations, Ph.D. Thesis, California Institute of Technology (1979)  20.  N.H. Ibragimov and A.B. Shabat, The Korteweg-deVries equation from the group point of view, Dokl. Akad. Nauk, SSSR, 24£ (1979) 56-61.  21.  A. Galindo, Conserved densities for nonlinear evolution equations I. Even order case, J. Math. Phys. 2_0 (1979) 1239-1243.  22.  M.J. McGuinness, Infinities of polynomial conserved densities for.nonlinear evolution equations, J. Math. Phys. 21 (1980) 2737-2742.  23.  A.S. Fokas_ A symmetry approach to exactly soluble evolution equations, J. Math, . Phys . / 21 - (1980). 1318-1325.  24.  A.S. Fokas and R.L. Anderson, Group theoretical nature of Backlund transformations, Lett. Math. Phys. 3 (1979) 117.  25.  For instance, B.C. 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Fris, Symmetry groups in classical and quantum mechanics, Soviet Journal of Nuclear Physics, 4 (1967) 444-450.  33.  A.A. Makarov, J.A. Smorodinski, Kh. Valisov and P. Winternitz, A systematic search for nonrelativistic systems with dynamical symmetries I, Nuovo Cimento 5_2 (1967) 1061-1084.  34.  G.W. Bluman, Proceedings of Conference on Symmetry, Similarity and Group Theoretic Methods in Mechanics, Ed. P.G. Glockner and M.C. Singh, University of Calgary Press, Calgary (1974).  35.  G.W. Bluman, On the transformation of a diffusion process to the Wiener process, SIAM J. Appl. Math., 39 (1980) 238-247.  36.  For instance, R. Courant and D. Hilbert, Methods of Mathematical physics, Vol.2, Interscience Publishers, New York (1962).  37.  A.R. Forsyth, Theory of Differential Equations, Vol.6, Cambridge University Press, Cambridge (1906) .  38.  S. Lie, Neue Integrationsmethode der Monge-Ampereshen Gleichung, Arch, for Math. 2_ (1877) 1-9.  39.  N.J. Zabusky, Exact solutions for the vibrations of nonlinear continuous model string, J. Math. Phys., 3 (1962) 1028-1039. C. Ferrari and F.G. Tricomi, Transonic Aerodynamics, Academic Press, New York (1968).  41.  B.R. Seymour and E. Varley, Soliton like interactions in a non-dispersive medium, Technical Report 80-12(August 1980), Institute of Applied Mathematics and Statistics, University of British Columbia  42.  V.L. Katkov, Group classification of solutions of the Hopf equation, Z. Prikl. Mek. i Techn., 6 (1965) 105.  43.  E. Hopf, The partial differential equation u +uu =yu , Comm. Pure Appl. Math. 3 (1950) 201-230. t x xx  44.  J.D. Cole, On a quasilinear parabolic equation occurring in aerodynamics, Quat. Appl. Math., 9_ (1951) 225-236.  45.  G. B. Whitham, Linear and Nonlinear waves, Wiley Interscience, New York(1974) .  46.  H.C. Thomas, Heterogeneous ion exchange in a flowing systems J. Am. Chem. Soc.. 66^ (1944) 1664-1666.  47.  A. Yoshikawa solutions to differential Kyoto Univ.,  48.  A.Hasegawa, Propagation of wave intensity shocks in nonlinear interaction of waves and particles, phys. Lett., 47A (1974) 165-166.  49.  H. Hashimoto, Exact solutions of :a certain semi-linear system of partial differential equations related to a migrating predation problem, Proc. Japan Acad., 5_0 (1974) 623-627.  50.  M.G. Sukharev, Invariant solutions of equations describing a motion of fluid and gas in long pipe-lines, Dokl. Akad. Nauk. SSSR, 175 (1967) 781-784.  51.  M.J. Ablowitz, D.J. Kaupman, A.C. Newell and H. Segur, The inverse scattering transform-Fourier-analysis:for nonlinear problems, Studies in Appl. Math., 5_3 (1974) 251-315.  and M. Yamaguti, On further properties of a certain semi-li-near system of parital equations, Publ. Res. Inst. Math. Sci., 9 (1974) 577-595.  APPENDIX 1,.. PROOF OF THEOREM 3.. ,  For brevity we write 0 V (X,Z,Z,...,Z) = 0 V (X,Z). 1 m v y V Suppose the system ^ Z = 0\admits L = '0 (X,Z)9 v . 1'hcn, . y z,  Proof.  since the system is linear homogeneous, L = 0 V ( X , e Z + U ) i s also a generator for an arbitrary solution of We write its power series expansion in e as 0 v (X,eZ+U)3 z V = h v ( 0 ; ) 9 z v  + eh  v  ^h  z V  + e2hv(2)9zV+  ... .  (Al) Since e is arbitrary, each term must be a generator of an invariance group.  In particular, the 0(e) term has the  form (1.43). Conversely, suppose that the system (1.43).  admit-  The function 0 v (X,Z) yielding (1.44) is not unique,  but if we restrict to 0 V (X,Z) which depends on X and only on those Z's which appear in the coefficients of 9 z v i-n (1.43), then 0 V is determined within an arbitrary additive function <}>V(X).  We let Z->-eZ in such 0 V (X,Z) and expand  0 V (X,eZ) in the series in  © V (X,eZ) = h v ( 0 ) - e h v ( 1 )  +  e2hv(2)  +  ... ,  where h ^ ^ = 0 V (X,O) which is a function of X alone.  We may  set h v ^ ^ = 0 since we have an arbitrary function cf>V(X) at  (A2)  our disposal.  With such a choice of (|)V(X), the operator  0 V ( X , Z ) 3 z V becomes an invariance group generator. this we let U+Z in (1.43), then let Z+eZ.  To see  The resulting  operator is still an invariance group generator of the same system.  We write its power series expansion in e as  L = egv(1) 8zV  +  £  2  gV(2)3zv+  ... .  (A3)  By the construction of 0 v (X,Z) above, we have i n (A3) is an invariance group  Since every generator of  we see that the operator  0 V (X,Z)3  v  = hvC1)3  v  +  hv(2;)3  is an invariance group generator of 4 V Z^=0.  v  + ...  APPENDIX 2. PROOFS OF PROPOSITIONS 9-12.  1.  Proof of Proposition 9.  LB V =B V L and LB V =B V L.  y y  From Proposition 2, we have  Then,  y y  V M V M [L,L] = (LB yZ - LB yZ )^8 Z 7 Vv = v( B L Z - B L Z ) 8v7 V ' y y ^ Z  V K y B Z - B V B K Z y ) 8v7 V K y K y ^ Z  (B v  = (CB,B]JV Z y )3 . y ZV v V  By hypothesis, we have [L,L]=L=(S^ Z M ) 8 z V .  y  Comparing these,  we obtain [B,B]=-B. •  2.  Proof of Proposition 10.  It is sufficient to prove  4vzy=0 of f =0. By condition (1.36), we have  for the explicit form the invariance 4VZy=0 y  -  L^VZy y  =  = 4 V Z ' y = 0, y  AVB KZ y  k y  V 1y V y i4 L 4 V Z y = 0, y Z = 0 - L Ay Z ' = Ly  v y v y i.e., . 4y Z = 0 + LL4 y Z =0.  Proposition 2 and  •  3. Proof of Proposition 11.  As in the proof of Proposition  L 4 v Z y = 4 v S y Z K = 0 . We also have B v 4 y Z K = 0 since y y K y K y K v y 4 Z = 0 . From these two we obtain (4 S - B v ^ y ) Z l < = 0 K y K y K u K provided A Z =0, i.e. U,B]Z = 0 provided AZ=0. • K 10, we have  4. Proof of Proposition 1-2. .We have LAZ=4LZ=-4VSyZX = 0 c li K for any values of X,Z,Z,... satisfying the equation ^.Z = 0. Obviously, Z=U(X), Z=U(X), ..., satisfy this equation and consequently, 4 v B y U K ( X ) = 0 , i.e. 4BU(X)=0. y K same argument, we find A(S) m U(X)=0. •  Repeating the  APPENDIX 3.  A DETERMINATION OF AN INVARIANCE GROUP OF (z ) 2 - 4z z = 0. ^ xy J x y  THE EQUATION ^  To illustrate the process of determining an invariance +  group of a differential equation, we consider the equation f = (z v ) 2 " 4z z ^ xy' x y  = 0.  (1)  To simplify notations in the following computation, we let Z X =P, z y =q,  z  xx=r'  z  xy=s  a n d  z  yy=t"  Then  >  Eq-C 1 ) becomes  s 2 - 4pq = 0.  (2)  We consider a generator of the form  1  = g(x,y,z,p,q)3z,  (3)  By operating its second extended form on (2), we get  £(s2-4pq) = 2sD D g - 4(D g)q - 4p(D g) A y  f  This equation  x  y  is' taken from Forsyth [373 page 198 ,  2{ +  §xys  +  gZyPs  Sxz^ 5 +  +  SzzP^ 5  Sxps2  +  §xqst  +  -SzpPs2 + SzqPst + § z s 2 2 +g rs + &g qrs + 6g rs + g r s t + 5g s s & Py pzH pp pq p x 2 2 3 2 +g qy s + &gq z qs + g qps +g qqs t + g qss y H - 2g y P - 2g z pq - 2g p ps - 2g q pt - 2g x q - 2g z pq - 2g p qr - 2g a qs}.  (4)  The invariance condition (1.36) demands that (5) vanish under the condition (1.34) which^in.this ckse takes the form  f = s 2 - 4pq = 0, D f = 2ss .X.  - 4rq - 4ps = 0,  -X  D f = 2ss - 4sq - 4pt = 0, y y We only need the first three equations since (4) involves coordinates only upto:-the third order. " We use these three equations to eliminate the coordinates s,s x ,s y from (4). The resulting quantity must vanish irrespective of values of x,y,z,p5Jq,r and t: rearranging terms,  0 = rtg + r(g +g q + 2g Fp 2Mq 2 ) + t(g +g p + 2'g p 2q 2 ) s PR PY p z H pp qx 5 q z F & q q F M +  2 g x y p 2 q 2 + 2gxzq(pq)2 + 4gxppq  +  ZgzyPtpq)*2  +  4  c  gqyP l  +  4  +  2 g z z (pq)^2" + 4 g z p p 2 q 2 (  gqzP l  3  +  8  (  Sqp ^P c l^  " 2 §yP " 2 g x q '  ^  We note that r and t appear only in the first three terms of (5).  Thus their coefficients must vanish:  g  = 0,  g  + g  g  + g  p + 2g  1, (pa) 2 = 0,  a + 2g  (pq)% = 0 .  It is easy to find that the most general solution to these equations is  g = a(x)p + b(y)q + e(x,y,z),  (6)  where a,b and e are arbitrary functions of their arguments, Introducting (6) into the rest of (5), we get  0  V  = eXyCpq)2 +  ^2  eyzp(pq)  V+ e x z q ( p q ) 2 + 2a x pq 3 J + ezz(pq)  + 2b y pq  (b y q + e y )p - (a x p + e x )q.  Since a,b and e do not involve p and q, coefficients of different powers of p^q11 must vanish.  The resulting  equations yield  e = az + g,  a, £3 arbitrary constants.  Therefore,  g = a(x)p + b(y)q + az + 6.  The final form of the generator is  I  = {a (x)p + b(y)q + az + 3)9 Z ,  (7)  where a(x) and b(y) are arbitrary functions and a and 3 are arbitrary constants.  We note the resemblance between  the generator (7) and the generator (3.35) of the Liouville equation.  As in the case of the Liouville equation, the  invariant solution associated with (7) will lead to the general solution of Eq.(l).  On the remarkable nonlinear diffusion equation (d/dx)[a(u + b) 2(du/dx)] - (du/dt) = 0 George Bluman and Sukeyuki Kumei  Department of Mathematics and Institute of Applied Mathematics and Statistics, University of British Columbia, Vancouver, British Columbia, Canada V6T1 tV5  (Received 18 July 1979; accepted for publication 14 December 1979) We study the invariance properties (in the sense of Lie-Backlund groups) of the nonlinear diffusion equation (<?/dx)[C (u)(du /dx)] — (du/dt) = 0. We show that an infinite number of oneparameter Lie-Backlund groups are admitted ifand only if theconductivityC(u) = a(u + b )' 2. In this special case a one-to-one transformation maps such an equation into the linear diffusion equation with constant conductivity, (d 2u/dx 1) — (du/dt) = 0. We show some interesting properties of this mapping for the solution of boundary value problems.  1. INTRODUCTION  In recent years nonlinear diffusion processes described by the partial differential equation (p.d.e)  J_ O ) du dx I~  dx  dt  (1)  with a variable conductivity C(u), have appeared in problems related to plasma and solid state physics. 1,2 Interest in such processes has long occurred in otherfields such as metallurgy and polymer science.3"5 Some exact solutions are well known for such equations. 6 These can be shown to be included in the class of all similarity solutions to such equations obtained from invariance under a Lie group of point transformations. 7 8 Recently, it has been shown that differential equations can be invariant under continuous group transformations beyond point or contact transformation Lie groups which act on afinite dimensional space.' These new continuous group transformations act on an infinite dimensional space. Such infinite dimensional contact transformations have been called Noether transformations 10 or Lie-Backlund (LB) transformations' 1 (Noether mentioned the possibility of such transformations in her celebrated paper on conservation laws'3). Well known nonlinear partial differential equations admitting LB transformations include the KortewegdeVries, 13 u sine-Gordon, 10 " cubic Schrodinger, 14 and Burgers' equations. 16 All of these known examples admit an infinite number of one-parameter LB transformations. Moreover, many of their important properties (existence of an infinite number of conservation laws, 13 M existence of solitons, 14 and existence17 of Backlund transformations 18) are related to their invariance under LB transformations. Any linear differential equation which admits a nontrivial one-parameter point Lie group is invariant under an infinite number of one-parameter LB transformations through superposition. Moreover, every known nonlinear p.d.e., invariant under LB transformations, can be associated with some corresponding linear p.d.e. With the above views in mind we study the invariance properties of Eq. (1). Previously,7 1 9 it had been shown that Eq. (1) is invariant under 1019  J. Math. Phys. 21(5), May 1980  a) a three-parameter point Lie group for arbitrary C (u), b) a four-parameter point Lie group if C(u) = a-(u + A) v , c) afive-parameter point Lie group if v = — j. [It is well known that a six-parameter point Lie group leaves invariant Eq. (1) in the case C(u) = const. 20 ] In the present work, we show that Eq. (1) is invariant under LB transformations ifand only if the conductivity is of the form  C ( U ) = f l-(U + i)- 2 ,  (2)  i.e., if Eg. (I) is of the form du dt  =  0.  (3)  Furthermore, this equation admits an infinite number of LB transformations. In this special case, there exists a one-to-one transformation which mapsEq. (3) into the linear diffusion equation with constant conductivity, namely, the heat equation  cFu  dx- '  ** =0. dt  (4)  In the course of this paper, wefind an operator connecting two infinitesimal LB transformations leaving Eq. (3) invariant. We prove that this operator is a recursion operator which generates an infinite sequence of one-parameter infinitesimal LB transformations leaving Eq. (3) invariant. Moreover, we show that no other LB transformation leaves Eq. (3) invariant. By examining the linearization of Eq. (3), we are led to construct the transformation mapping Eq. (3) into Eq. (4). It is shown that this transformation maps the recursion operator of Eq. (3) into the spatial translation operator of Eq. (4), giving a simple interpretation of the transformation relating Eq. (3) to Eq. (4). We use this transformation to connect boundary value problems of Eq. (3) to those of Eq. (4). We construct a new similarity solution of Eq. (3) corresponding to invariance under LB transformations.  0022-2488/80/051019-05S01.00  (T 1980 American Institute of Physics  2. DERIVATION OF THE CLASS OF NONLINEAR DIFFUSION EQUATIONS INVARIANT UNDER LB TRANSFORMATIONS  lead to determining equations Cl/„.„= 0,  LB transformations include Lie groups of point transformations andfinite dimensional contact transformations. 1 1 The algorithm for calculating infinitesimal LB transformations leaving differential equations invariant is essentially the same as Lie's method 8 for calculating infinitesimal point groups. Consider the most general one-parameter infinitesimal LB transformation that can leave invariant a time-evolution equation, 21 namely; u* =  u  + eU(x,t,u,u,  a„) + 0(e*),  (9)  "C'U„u, = 2C X  •  (10)  ;•= o  Solving Eqs. (9) and (10) wefind that l/ = a ( C ) " / 2 , X + £(u,u„,..,u„_, ),  (11)  where f i s undetermined, and a = arbitrary constant. The substitution of Eq. (11) into the remaining terms of Eq. (8) leads to a polynomial form in u„ whose coefficients of (u„ Y and u„, respectively, lead to determining equations CE n _,.„_, = 0 , (12)  (5) t* = t, where u, =ffu/dx',  u, =C'(u,y  + (1 - n)C'£„_, u, - —n(n +3)C' (C)  i = 1,2,-. Let du/dt = u,, du,/dt  = u„, au/du = U 0, dU/du, = {/„ d 2U/du,du C • = dC /du, and C " = d 2 C /du 2. In the above notation Eq. (1) becomes  t  = U iJt  (13)  Under Eqs. (5) the derivatives of u appearing in Eq. (6) transform as follows: , 2  («,)* = «, +eU + 0(e ), (u,)' = u, + eU' + 0{ e2), (u 2)* = u2 + eU" +  where  0(e 2),  i  (7)  U* = D,U=^+ ±U iUi+,, dx  iE o  rtt 2  Solving Eqs. (12) and (13) wefind that U=a  [ ( C ) ( , / 2 ) X + ln(n +3)C' (C)" + F(u)u„_, +G(u,u u„_2),  /2, n  -'  u,u„_, ] (14)  where Fand G are undetermined and, more importantly, for a ^ 0 it is necessary that the conductivity C(u) satisfy the differential equation 2CC' = 3(C') 2 -  au at  "u2  + a [ j n 2 ( C ' ) 2 ( C ) n / 2 1 " - j n ( n + 2 ) C ' ( C ) " / 2 > " ] ( u , ) 2 = 0.  (6)  + Cu 2.  ,1/1 )  4  [To dx  Hence, it is necessary that C (u) — a-(u +  (15)  b )-2,  (16)  where a and b are arbitrary constants for the invariance of Eq. (1) under LB transformations. Without loss of generality we can set a = 1, b = 0, i.e., from now on we consider the equivalent p.d.e.  =  J-( u-'*!L)  (17)  + X + .">+! + S U'"<*2- at dx\ dx) This particular equation has been considered as a model i.j=0 (=0 4 5  /), and Z), are total derivative operators with respect to I and x, respectively. The transformation (5) is said to leave Eq. (6) invariant if and only if for every solution u = 6 (x,t) of Eq. (6) U ' = C " U ( u , y + 2C'tf*tt, + C , t / u 2 + C£/". (8) The fact that U must satisfy Eq. (8) for any solution of Eq. (6) imposes severe restrictions on V. Using Eq. (6) the derivatives of u, with respect to /, i.e., u„, can be eliminated in Eq. (8). Since the invariance condition (8) must hold for every solution of Eq. (6), Eq. (8) becomes a polynomial form in u„ + , and u„ + 2 . As a result the coefficients of each term in this form must vanish. This leads us to the determining equations for the infinitesimal LB transformations (S). If in Eq. (5), n<2, we obtain the Lie group of point transformations leaving Eq. (6) invariant. Without loss of generality we assume n>3 in Eq. (5). It turns out that for n> 3, U is independent ofx and I. In our polynomial form, the coefficient of u„ + 2 vanishes and (he coefficients of (i/„ 4 , )' and u„ t l , respectively,  equation of diffusion in high-polymeric systems.  3. CONSTRUCTION OF A RECURSION OPERATOR; AN INFINITE SEQUENCE OF INVARIANT LB TRANSFORMATIONS OF EQ. (17)  For n = 3 it is easy to solve the rest of the determining equations and show that the only LB transformation leaving Eq. (17) invariant is {/= U 1'' = u'*u3 ~9u~4u,u2 +12«' 5 (U|)\ For n — 4 we obtain two linearly independent LB transformations U' 1' and V' 2' = u-*u< -14u- 5 u,u, -10u" 5 (u,) 2 +95u- 6 («,) 2 u 2 -90u- 7 (u,) 4 .  (19)  The existence of U " ' and V' 2', combined with the work of Olver,' 6 motivates us to seek a linear recursion operator & leading to infinitesimal LB transformations C"k 1 defined as follows:  (18)  - 107 -  (3)kB=U {k\ k = 1,2,—. (20) /= f-(uS), 2 The character of \B,U"\U' '\ leads one to consider for 3 dx the form  &  we obtain from Eq. (29) the equation  pD, + q + KD x)\  (21)  l\ dx) where D„ is a total derivative operator, (D X)-(D X)~' is the if we set identity operator, and J p,q,r] are functions of\u,u„uand \ . 2 Then one can show that 2B — U' 1' if and only if d_ p = u\  and  •*•"  2  (22)  2  dx  q[u u2 —2u"'(u ,) ] + ru" u, = -3u-4uiU 2+6U- 5(U i)3. 2 2 Furthermore, (3 ) B=U' ' if and only if 2  q= —2u" u„  (23) •  —  (24) dx  dt]  (31)  (32)  dt  Eq. (31) becomes  and  r = -u- 2 u 2 +2u- 3 («,) 2 .  =  dt  dx  d .3 J_ dx' u u.  3  2  dx  2  (33)  H i =0 dT  Since/= 0 is always a solution of Eq. (29), the relation (30)  (25) suggests that we set uu — constant. This and Eqs. (32) lead  A more concise expression for the operator is us to the transformation dx — u dx + u"2u, dt, & =(D xf.(u {).{D xyK (26) dt = dt, (34) We now show that the constructed operator 3 is indeed a recursion operator. Let the operator relating solutions u = 8 (x,t) of Eq. (17) to solutions A = £ B,(DJ u = 6 (x,7)ofEq. (33). Choosing afixed point (x 0,l 0), we have the following integrated form of Eqs. (34): i=0 = u\D x)2 -4u-3u, D, +6!/""(u,) 2 -2u}u2 = (D,)W 2,  (27)  where B, = (d/du,)B. Olver's work 16 shows that cursion operator for Eq. (17) if and only if the commutator  (35)  3> is a re-  It is easy to check that Eqs. (35) indeed transform Eq. [A-D,,3] = 0, (28) for any solution u = 0(x,t) of Eq. (17). Moreover, if 3(17) is ato Eq. (33), and define a map relating the solutions of define recursion operator, then the sequence | U '",£/' 2 ',••• j given by Eqs. (17) and (33). Moreover, if u > 0 (u > 0), Eqs. (35) a one-to-one map since dx /dx > 0 for each fixed t. 22 Eq. (20) is an infinite sequence of LB transformations leavWe now show that under the transformation (34) the ing Eq. (17) invariant. It is straightforward to show that A recursion operator 3 of Eq. (17) is transformed into the and 3 satisfy Eq. (28). operator The nature of U' 11 and the form of a general V givenrecursion by Eq. (11) show that for n = / +2, there are at most &</ lin3 — Ds, (36) early independent LB transformations leaving Eq. (17) inleading to an infinite sequence of LB transformations of the variant since U must depend uniquely on u, + ! . heat equation (33). The proof is as follows: The proof that 3 is a recursion operator demonstrates An LB transformation of the form (5) induces an LB that k = 1 and hence we have found all possible LB transfor- transformation on the variables \x,t,u\ through Eqs. (34), tl mations leaving Eq. (17) invariant, namely, | V '), namely, k= 1,2,-. 2 x* =x + ei+0(  4. A MAPPING TO THE LINEAR DIFFUSION EQUATION  As far as we know all p.d.e.'s invariant under LB transformations have a recursion operator and, moreover, can be related to linear, p.d.e.'s. This suggests the possibility of seeking a transformation relating Eq. (17) to a linear equation. This leads us to consider the linearization of Eq. (17), namely, (A -<9/<?r)/=0, where/4 is given by Eq.(27)foranysolutioni/ = 8 (17). Introducing a new variable u by  (29) )ofEq.  e ),  r* = 7,  (37)  u* = u + etj + O (r),  where g and rj are defined by d£ = s/ dx + 33 dT,  .</ = uV, & =u ;U+(u)\U*-2U), (38) V — ~(M?V. It turns out that for any solution u = §(x,T) of Eq. (33), ,c/ and satisfy the integrability condition D-sJ = D, .-:'<, so that dg is an exact differential. The integrated form of; is  i= - {D,Y'[rj<u)-']+c,  (39)  where c is an arbitrary constant. Since U" + " = 3> U"\ where .£/' is given by Eq. (26), for c — 0 we get a corresponding infinite sequence of invariant infinitesimal LB transformations j t/" 1 ) for Eq. (33), namely, in  U = f - u-,i',  u = 0(x,r)= ^—-!  on the domain Jt€(*,(/), x2(t)), / > 0, where xl(t)  (40)  u = 8(xj) =  (41)  = (D iyu-2,  ' = 1,2,-.  x = X(x,t)  5. PROPERTIES OF SOLUTIONS OF EQ. (17) FROM THE MAPPING  ,'  -  -  =  f 0(x\t  Jo  )dx'  (48)  df,  with the corresponding definition of the function  X\x,7) = x.  Example 1: The source solution of Eq. (33), i.e.,  u = 6 (x,7) — a(4ir7y' r2e ~  ' on the domain  — oo <x < x ,t > 0, is mapped by Eqs. (45) and (46) into the following separable solution of Eq. (17): u = 0(x,t)  =  a'(4rrt)'  /2  e'\  on the domain — Ja <x < Ja, t > 0, where u(jc) is defined (49) by  x=  V7-•r  Note that l i m ^ _ ± i a 0 ( x , t ) = + oo. Example 2. The dipole solution of Eq. (33), i.e.,  We now consider the use of Eqs. (34) in constructing solutions to Eq. (17). It is easy to show that Eqs. (34) are equivalent to  dx = u dx + u, dt, dt = d7,  — _  d (0(x,!'))-' dx  (42)  D, corrsponds to the obvious invariance of Eq. (33) under translations in x. It is interesting to note that the recursion operator for the invariant LB transformations of Burgers' equation is also mapped into the space translation operator under the HopfCole transformation relating Burgers' equation to the heat equation. Moreover, we can obtain the Hopf-Cole transformation by examining the linearization equation (29) corresponding to Burgers' equation.  (46)  X(x 2,t).  1 (47) x,(n = X(x„7), x 2(7) = X(x,,7),  leading to Eq. (36). Moreover, U< n = D;(ug')  =  on the domain JPe(Jc,(/ )Jc 2 (t)), t > 0 where  and u7 = (d/dx )'u. From Eqs. (40) it is simple to show that  =DiU"\  = X(x„tU 1(.t)  In a similar manner, Eqs. (35) map a solution u = 6 (x,t) of Eq. (17) to  where V'=-(i7)2U°\  ,  0(X'(x,t),t)  u = 0(x,7) = -  dx  —la(4v7y  V2  e-ii:">T ,),  on the domain 0 <x < oo, F> 0, is mapped by Eqs. (45) and (43) (46) into the following self-similar solution of Eq. (17):  « = («)-•, with an integrated form  := J_ udx + ' = '-h,  (uT h., dt\ (44)  « = («)-', for somefixed point (x 0,t 0). In the following, we assume u > 0 (u> 0). Without loss of generality, we set = ?„ = 0.  on the shrinking domain 0 <x  (50)  <a(4-t ) " 2 , t > 0.  B. Connection between Initial conditions; connection between boundary conditions  The mapping formulas (34) and (43) demonstrate a one-to-one correspondence (within translation of at,;) between initial conditions for Eq. (17) and those for Eq. (33). A. Explicit formula connecting solutions; examples As for the connection between boundary conditions, from the same formula it is easy to see that x = s(f) is an insulating First we consider the problem of giving a more explicit boundary ofEq. (17), i.e., [dd(x,t)/dx] T ,.« { , = 0, if and formula for relating solutions of Eq. (33) to those of Eq. (17). Let a = 0(x,7) be a solution of Eq. (33) on the domain 7>only 0, if the corresponding boundary if = s(t ) is an insulating boundary of Eq. (33), i.e., the corresponding solution xe(x,^ 2). By Eqs. (43), u = 6(x,7) satisfies [dS(x,7)/dx] i _ s,-> ='0. Moreover, x = X(x,7) = pi*-,f) d? + £ =o '.(45) s(t) = const if and only ifi(() = const, i.e., there is a one-toone correspondence betweenfixed insulating boundaries of This uniquely determines the function A""1, x — X~'(x,t), Eqs. (17) and (33). where t = t. Now Eqs. (44) lead to the following solution of In general, a noninsulating boundary condition for Eq. (17), on afixed boundary x = const = c, is mapped into a Eq. (17):  noninsulating boundary condition of Eq. (33) with a corresponding moving boundary x = ) # const with speed  where, as previously mentioned, u = 6 (x,t) > 0.  (e) Since Eq. (1) admits an infinite sequence of LB transformations if and only if C(u) satisfies Eq. (15) with associated mapping (34) whereas Eq. (4) admits an infinite sequence of LB transformations, there is no point transformation of the form x = K(x,t,u), 7= L (x,t,u),  6. CONCLUDING REMARKS  (a) From invariance under the LB transformations \V' n],i= 1,2,—Ahere exist similarity solutions of Eq. (17), i.e., u = 8(x,t\n), whose similarity forms satisfy c*C/ ( " = 0,  U<"'+  (52)  A=I where |c,,c,,...,c„ ) are arbitrary constants, n = 1,2,—. For example, for n = 1, Eq. (52) leads to the similarity form u = 0(x,f,i)  + b(!)) 2 + c ( / ) ] " 2 ,  =  U' = u,,  U b =xu,  V = B.  ACKNOWLEDGMENTS  It is a pleasure to thank Professor C.E. Wulfman for stimulating discussions. We thank the National Research Council of Canada for financial support.  (53)  where (o(r), b (t), c(r) | are arbitrary. Substituting Eq. (53) intoEq. (17) wefind that Eq. (53) solves Eq. (17) if and only if a = a, b = /3. and c = ye2°', where \aJ3,y\ are arbitrary' constants. This solution is not contained in the class of similarity solutions of Eq. (17) obtained from invariance under a four-parameter point Lie group. 7 8 (b) The infinitesimal transformations (5) of the fourparameter point group of Eq. (17) are given by V" = u + xux,  u = M(x,t,u), relating solutions of Eq. (1) and those of Eq. (4).  + 2tu,,  (54)  'J.G. Berryman and C.J. Holland, Phys. Rev. Lett. 40, 1720 (1978). -P. Baeri, S.U. Campisano, G . Foti, and E. Rimini, J. Appl. Phys. 50, 788 (1979). 'Y.S. Touloukian, P. W . Powell, C.Y. Ho. and P.G. Klemens. Thermophysical Properties of Matter (Plenum, N e w York, 1970). Vol. 1. H . Fujita, Text. Res. J. 22, 757, 823 (1952). •R H Peters, in Diffusion in Polymers, edited by J. Crank and G.S. Park (Academic. N e w York. 1968); R. McGregor, R.H. Peters, and J.H. Petropolous, Trans. Faraday Soc. 58, 1054 (1962). "J. Crank, The Mathematics of Diffusion (Clarendon, Oxford. 1956). 'G.W. Bluman, Ph.D. Thesis, California Institute of Technology, 1967. "G.W. Bluman and J.D. Cole, Similarity Methods for Differential Equations (Springer, N e w York, 1974).  4  Under the mapping (34), these are transformed, respectively, to corresponding infinitesimals of invariant point group °R.L. Anderson, S. Kumei, and C.E. Wulfman, Rev. Mex. Fis. 21, 1, 35 transformations of Eq. (33): (1972); Phys. Rev. Lett 2«, 988 (1972). I0 H . Steudel, Ann. Physik 32, 205 (1975). U" = a, U h = xu- t + itUj, " N . Ibragimov and R.L. Anderson, J. Math, Anal. Appl. J9. 145 (1977). ,! d E . Noether, Nachr. Akad. Wiss. Goettingen Math. Phys. K1.2 234 V' = 0, U = W=u }. (55) (1918). " H . Steudel, Ann. Physik 32, 445 (1975) Conversely, the mapping (34) transforms the six-parameter "S. Kumei, J. Math. Phys. 18, 256 (1977). point Lie group of Eq. (33) as follows: The three-parameter "S. Kumei. 3. Math. Phys. 16, 2461 (1975). subgroup of infinitesimals given by Eq. (55) transforms to '"P. Olver, J. Math. Phys. 18, 1212 (1977). I7 \U",V'\U d \ given by Eqs. (54) and U = transforms to A . A . Fokas, Ph.D. Thesis, California Institute of Technology. 1979. Chap. V. U = 0; the remaining infinitesimal point group transformar Backiurtd Transformations, edited by R,M. Miura (Springer. N e w York. tions U'' = xu +2 Tu; and U = + \7)u + xiut + Pu,1976). are mapped, respectively, into infinitesimals which depend '"L.v. Ovsjannikov. Dokl. Akad. Nauk SSSR 125, 492 (1959). : on (;t,r.u.u|! and integrals of u. "S. Lie, Arch. Math. 6, 328 (1881). *'An infinitesimal L B transformation (c) Generally speaking, the action of a recursion operax' — x + €i + 0(r 1, tor 9 ' on any infinitesimal invariance transformation Uof t ' = / tT •+ 0(r'l. the form (5) (whether of point group or LB type) yields a new u' = u + f/i + O (r). acts on a surface F(x.t.u) =-- 0 in the same manner as infinitesimal transformation U' = .9' Vif 'J U # 0 . For Eq. x" = * (17). we can show that = 3 V = 3 V = 0. <• = t. (d)The heat equation is a special limiting case of Eq. (3) «• = « + <l' + 0(r). obtained by setting a — b: and then observing where I' — ft — £u. — tu,. !: 2 G . Rosen, Phys. Rev. B 19, 2398 (1979). After submitting this paper we lim h . r b ~(u + b)'- = 1. As one might expect if a = b , for discovered the above reference through Nonlinear Science Abstracts. the corresponding recursion operator 3 , limA . a £/' Here Rosen discovered transformation (35) and worked out some exam= d/dx. and the mapping formulas reduce to identity ples. W e are also grateful to the referee for bringing the above paper to our : attention. mappings, '  Invariance transformations, invariance group transformations, and invariance groups of the sine-Gordon equations* Sukeyuki Kumei Department of Physics, University of the Pacific, Stockton, California (Received 5 May 1975)  95204  We investigate a structure of continuous invariance transformations connected to the identity transformation. The transformations considered do not necessarily form a group. We clarify the relationship between the infinitesimal invariance transformation and the finite invariance transformation by showing explicitly how the infinitesimal transformations are woven into the finite one. The analysis leads to a new method of finding generators of the invariance group transformation. The results are useful in the study of symmetry properties, or group theoretic structure, of differential equations. We use the results in studying the group properties of the sine-Gordon equation u x t = sin u, and indicate that the equation is invariant under an infinite number of one-parameter groups; the groups obtained are of a more general' type than that dealt with by Lie. These findings are used to prove the group theoretic origin of the wellknown conservation taws associated with the sine-Gordon equation.  INTRODUCTION The discovery of the puzzling behavior of nonlinear wave "solitons" in various fields of applied science has triggered extensive study of nonlinear dispersive w a v e s . 1 One of the basic properties of the system which admits a soliton appears to be the possession of an infinite number of conserved quantities. As has been shown by L a x , 2 the existence of such conserved quantities is closely related to the soliton behavior of the waves. In spite of their importance in elucidating the nature of nonlinear waves, it seems that no one as yet has obtained a clear understanding of the origin of such conserved quantities. 3 It is well known that both in classical and quantum mechanics the conservation law reflects the existence of symmetry in the system. In classical mechanics, Noether's theorem associates one conserved quantity with each invariance group of the action integral. In quantum mechanics, we can associate one conserved quantity Q, which satisfies the equation [y, H) + icQ/dl = 0, with each invariance group of the time-dependent Schroedinger equation. 4 From these experiences, it i s natural to wonder whether there exists an invariance group associated with each conservation law of nonlinear waves. In the present and in future communications, we will investigate this question by applying Lie's infinitesimal analysis 5 and its generalization 5 " to the differential equations governing the waves. In this paper, we approach the question by studying the group theoretic aspect of continuous invariance transformations, which has been proved useful in systematically deriving a series of conservation laws. In Sec. n, we analyze continuous invariance transformations (not necessarily a group transformation) connected to thf identity transformation, to clarify the relationship between local and global invariance transformations. The results will be used in Sec. Ill to elucidate the group theoretic aspect of continuous invariance transformations of differential equations. In Sec. IV, we apply the generalization of Lie's theory to find some invariance groups of the sine-Gordon equation, 2461  Journal of Mathematical Physics. Vol. 16, No. 12, December 1975  u,, =sinu. In Sec. V, by using the result of Sec. in, we develop a new method of finding generators of an invariance group of differential equations. The method will be used, with the aid of the Backlund transformation, to show that the sine-Gordon equation is invariant under an infinite number of one-parameter groups. In Sec. VI, we investigate a relation between these groups and a series of conservation laws of the sine-Gordon equation.  I. PRELIMINARY We consider a partial differential equation of the form F(z', u,ujt « , „ • • • ) = 0 , l  (1)  2  where z' = (z ,z z"), us = 0 , « , . . . , 5„«), etc. Let's suppose that there exists a solution u = u(z', a), which depends on a parameter a continuously. Assuming that it i s analytic near o = 0, we expand the solution in a Taylor series in a, u=££u>(z<),  «*={0o)M«-  (2)  Putting this solution into equation (1), we obtain a sequence of partial differential equations which will successively determine a possible form of the u*. In particular , the first term u° must be a solution of equation (1). If the equation is linear, all the u"s must also satisfy the same equation. In the case of nonlinear differential equations, however, all the equations are different. First, the differential equation for ul becomes homogeneous linear and involves the first solution u°; we then obtain a nonhomogeneous linear equation for the u*, k > 1, which has the same homogeneous part as the u 1 ; the nonhomogeneous term depends upon the u°, «',..., and their derivatives. By a deductive argument, we expect that if only the nonhomogeneous solution is taken for u 2 , u'..., u*~\ the nonhomogeneous solution for a* will have a strong functional dependence on the u°. We consider the sine—Gordon equation uIt - sinu = 0 as an example. The equation for u°, u \ and u 2 i s found to be Copyright © 1975 American Institute of Physics  2461  u'cosu"=0,  uj, - stn«°=0, (4, ul, -u ! cosu° = - (u'l'slnu 0 .  We analyze the effect of this transformation on an analytic function fix') defined in R". Expanding fix') in a Taylor series In a , we obtain  It is surprising that we can find many solutions for u 1 and u 2 which can be expressed as simple functions of u" and its derivatives; a few examples are  u' = u°  and  Ax')= t ^ <*>,(*'), *, = {(3.)*/(5')}»o = A,/(x'),  u2 = u°,, (3a—d)  = + and 2 +3 2 " = »L,,« K) "°„I  where  i  •A„= 1 and  The existence of such solutions is directly connected to the origin of the infinite number of conservation laws, and the study of the origin of such solutions will provide a key in understanding the origin of the conservation laws. We ask how a nonhomogeneous solution for u* will depend on the u° if we take only the nonhomogeneous solutions for a 2 , . . . , u*"1; this problem requires a careful analysis of invariance transformations. II. RELATION BETWEEN AN INFINITESIMAL AND A FINITE INVARIANCE TRANSFORMATION  where ps and q : are the integers satisfying the conditions ppjq^k,  for »<;, 1 « q r  Here, we apply the summation convention with respect to the indices rj. The choice of the sets ipl ps) and satisfying the conditions is not unique, lt and the sum in (9) is to be taken with respect to each of such sets. Using the differential operator At, we can write the effect of the coordinate transformation on the function fix') as  (q ... ,q,)  We have considered an example in which one solution u is continuously connected to another solution u° through a parameter a. This may be considered as a continuous transformation of to u; it is a special case of the continuous invariance transformation which is connected to the identity transformation.  /(*')= (l+t £A^/ix') = Ti a(10) )fix l). Tia)x l = x' + a£j +  We note that the definition we started with.  We consider a set of transformations of the coordil nates of the n-dimensional vector space which analytically depends on the parameter a , and becomes the identity transformation for a = 0:  R"(x , ...,*")  +• • •=  recovers  Now we suppose that the continuous transformation T(a) leaves the equation Fix') = 0 defined in Rm invariant in the sense defined above. Then, the following statement will be obvious:  (4) x'-x'^X'ix'.a), x' = X'(x',0).  We also consier an equation f(*') = Fix 1,l• • • ,x") = 0 which is defined in the subspace ,*") of The equation F(x') = 0 defines a manifold S, or hypersurface in Rm. We define the invariance transformation in the following way;  R"(x ,...  Transformation (4) i s a continuous invariance transformation of the equation F{x i) = 0, if the condition F(X'(x',a)) = 0 i s satisfied for the continuous values of a on the manifold S defined by F(x') = 0. Geometrically, this implies that an invariance transformation carries a point on S into another point on S. We first investigate this invariance condition in detail, and will come back to the invariance transformation of the differential equation in the next section.  R".The transformation T(a) is a continuous invariance transformation of the equation Fix') = 0, if ... and>only If A„F(x') = 0 on the manifold S defined by Fix>) = 0. Although this provides the condition for a transformation to be an invariance transformation, it is very difficult to get any clear view of the structure of the transformation unless a considerable simplification of expression (9) is made; it is crucial to oberve that we can re-express (9) as  rfc  07,)'.  .(11)  where we take the same rule of summation as for (9). The remarkable feature of this expression is the fact that Under the condition we have imposed 1 on the transall in the V k's are first-order differential operators. We formation, we can expand in a Taylor series write down the first four generators in this form: a by :  X'ix , a)  x'=X'(x>,a) = x' + t  A,= U A 2 = (£/ 1)2 +1^, A 3=(C/ 1)3 + 3C/ 1i72 + r' Ci = {0„m*',lt«)}„.„• A t= (UJ* +6(V,) 2'U 1 + 3(V 2)2 + + U t, ' (12a)—(12d (5) where  Defining the differential operator U t by  U^pt-'d, , we can write (5) as  (6)  T/l=C/l=?<a(, V,= U,-SU{.,8, V,= U, + {-3( +2{J ?;t(?1>tyl + 2{i?;5f,J3„  +35k;. +H k'Ai -  .{i. i€f.»  dimensional space ^'"'(x 1 ,.. .,«*"') with k<m, assume the following:  and  The coordinate x* 16 determined by the relation - 12?!«!?!. ,t!. „ - St k {«»,«»'. , • (13a)—(x* 13d)= u(x'), and the coordinates At**1  The importance of the decomposition into this form will be recognized if we remember the basic lemma used in the theory of continuous group transformations:  If two first-order differential operators Vc = ^ d , and = ^ J5 j ^ t = 1,2 n, satisfy the conditions Ujfix') = 0 , j = a,b, on the manifold defined by /(*') = 0, then we have U aU tf(x') = 0 on the same manifold. Successive applications of the lemma to the invariance condition (A), lead to the conclusion that all the operators Uk in the expression (11) must satisfy the condition: U,F{x i) =0  on the manifold S defined by Fix') = 0. (B)  This allows us to draw the following conclusion: All the A„'s are constructed from first-order differential operators Q, which satisfy the condition yf(x') = 0 on F(x') = 0.  (C)  In Lie's theory of group transformations, the operator which satisfies condition (B) is called a generator of the invariance group. We suppose that the largest invariance group of the equation .F(x') = 0 is an r-parameter group with generators Q t . Then, all the operators which satisfy condition (B) can be written as Uk = t a ' t Q t .  (14)  i.l  In particular, if we let aj = 0 for fcs 2, we obtain A, = (V1)* from (11), and the operator r(ar) in (10) reduces to an exponential operator, r ( a ) = E £([/,)* =  fc>0 * '  (15)  Result (C) is significant in studying the structure of invariance transformations because it clarifies the constraints on and arbitrariness of an invariance transformation. The vital fact is that if we have a complete set of generators of the invariance group of the equation F(x') = 0, then any continuous invariance transformation connected to the identity transformation can be constructed from these generators. Now, the problem is how to find such generators for a given equation F(x') = 0. The basic idea of deriving the generators was established by Lie, and we will illustrate it briefly after the discussion of differential equations.  III. INVARIANCE TRANSFORMATIONS OF DIFFERENTIAL EQUATIONS We have considered a set of coordinate transformations in R" which leave the equation F(x') = 0 defined in m R invariant. We now introduce some functional relations among the coordinates, which are compatible with the equation F(x') = 0; such relations will restrict further the domain of manifold S. We consider a function u(x') defined in the {k - 1 ) -  x"  determined as the derivatives of u(x') with re. . spect to the coordinates x 1, For u inl stance, s t 1 x^'^a.aju,  are  x*" = B u, ... ,x ' = d . u, *J'=3 151u,  We suppose that R" is chosen in such a way that if it contains a coordinate corresponding to a jth derivative, then coordinates for all the other jth derivatives also appear in R". The condition we have imposed are compatible With the equation F(x') = 0 only if the function u(x') i s a solution of the equation F(x') = 0, interpreted as a partial differential equation by considering x"s as derivatives defined by (16). Each solution will define a submanifold 3 of the manifold S, called the solution surface. Now, we consider a continuous coordinate transformation (4) under which a manifold satisfying condition (16) is always mapped onto a manifold which also satisfies condition (16), with x* = u ( x ' , a ) . In analyzing such transformations, it is convenient to introduce the following definitions: Basic coordinates and jth order coordinates We call the coordinates x 1 , . . . basic coordinates; and the coordinates corresponding to the jth order derivatives, jth order coordinates. For instance, in (16), x** 1 ,... . x 2 ' " 1 are first-order, and x a \ x 2 " 1 are second-order. Basic space and jth extended space We call the vector space ( * ' , . . . , x'), jth extended space if it consists of all and only the basic coordinates and a complete set of the first through the jth order coordinates. In particular, we call the 0th extended space (x 1 x k), the basic space. Basic transformation We call the transformation of a set of basic coordinates, the basic transformation. Basic operator and jth extended operator We call the operator of the transformation in the jth extended space the jth extended operator. The 0th extended operator will be called the basic operator. It i s clear that under condition (16), the transformation of the basic coordinates will determine the transformation of the rest of the coordinates. In particular, If a basic operator is given, we can determine all the extended operators. Now, we require that such transformation leaves the equation f ( x ^ = 0 invariant. The geometrical meaning of the invariance transformation i s more important; the invariance transformation maps one solution surface S to another solution surface 5' (or onto itself), both of which are on S. A discovery of such a transformation will lead to a new solution of the differential equation. The transformation studied most extensively Is the group transformation. Lie considered an invariance group transformation of the form  x'-x^e^x'-x' x=(x*  + a('(x) + •••, i = l,...,k,  x*),  (17)  in which infinitesimal terms of the basic transformation depend only on the basic coordinates. It is important to note that, under such assumptions, a finite transformation of the coordinate x* does not involve any coordinate whose order is higher than the order of xp. This guarantees that the jth extended space is closed under the transformation. The existence of such a closed space enables us to elegantly construct a finite group transformation, via the method of characteristics, from a generator of the group., Anderson, Kumei, and Wulfman, however, found that there exist invariance groups of time-dependent Schrodinger equations which are not of Lie's form." They generalized Lie's theory by allowing infinitesimal terms of the basic coordinates to depend on the coordinates of higher order:  x' — x' = e°°x' =x' + ' (x) + • • •, i = l,... ,k, x=(x\...,x'), ' (18) Here, the order of the coordinates in is not restricted, and coordinates of any order may appear. 6 We note, however, that we no longer have any closed finite-dimensional space under such a group transformation. 7 Although this does not cause any problem in finding generators of invariance transformations, we can no longer apply the method of characteristics in finding a finite transformation. This generalization, however, is absolutely necessary to uncover all the invariance groups inherent in the differential equations. 6 Before we show that the sine—Gordon equation admits such invariance groups, we answer the question raised at the end of Sec. I. To put the problem into our present language, we rewrite (1) as  F(x')-0  with x' = z',  t= l  x"-' = u, x"-2 = « , , - - - ,  (1')  and (2) as x'=x',  IV. SOME INVARIANCE GROUP TRANSFORMATIONS OF SINE-GORDON EQUATION We now go back to the analysis of the solutions (3a)—(3d). According to result (D), these solutions clearly indicate the existence of invariance groups, or symmetries, of the sine—Gordon equation. We will systematically determine generators of the invariance group transformations and will reveal new symmetries of the equation. We first specialize the general formulation given above to a case in which we have three basic coordinates x 1 , x 2 , and x 3 , and F(x') is chosen as F(x\ x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x») = x 7 - sinx 3 . As stated in (16), we establish the following constraints on the coordinates: x 3 = it(x',x 2 ), 7  x =  M12,  X"=«I  2 2  X = K22  X  ,  X 1 5 ~ "1122,  =  W  ~ <'11112, X  9  X = H  2 2 2  XL6 = «  -19 — ,, X  X4 = M1, X s —u2,  8  ,  1 2 2 2  X ,  U 1 =  ,  XS = H 11 ,  X'° =  MIIII,  I/U2, X  — U1112,  X 1 —U 2222, X ~ I'm 11 ,  V20 — „ V21 — u — "iu22, X — «11222,  r22 — II —"12222,  X 23 = « 2 2 2 2 2 , (19) where subscripts 1 and 2 indicate the derivatives with respect to x' and x 2 . We now consider a transformation, of the generalized form (18), in which x' and x 2 are unchanged and the infinitesimal transformation of .v3 depends on x 3 and the first through the third-order coordinates:  x' = x', x2 = x2,  x 3 = x 3 + o?3(x3,x4,x5,x',x8,x9,x'2). (20) We note that the inclusion of the coordinates x", x u\ and x " is redundant because we can replace them by the coordinates in £ 3 after we have introduced the condition F = 0. The infinitesimal transformation, induced by (20), of the coordinate corresponding to the derivative (S 1 ) m (d 2 )"u is calculated a s 9  x W + aO.Ha^^x' + a?'.  t= l  ?"=*•"(x',a)=x""  +  <2<)  From a transformational viewpoint, the statement that ii is a solution of equation (1) is equivalent to saying that transformation (2') leaves equation (1') invariant. For such a transformation, as we have found, we can write (or u"=Atu" in the old notation). This leads to the conclusion, If a differential equation F(z', u, ut, u,„ • • -) = 0 admits a solution u(z', a) =£^=0 )u* (z') which depends analytically on a near a = 0 , then u* is always 0 i written as u"(z')=A llu (z ), where h° is a solution of the same equation and the operator Ak is constructed by (11) from the generators Q of invari- , . ance group transformations of F=0 by which the independent variables z 1 are unchanged. In particular, uJ = Qu°. Furthermore, if only the inhomogeneous solution is taken for every h* , k > 1, then «* = (Q)*«°, and a resulting solution is expressed as  h(z') = e'^u'U')•  (21)  Here, the partial derivative should be interpreted as  (a1r02)"53 = 01)"02)''  X «»(«(*',jr*),^1,*2)  «222(x',x2))  (22)  For instance 5  4  = 5 3 X 4 + ? 3 x « + ?|X7 + 5 3 X 9 + | 3 X " + ^ , 3 + ^ 2 X , s ,  (23)  where is the derivative of £ 3 with respect to the coordinate x' contained in | s , and should not be confused with the same notation used in Sec. n. As in Sec. II, we write the infinitesimal transformation in the jth extended space as  xi = (l + a4)x'i-i,  with {'=^ = 0. (24)  Now, we assume that the equation F = 0 is invariant under the group transformation whose infinitesimal form is given by (20); the condition is 4 i r = ? , - ? 3 c o s x 3 = 0 on x 7 - s i n x 3 = 0,  (25)  which is the partial differential equation for £ 3 . The equation QF = 0 will split into a set of partial differential equation because some of the coordinates which appear in the equation are independent from the coordinates in £ 3 (Appendix). By solving these equations, we find four independent solutions:  od although the completeness of the set of generators obtained is still not assured.  Hie idea of the method is to reverse the result in Sec. II. We found that the operator Q which satisfies a condition QF = 0 on F=0 is the building block of any t*=jr<, 5> = *» + *(* 4 ) 3 , (26a) invariance transformation connected to the identity transformation. By reversing this, we argue that if we C= d = * , 2 + i<*5>3(26b) have an invariance transformation connected to the „ identity transformation, then we can find at least one. Obviously, the test two solutions can be obtained from such operator. More precisely, we proceed in the folthe first two. by interchanging the roles of x' and x2. lowing way. The second extended operators associated with £ 3 and {{j are calculated from (21) and (24) as Suppose we have an invariance transformation of the equation F(x') = 0, Qa = .v < c 3 + A-6c4 + x , c 5 + * 9 t 6 + j - ' 0 c 7 + j - n c e , (27a)  Q t=ix° + i(x") }o 3  3  J W + E^A/tW + Eit?!, '= 1  + {x>3 + f (x 4 ) 2 x 6 }5< + {*»  + j ( . V ) y } s 5 + {x' e + 3jr-cV) z +4 ( x 4 ) V } a 6 4  0  + {a-" + 3,vVV + f Cv )V }5 7 + {x  20  4  7 2  + 3x (x )  + i(A- 4 )V'}£ 8 .  (27b)  We can easily check that they satisfy condition (25): thus, the sine-Gordon equation is invariant under the group transformations generated by these operators.  in which all the £ j are known. Using the result (11), we can write,  (> =A sx< =  „ 2 = § a .v = .v<  where all the Vk are first-order differential operators. From the first equation, we obtain  • (28a)  and  h| = W„)V = A  , s<=As> = {&>)' + U,}x<,  li=Aje> = {(t/,) 3 + 3 U & + t'3}*<, • • •,  This result explains the origin of the solutions (3a)—(3d); they are obtained from result (D): 3  (28b)  «> = §„ x' = x° + i(x<) 2  Feeding this into the second, we get which provides l/2=£UM<&i>2*'}]?( •  (31)  tel  Next we substitute these for the Z\ and U2 in the third equation to determine U 3. Continuing this process we can obtain a series of operators U t , all of which satisfy the invariance condition on We note that if the starting transformation (29) happens to form a group, then we only get Z\ and all the others are equivalently zero for the reason discussed in Sec. n. We may consider the starting transformation (29) as a generating function for generators of an invariance group. The upshot of the method is that only algebraic computations are involved in the process and a computer can be used, whereas the construction of the solutions of the determining equations by computer is very difficult. Obviously, this method can be used to find generators of an invariance group of a differential equation if the constraints (16) are taken into account. We apply the method to the sine-Gordon equation to find additional generators.  "QF=0  «l = (Q„)V = x24 + 3 (x 4 ) 2 *' 3 + |(x 4 )"x 6 (28d)  We note that we need the extended operators and Qb for calculating the second term u2. As we stated in Sec. HI, this is the general character of the generalized transformation (18) and we need the Km - l)th extended operator to calculate the nth term u" if the basic transformation contains the fc-th order coordinate. V. GENERATING FUNCTION FOR GENERATORS We have obtained four generators of the invariance group of the sine-Gordon equation by considering a generalized Lie transformation (18). However, if we had assumed a more general for we might have been able to produce more generators. It is unfortunate that we have no theory which tells us which coordinates we need in of (18) to obtain a complete set of generators, hence we must make some assumptions on the form of £'. In practice, it is not possible to retain too many coordinates in because the determining equations for become too huge to solve. Therefore, it is highly desirable to have another method for producing the generators, which does not require either such assumptions or the construction of the solutions of determining equations. Here, we provide one such meth-  form  (30) U 2x'= t,'2-(U 1)2x',  (28c)  and  + 9.vVx B + 3(x e ) 3 .  <29>  F—0."  We start with the well-known Backlund transformation of the sine-Gordon equation, 1 J 4 - X 4 = 2a sin£(* 3 + .r3),  a (if 5 + x s ) = 2 sinj(x 3 - .v3), (32)  with the convention established in (19). This transformation guarantees that if x 3 is a solution of the sine-Gordon equation then so is x 3 for a continuous value of a . A principal use of the Backlund transformation is to construct a new solution x 3 from a known solution A-3 by solving a set of first-order differential equations (32). We assume that the new solution A 3 , is an analytic  function of a in the neighborhood of a = 0 , and so are its derivatives. Then, it is clear from (32) that the transformation is connected to the Identity transformation; x 3 — x 3 as o — 0. The analyticity assumption allows us to expand the solution x 3 in the Taylor series in a near a = 0. Such an expansion is found In the paper by Scott et at.,1 and we rewrite their result: =  +  (33)  fc»l  with: t,\ = Zx\ £ | = 4x',  Is Invariant under an r-parameter group with the property: x' = r , ( a ) x ' = x ' for x = l , . . . , f c - l T  ,  (  a  )  =  e  = rt(a)/'(x'  5 3 = 48x 1 3 + 48(x 4 ) 2 x 8 ,  (  3  8  )  We suppose that transformation (38) exists If la !<6. Here, 6 Is a positive number. Under such assumptions, x* represents a new solution of the equation and a corresponding flux, f = ( / , / 2 , . . . ,/""*) is written as /'=/'(x1)...,x*-I,x*  t» = 12* B + 2(x < )',  x  with  x')  x*"',x*  x').  (39)  The Implication of the new flux Is the same as the old 4 = 240x + 360(x ) x® + 600x (x ) +18 (x ) , 24 one, except that it is now for the new solution, However, 4 2 13 + 2880(x ) x its power series expansion in a tells us something new about the starting solution x*; because we have assumed +12 9 6 0 x 4 x V + 3840(xV + 1440(x 4 )V that the transformation T t(a) exists at least for some { 3 = 10080x 3 1 + 176 4 0 0 ( x 8 ) V +95 760x , (x 9 ) 2 range of la I, It acts as a generating function of fluxes; 4 8 13 4 2 18 a ' also + 141 120x x x + 25 200(x ) x (34a-g) each term of the expansion of (39) in a ' forms a divergent free flux. We state this as follows: + 63000(x 4 ) 3 (x 5 ) 2 +18900(x 4 ) 4 x 9 +450(x 4 ) 7 , • • If a differential equation F(x') = 0 admits an inwhere we have adopted the convention (19) and x 3 1 variance group with property (38), and if a flux M = iuuu'his specific Backlund transformation, f of the form (37) exists, then, for any polynomial (E) 1 2 the coordinates x and x are unchanged, i . e . , function G(Q l Qrj of the generators of the x ' = x l , if 2 = x 2 or = ^ = 0 for i » l . (35) group, the vector Gf forms a divergent free flux. 3  18  4 2  4  6 2  4 5  = 1440X  The transformations (33) and (35) form the basic transHere, we see two basic patterns for a series of diverformations, and they provide all the necessary informagent free fluxes to arise: one associated with a series tion to follow the above prescription to find V t . We list Qjf, i l and one associated with a series (Q,)"t, the results up to £/,: n = 1,2, •• • . It will be reasonable to say In general, that 4 4 3 the former is more fundamental than the latter because &2=&,= = 0, £/1 = 2x 3„ s {4x» + 2(x ) }3s,the series of the second type can be mechanically constructed If Q t Is known, although the reverse is not &, = {48x l s + 120(x 4 )V + 120x 4 (x 8 ) 2 + 18(x 4 )'}3 a , 31 possible. One, however, should not think that the fluxes 2 s 4 2 + 25 200(x") x +15 120x (x") of the second type are t r i v i a l . 1 0 + 20 160x 4 x 8 x 1 3 + 5040(x 4 ) 2 x 1 8 +12 600(x 4 ) 3 (x') 2 We now apply this analysis to the sine-Gordon equa+ 6300(x4)V+450(x4)7}33. (36a-e) tion, J = x 7 - sinx 3 . The equation can be put into the conservation form by multiplying by x ! ; Here, we have given the operators in the basic form; = 0 2 the operators in the extended form can be obtained from 3l/'+32/2 wi'h f=(/',/ ) = (i(x') 2 , cosx 3 ), (21) and (24). By continuing this process, we will be and the generators (36b)—(36e) can be used to derive able to find an infinite number of operators which satisfy new fluxes. We list a few of them, (using the notation the Invariance condition (25). We can associate one in-  =  t/,  r  Z/ =  U, = {1440X  variance group transformation of the sine-Gordon equation with each of these operators.  2  4  4 2 14  U -fi = 24{2x ' + 1 0 x x V + 5(x ) x  In this section, we use notation (16), hence x* represents a solution of the differential equation Fix 1) = 0. We consider an equation f(x') = 0 which can be put Into a conservation form: x*"1,x*,...,x'),  f , : / { = 2 x V , / f = - 2x 4 slnx 3 , f, :/J = {4x 14 + 6(x4)2xV.  fl = ~ {4x  VI. A SERIES OF CONSERVATION LAWS AND INVARIANCE GROUPS  S 3 , / ' = 0, / ' = / ' ( x '  t, = b,t)•  4  4  + 5x (x")2  + 10* xV°+^(x )V}x , / ; = - 2 4 { 2 x 1 0 + 5(x 4 ) 2 x» + 5x 4 (x 8 ) 2 + | ( x 4 ) ! } s l n x 3 ,  4 J 7  4  6 2  4  + {3(x ) x + 9 x x V } x  where the derivatives are to be taken by considering x* x' as functions of x 1 , . . . , x*"1. The vector f = ( / , / 2 , . . . ,/*"') establishes a divergent free flux in the space Ri~ L(x l x*"1) for each solution of the equation. Now, we assume that the equation F(x') = 0  + 2(x 4 ) 3 } slnx 3 ,  7  !  • / i , = ( ^ ) V ' = ISlx 5 .* 32 + 3(x 4 ) W (37)  8  5  14  s  + (x 1 4 ) 2  + 6xVxV3  + {9x (x ) + 9 x x V + | ( x 4 ) V } x 1 0  ( 4 0 a  + 9 x V x V + 9 ( x 4 ) 3 x V x 7 + |(x 4 ) 4 (x 7 ) 2 ] . Here, we have listed only the first component for f,,,. Among these fluxes, the first flux, tl is trivial because It Is the derivative of f with respect to x ' . 1 2 We analyze the known results from our viewpoint. Our  _d)  results are clearly different from the fluxes given in 5 <2! u 1 the paper by Scott el al." Their results, however, can be obtained by taking a linear combination of fluxes ACKNOWLEDGMENTS with the form (£/,)"(!/,,)' • • • In fact, by using (11) r and (36a)-(36e), we find that A,t and AJ recover their I wish to express my grateful acknowledgment to results. For instance, Professor Carl E. Wulfman for helpful discussions, AJ 1 = {(T/,)3 + 37/,% + U,}/ 1 = 6 { 2 x V 4 + 4 x V ° + (x*) 2 xV} his continuing encouragement and many valuable comments on the manuscript. I also thank the Research 3 Cooperation and the Physics Department of U . O . P . ' 2 111j L2 llz 1 2 12 for the support of this research. where we interpret TJ as a generator extended to a  T(a) = e" ' e  e" '  e""- *4®* ...  (U Yt.  = 6{2u «  + iu u  + (u ) u u },  l  necessary order. Now, we ask which fluxes are most basic among these. Although this question is very important in analyzing the nature of conservation laws in general, the answer depends on the measure one uses. However, as we have indicated above, the hierarchy becomes quite clear within the framework of group theory, we classify fluxes into two categories: (1) Basic fluxes: f, Q,f,  i= l  and (2) Associated fluxes: W^FifQ, )"*• • • (Qif)"'t with it = 1 , . . . , r, and n, +  + • • • + n, > 1,  and we use the basic fluxes to characterize the conservation law associated with a solution. The remarkable feature of the sine-Gordon equation is that it p o s s e s s e s a series of basic fluxes. SUMMARY  APPENDIX: DETERMINING EQUATIONS OP GENERATORS Although our transformation is more general than that of Lie, the basic idea for obtaining the differential equations (determining equations) for is the same as Lie's, and for a detailed discussion of the Lie method we refer the reader to the book by Ovsjannikov" or the book by Bluman and C o l e . 5 c U s i n g / f o r £ 3 , the detetermining equations for our problem are the following:  /». V^ +A +/., +/.. 1**° +/„. i^1 +/». i2*16=o, /s.o*5 +US +/,.»*a +/„,»*" = 0, 7 19 A* +A*" + V +A* + fi, (A.+/„,,*7 +/5.+/.,,*10 +/,.,*12 6, 8,12  +/,.,*»)=o, with supplementary conditions:  To conclude this paper, we briefly summarize the results obtained in the present study. In Sec. n , we studied a structural aspect of continuous invariance transformations connected to the identity transformation, and we stated the explicit relation between a continuous invariance transformation and a continuous invariance group transformation [(A), (11), (C)]. In Sec. HI, we used the result of Sec. n to analyze invariance properties of differential equations and we uncovered the group theoretic structure, inherent in any solution which depends on a continuous parameter [(D)], In Sec. V, a new method was given for obtaining generator of an invariance group and it was used to find a series of new generators of an invariance group of the sineGordon Eq. [(36b)—(36e)]. In Sec. VI, we gave a group theoretic criteria for the existence of a series of conservation laws associated with solutions of a differential equation [(E)], and this was used to provide a group theoretic explanation of a series of conservation laws of the sine-Gordon equation. The results (40a)—(40d) explicitly indicate that there exist conservation laws whose existence is inexplicable within the Lie's framework of group theory, but still can be explained by group theory if the generalized theory (Ref. 5d) is used. In the next papers, we will show that the conservation laws of the Korleweg—deVries equation and the cubic Schroedinger equation are also related to invariance groups of the generalized Lie t y p e . 1 3  • T h i s work was supported by a R e s e a r c h Cooperation G r a n t . ' A . C . Scott, F . Y . F . Chu, a n d D . W . McLaughlin, P r o c . IEEE 61, 1444 (1973) (review a r t i c l e ) . l P . D. L a x , Comm. P u r e Appl. Math. 21, 467 (1968). s I t has been suspected that s o m e transformation property of the differential equation governing the wave motion is r e s p o n sible for the existence of a s e r i e s of conservation laws. In fact, the r e s t r i c t e d Backlund transformations (R. B. T.) have provided a s y s t e m a t i c way of deriving a s e r i e s of c o n s e r v a tion laws. However, the derivation involves a p r o c e s s of power s e r i e s expansion of a solution with r e s p e c t to some p a r a m e t e r . Such a method only exemplifies the existence of a s e r i e s , but explain of individual c o n s e r v a tion law. On the discussion of R. B. T. in the theory of s o l i t o n s , we r e f e r to (a) G. L. L a m b , Rev. Mod. Phys. 43, 99 (1971); (b) D. W. McLaughlin and A . C . Scott, J . Math. Phys. 14, 1817 (1973); (c) H.D. Wahjquist and F . a E s t a b r o o k , P h y s . Rev. Lett. 31, 1386 (1973).  defined by (11), (12a—d) has been found to be the power series expansion, in or, of the expression  ' R . L . Anderson, S. Kumei, and C . E . Wulfman, Rev. Mex. F i s . 21, 1 , 3 5 (1972); J . Math. P h y s . 14, 1527 (1973).  x 7 = sinx 3 , x  14  x10=x4cosx3, 3  4 2  x" =** cos* 3 ,  3  = x " coax - (x ) sinx ,  x " = x*cosx' - (x 3 ) 2 sinx 3 , x " = x ° cosx 3 - 3x 4 x f sinx 3 - (x 4 ) 3 cosx 3 , x 2 2 = x " cosx 3 - SxV sinx 3 - (x 5 ) 3 cosx 3 , where f, = 3,f and  , = d fij.  does not  the origin  We add in proof the following papers on the Backlund transformations' . (10) Math. P hA„ y s . 15, 2157 (1974); M. Wadati, H. Sanuki, Note added in proof: The transformationJand with K. Konno, P r o g r . T h e o r . P h y s . (Kyoto) 53, 419 (1975).  *For L i e ' s work and its l a t e r development, we refer the r e a d e r to (a) S. L i e , Transformationgruppen (Chelsea, New York, 1970), 3 Vols. , R e p r i n t s of 1888, 1890, and 1893 eds. , S. Lie, Differentialgreichungev (Chelsea, New York, 1967), r e p r i n t of 1891 e d . , S. L i e , Continuierliche Gruppen (Chelsea, New Y o r k , 1967), r e p r i n t of 1893 e d . (b) L . V . Ovsjannikov, Group theory of differential equations (Siberian Sec. Acad, of S c i . , Novosibirsk, USSR, 1962). This book has been translated into English by G . W . Bluman, Department of M a t h e m a t i c s , University of British Columbia (unpublished). L . V . Ovsjannikov. Some problems arising in group analysis of differential equations (Proceeding Conference on Symmetry, Similarity and Group T h e o r e t i c Methods in Mechanics, edited by P . G. Glockner and M . C . Singh (University of Calgary P r e s s , Canada, 1974). (c) G.W. Bluman and J . D . Cole, J . Math. Mech. 18, 1025 (1969). G.W. Bluman and J . D . Cole, Similarity Methods for Differential Equation (Springer, New York, 1974). (d) R . L . Anderson, S. Kumei and C . E . Wulfman, P h y s . Rev. Lett. 28, 988 0972), ®We note that the well-known contact transformations of o r d i n a r y differential equations, which w e r e extensively studied by L i e , a r e a realization of the derivative-dependent t r a n s formations in which only the f i r s t - o r d e r derivative a p p e a r s , 'if the equation is an ordinary differential equation, it is always possible to find a closed s p a c e . 8  S e v e r a l y e a r s ago, Professor G . M . Lamb kindly r a i s e d the question of the relation between this generalization and the Backlund transformation, which depends on f i r s t - o r d e r d e r i v a t i v e s . The basic difference is the fact that the Backlund  transformation is not a group transformation in g e n e r a l , whereas our generalization allows us to construct a group transformation. We should consider that a Lie type t r a n s formation and the Backlund transformation a r e complement a r y in the sense that neither of them s u b s u m e s the other. L i e ' s infinitesimal approach, however, will be s u p e r i o r in the s t r u c t u r a l analysis of continuous invariance transformations. *The general formula of the expression of the extended o p e r a t o r will be found in the paper by R. L. Anderson and S. Davison, J . Math. Anal. Appl. 48, 301 (1974). 10 We define " t r i v i a l " flux in the following way. We consider .a set S { f , , f 2 , . . • ,fj} which c o n s i s t s of divergence free fluxes, and their derivatives of any o r d e r . We note that the t derivatives a r e a l s o divergence free. Now, a flux f is said to be trivial with r e s p e c t to the set S, if f can be e x p r e s s e d as a linear combination of the m e m b e r s of the set S. In this s e n s e , the flux <?*f is, in general, nontrivial with r e s p e c t to the set S{f,Q(,Q :t t... .Q^'f}. For instance, the flux f 3 | , of (40) is nontrivial with r e s p e c t to the set S{f,Ujf}. " E q . VI. B. 7 in Ref. 1. l2 T h i s is due to the special c h a r a c t e r of the operator the operation of Uj on the variable x{, i >2, is equivalent to the differentiation of the function u with r e s p e c t to xA. For ins t a n c e , l\x z-xit (C/|) 2 x 3 =ar 6 and I/Jx 5 = x" a r e the t r a n s f o r m a tions w — u , , u~u|i and t/ 2 — « 1 2 . Because of this p r o p e r t y , the fluxes obtained from (i/jJ'f a r e all t r i v i a l . 13 S . Kumei, "Group theoretic studies of conservation laws of nonlinear d i s p e r s i v e waves" (II, III, IV) (submitted for publication).  f,..., f|  Group theoretic aspects of conservation laws of nonlinear dispersive waves: KdV type equations and nonlinear Schrodinger equations* Sukeyuki Kumei Department of Physics, University of the Pacific. Stockton, California  95204  (Received 5 January 1976; revised manuscript received 3 September 1976) * ' Group theoretic properties of nonlinear time evolution equations have been studied from the standpoint of a generalized Lie transformation. It has been found that with each constant of motion of the KdV + a(/)/,+/, = 0 and of the coupled nonlinear Schrodinger equation / „ + Q(J,g)+  equation  Su + a (&<f) — 'Si ~ 0  o n e  type  if, = 0,  invariance group of the equations is always associated. The well-known series of  constants of motion of the K d V equation and the cubic Schrodinger equation will be recovered from the invariance groups of the equations. The doublet solution of the K d V equation will be characterized as the invariant solution of one of the groups. In a more general context, it will be shown that the well-known equation of quantum mechanics  (d/dtX U)  = <[iH, I/] + dt//9/> can be generalized to a class of  nonlinear time evolution equations and that if V is a generator of an invariance group of the equation then (d/dtK  C/> = 0 . The class includes equations such as the K d V , the cubic Schrodinger, and the Hirota  equations.  INTRODUCTION In this paper, we study group theoretic aspects of time evolution equations of nonlinear waves, particularly of the Korteweg—de Vries (KdV) equationf,„+ff x+f, = 0 and of the cubic Schrodinger equation  f„+fj*  + ¥t = 0.  Some time ago, Anderson, Kumei, and Wulfman proposed a generalization 1 of the Lie-Ovsjannikov 2 " 4 theory of invariance groups of differential equations, and applied it to a number of quantum mechanical systems to systematically study dynamical groups. 5 Recently it has been shown by Ibragimov and Anderson 6 that this generalized transformation is an infinite dimensional contact transformation. It has been shown in the preceding paper' that the sine-Gordon equation f xt - s i n / = 0 admits an infinite number of one-parameter invariance groups of this new type, with each of which one can associate a series of conservation laws. Although the generalization appears to broaden the usefulness of group theoretic analysis of differential equations, particularly of nonlinear ones, the physical implications of the new type of symmetry are still unclear in many respects. The aim of the present paper i s to investigate some of the well studied equations of nonlinear waves 8 from the standpoint of the generalized theory, and to gain a clearer insight into the physical significance of the presence of the new kind of symmetry. It will be shown that some of the fundamental properties of the KdV and the cubic Schrodinger equations are the direct results of the existence of new groups. In Sec. I, welbriefly review a few basic ideas of infinitesimal invariance transformations to fix notations.  number of invariance groups. (2) The doublet solution, as well as the singlet solution, of the KdV equation is the invariant solution (or generalized similarity solution) of one of the groups. In Sec. Ill, we prove that with each conservation law of the coupled nonlinear Schrodinger equation / „ + «(/, ir) + if, = 0, g„ + a(g, /) - ig, = 0 one can associate one invariance group. The constants of motion of the cubic Schrodinger equation due to Zakharov and Shabat8 will be recovered from the invariance group of the equation. In Sec. IV, we investigate some general properties of generators of invariance groups of time evolution equations f ilt••')+/, = 0. It will be shown that (1) A generator U of an invariance group of H + / , = 0 always satisfies the relation f] + = 0, where ff is a Lie operator associated with H\ (2) For a class of nonlinear time evolution equations, the equation I/] can be generalized; in particular, if is a generator, then = 0.  H(t,x',f, ft, [Hi  (d/dt)(V) = ([//, + ?U/ct> U (d/dt)(V}  I. INFINITESIMAL INVARIANCE TRANSFORMATIONS  We denote ?n-dimensional real and complex vector space by Rm and C", respectively and we consider the following infinite direct sum of the spaces; by denoting c ' J " , 1 , > by C,  v=R"s-cececec'e---ecec'e---.  (d  0 0 1 1 It k The prime is to distinguish two spaces of the same dimensions. We denote the elements of C and C' by u and v, thus the elements of V are *  z = ix,u, «,K, J), ...,H,t), •••), x^R"' 1.  The components of «, v are written as «>t»j-«»,> where each index runs from 0 through N'°:  u = {u), v=(v), U = («„,«(,... ,U K), I) = (t y = («oo, u0/r,...,uin,u„i «„),  In Sec. n , we investigate group theoretic properties of the KdV equation and the related equations. The main results are: (1) With each constant of motion of the KdV type equation / „ , + x + / , = 0, one invariance groupV = (WoO> Vol 2 i s associated, hence the KdV equation admits an infinite  a(f)f  iV/dt  v  0Jf  v  JtO> vNi> " ' I  v  NK)I (3)  -  119  -  Now we consider an infinitesimal transformation in V  The operator U is then a generator of an invariance  and the components of TJ and £ are to be determined by the formula  where the asterisk represents a complex conjugation.  group. z=z+(Z, Z = (0,7], C,n, [,...,!),{, •••), (4) 11 > » We define ((-conjugation of a quantity A(z;c) =A(x,u,v,...,u,v;c) by where j) = ] i(z;c), £ = E(z;c), (5) A(z;c)*=A(x, v,u,... ,v, u;c*),  (6) Ah important subclass of Eq. (10) i s  Vi-'.^'iV"'."' ''.v-'r^v','' F'(z;c) = 0 with F = ( F ) , D tl...m =D,D, •••D m with ri * « v. v =f(x) , . «=/(*), D< = + M. + v,8 v) + (un d U j + v„d For +equation, • • • the generator U takes the form V j) this 2  l #  '  where  + («U—- S.j...m +  + - -' •  C)  In this paper, the summation rule will be assumed for repeated indices. In (5), c denotes a collection of all the real and complex numbers appearing in the expression of TJ or £. We write (4) compactly in the usual way as  (8)  Z=(l + eC/)z,  (16) In this paper, we consider the infinitesimal transformations of the type (4) which involves no transformation in x. This transformation, however, i s not as special as it might look. Let us consider an infinitesimal transformation of a more general t y p e 2 - 6  1 v i=z+tz, z=(i,1 i,t, 15,£,•••), e = I,...,i-),  with foa. + J3„) +  +  + f,3„ () + • • •+ (17,...,3  +  (17)  (9)  where  The operator U has the following property (see Appendix A for the proof): Lemma 1: If a function A(z) is twice differentiable with respect to all the variables, then ( D t U - UD t)A(z) = 0 for i = 0 , 1 A'. We consider a set of differential equations for functions /(*) and g(x),  (18) It can be proved 11 that if we know the transformations of type (4), then we can also obtain the more general type (17): Lemma  2: If (4) is an infinitesimal invariance trans-  formation of f, ij, F'(z;c) = 0, i = l,2, (10a) of (10), then for an arbitrary choice and £ subjected to the conditions TJ — £ = rj. i -V ', = £> «=/(*), v=g(x), '<={(*), y=g(x), ft = 1,2, (17) is also an invariance transforthe transformation F  1  (10b) where fix) and g(x) are functions of the (N+ l)*-tuple  mation of Eq. (10). Conversely, if (17) is an invariance  2  They, however, must be always interpreted as their infinite prolongation. Also, we use the following abbreviation:  transformation of (10), then so is (4) for TJ = FJ - L'U k k C = S-Vr,. f{x) = (/„, /, f„), g(x)=-(g 0,g g„), l In the following sections, we write the operators (9) (11) 1 1 and (16) as (19) fix) = (/oo fsn), g(x) = (g m, . . . ,gff.y), C/ = I)3. + £3„, U = TJ3„ + TJ*3V 2  w i t h / , . . . / = 3,i••• 3,/fix), # , . . . / = 3 , , - * ' 3 ^ W . C in (10a) represents a set of parameters (real or complex) appearing in the differential equation. Each solution of Eq. (10) defines a manifold in V which we call a solution manifold. It is well known 2 " 4 that a group transformation e°" maps a solution manifold of (10) into another (or the same) solution manifold if and only if  . ~UF  l  (z;c) | j = 0, i = l , 2 ,  vhere (• •) I j indicates to evaluate the quantity under he conditions  T,  [A(z)]„^„, = [A(n,t>)]/f and  / [A(u, v)\,dx = JA(u, v)dx. II. A GROUP THEORETIC ANALYSIS OF THE KdV EQUATION • (12) 12-16 The equation of our interest i s ut t +/0 = 0. The equation i s a particular case of (10) for which F 2 = 0, g= 0. In this section, we use t, x for x a n d write coordinates such as u t , » I 0 , • • • as u„ n I t , ' " . Similarly, we write RJ0, IJ10, •• • as TJ„ IJ,„ • • •. Thus, by  F 1 — Of D, r., tF< = 0, i = 1,2,fe = l,2  f  ». (13)  +ff  definition i), = D(t), r] lt =DIDtrj, etc. Also, because the equation involves a single real function, all the f's in the first section are to be ignored.  ear equation  A. A Lie algebra of an Invariance group of the KdV equation  where//„(/) = (a,HL/> etc. and 0"" = (d,r<pix, I). We  We write the equation as  F = u„x + uux + u,= 0, u=f(x,t), u^=_f x(x,l), «,=/,(*,/),•  (20)  We look lor an operator £/ = j)B„ which satisfies condition (12) for this equation. We assume the transformation to be a generalized Lie type 1 with i) = i}ix, t, it, ux, u„, « „ , , i<xxxx,  u x x x x x ) . T h e a b s e n c e in 7j of  coordinates corresponding to t derivatives may be justified for time evolution type equations in which the only I derivative contained is / , . The application of Lie's algorithm 3 ' 4 for finding generators leads to the following results:  lA = (tux- i)a„, (21)  4  £/ = K,a„, I/ = (*„, + ««,)a„, 5  u = (i  2i  + «»*** + v„ + W«  f + H,(f)  0 , + • • •+#.<„,(/) 0("> + 0 , = 0,  (24)  note that tj(/) of a generator of an invariance group of the equation H(f) + / , = 0 is a special realization of 0 , The effect of the transformation on the constant of motion 1(f) is /(/+£0)=/(/) + t(r(/),0),  (r (/), <f>) = 3, /(/ + £0) | ,.o.  (25)  The function T(/) is a gradient of the functional 1(f). For the constant of motion of integral type, i. e . , /(/) = fp(f)dx, the gradient has a simple expression: Assuming p(it) =p{x,t,u k"'), ls  r(u) =P,-D,p, x + Dip^ x +••• + (-  (26)  In this case, we have  (T(f),<p)= f T(f)<t,dx.  (27)  Lax observed  V 2 - j {arw, - 3/(h„, + mi,) + 2ii] a„, 3  H.if)  x)  a„.  The generators form a nonsemisimple algebra (see Appendix B for the definition of a commutator) [t/ 1, U 2] = iC/ 1, [U 1,t/ 3] = 0, IV,U>] = U\  [lfi,lfi] = V\ [£/ 2,t/ 3] = iU 3, [[/ 2,J/ 4] = U\ (22) [U 2,lfi) = \lfi, [t/3,£/4] = 0,  [v3,ifi}=a, [f1,t,5]=o.  By making use of Eq. (20), and by applying Lemma 2, one can cast the first four generators into "genuine" Lie generators: They are equivalent to  T71 = - - a„, TT-=H- xi,- 3/a, + 2na„), T73 = - a„ I7J = a(.  (23)  This set of generators is well k n o w n . 1 4 , 1 7 The generator ifi, however, is new and its properties will be analyzed later. Let us consider operators dU/BI = (5,tj) 3 u and / / = ("„, +mi,) a„ = t/ 4 . It is remarkable that all the U' of (21) satisfy the relation [H, U'} + dU i/dt = 0. In Sec. IV, it will be shown that a generator of an invariance group of time evolution equations always satisfies such a relation. It is well known 13 that the KdV equation admits an infinite number of conservation laws. To study a possible connection between the present groups and the conservation laws, we need to know effects of infinitesimal invariance transformations on constants of motion.  (r(/), 0) is a constant of the motion.  B. Constants of motion of f KMX + a{f\f, its groups  (28)  + f, = 0 and  Now we prove a theorem which establishes a relationship between a constant of motion of the KdV type equation and its invariance group. We consider an equation h  "if)fx +/< =  (29)  where aif) is a function o f / . We assume that an initial value problem for this equation is well posed for a periodic boundary condition fix, t) =fix + * 0 , /) or for a condition/(I) =/(°°, t) = 0. Let us suppose that the system has a constant of motion of integral type /(/) = !pif)dx. The limits of the integration are either over the period or from - « t o ® . We prove:  Theorem 1. If r(n) is the gradient of a constant of motion 1(f) = fp(f)dx associated with the equation/,,, + o ( / ) / , + / , = 0, then the operator 0 = r]?u which has r\(ii) = Dxr(u) is a generator of an invariance group of the equation. Proof: It is sufficient if we prove {£/(«„, +a(n)", + »,)}/ = 0. We consider a transformation of a solution/ to a solution/ + £0. Then, by (24), 0 „ , + o ( / ) 0 , + a„(/)/,0 + 0 , = o. Thus, O = / r ( / ) ( 0 , „ + (j0, + o B f , 0 + 0,)djr. Integrating this by parts and assuming null contribution from the boundary terms, we obtain 0 = / { - D 3 r - Dx(aT) + Tajt x-D,T} l^dx + (d/dl) IT <j>dx. The second term vanishes because of (28). Because we can prescribe an arbitrary admissible function for 0 at initial time t 0 , this equation implies {D 3 r + Dx(aT) - Tajt x + D,!"}, = 0. Differentiating this with respect to x, and defining V=DJ~, we find {£>3rj + r)aji x + (£> xtj) a + D,n}, = {f (n„,  + aux + «/,)}, = 0.  This theorem establishes a relationship between conIn his analysis of constants of motion of the time stants of motion and invariance groups of Eq. (29), evolution equation H(x, t, u,ti,, u „ , . . . , « ' " ' ) + « , = 0, «'"'=«Srt7T , u=f(x,t), Lax 1 5 considered an infinitesimal l(f) = f p(f)dx — r(u)—~{U=»)3., j)=i»,r}. transformation of a solution fix, t) into a solution u =f(x, t) + «t>ix, t). The function 0 must satisfy the lin- The process from U to I involves an integration process  (30)  and not all the generators are integrable to I. In Sec. IV, we provide another scheme to connect a group to a constant of motion which can supplement such a nonintegrable case. The application of the theorem to the generators (21) leads to (within constant factors), l' = f(itu 1-xu)dx,  I^fi^dx, (31)  P = j&J-u\)dx,  5  1 = j au'-3uul + l„l)dx. 1  The generator l/ is not integrable. The constants (31) coincide with members of the set of constants of motion due to Miura, Gardner, and Kruskal. 13 The simplest constant / = !udx is missing; the reason i s that it gives r = l, hence U = 0. In the last section, however, we show that one can associate this with the generator V 2. Thus, we write f = I udx. The fact that there exist an infinite stants of motion for the KdV equation equation is invariant under an infinite the situation is similar to the case of equation / „ - sin/ = 0.'  number of conmeans that the number of groups; the sine-Gordon  C. Properties of gradients (Lax and Gardner) Lax has proved that the gradients associated with the constants of motion of the KdV equation has the following unique properties: 2, then  = s(x-ct)  (32)  is an eigenfunction of the gradients r(s) = y(c)s, r(c) = eigenvalue.  (33)  In the study of doublet solutions of the KdV equation, Lax, as well as Kruskal and Zabusky, 12 focused his attention on three constants I 3 , i 4 , and T5. For these constants, the gradients are  r 3 = u, r* = u1 + 2uxx, r 5 = a 3 + 3i4 + 6K«„ + f u„„, and correspondingly,  r3(s) = s, r4(s) = 2cs, r i (s) = f c ! s .  (3) If we define an operator W' associated with r'(u) off', i> 2, by  then [ W , W"] = 0.  This is obviously the reflection of the fact that the KdV equation is a completely integrable Hamiltonian system.18>19 Incidentally, It is often useful to note that: If 1(f) = fpif) dx is a constant of motion associated with the differential equation Fix, t, / , / , , / „ / „ , / , „ / , „ • • •) = 0, and If V Is a generator of an Invariance group of F = 0, then the quantity /' = J {Up(u)}, dx is also a constant of motion of the same equation. The application of this scheme to the KdV equation, however, fails to generate a constant; indeed, by making use of Eq. (26), Lax's result (1), and Lemma 1, we find  (DW)p>„ )dx= / E v'(-o x)V <tl  dx  = /n'v<dx=J ipjr')r'dx=JD xj"dx = o. (38) Although the method fails to generate a string of constants of motion, it has been found that U' gives rise to the following recursive relation: 0= / u l p i d x ^ c j t l " \  (39)  This relation has been checked up to t = 4.  If u —fix, t) is a solution of the KdV equation, then, by construction, a function v=fix,t\a) = {e° v «}/ is also a solution provided a series ST.o {»*/&*Hi/')*}, exists. First, we show that this group transformation does not alter the values of the constants of motion I 1,  ${?>(«)},dx= $ {p>MY fdx,  Proof: First, by (38), I {U'p'} fdx = 0. This must hold at initial time for it is a constant of motion: / {£/ V}» (,> dx = 0 for any admissible initial condition fix, 0) = (i(*). It can be proved that this is possible (34) only if U'p 1 =Dth"M, h" = polynomial in « , < / „ « „ , • • • . Then, by using Lemma 1, ( t / ' ) V = ( V ' ^ D . h " = Thus, !{p'b<ix=l{p l +D,Zl t(a'/ (35) t!)( U') k-1h"},dx = !{p'} idx.  Another remarkable property of r (u) of the KdV equation is due to Gardner,  W = (15,1") 3„ + (Djr 1 ) 3., + (Djr 1 ) 3^, + • • •,  (37)  E. Properties of e M '  (2) Every solitary wave solution h = 3c sech^VtT (x-ct)  [ l / ' , t / ' ] = 0, i,j> 2.  / vV </* = /£  Now, we study properties of the groups associated with constants of motion of the KdV equation. First we review a few important properties of the gradient found by Lax" and Gardner. 16  (1) If r'(r<) is a gradient of / ' = Jp'(f)dx, r'(u)D xr'(u)=J" with J" = polynomial in «> »*> "xx, • " •  0. Properties of W , i > 2 We note the similarity between the generator U' = (DJ"') 3, and Gardner's operator W'. They, however, are different in that the prolonged U* involves terms such as (•) 3, , (•) 3« i( whereas W' does not. Nevertheless Gardners result implies that two generators U' and U' associated with l ' and I ' commute,  (36)  This result reminds us of quantum mechanics where group operations e'° A, e' bB do not alter the values of observables (A) and (B) provided [A,B] = 0. Here operators V and observables / ' are related by (30) and in fact the U " s commute by (37). The relation (40) indicates that both solutions fix, /) and/(x, f; a) will break up into the same set of solitons. To prove this we start from Lax's result (2). We suppose r to be a linear combination of r ' associated with  (40)  the constants of motion / ' of integral type. Differentiating Eq. (33) by x and using the relationship between T and 11, we obtain {t/nj^ytcjs,,  s = s(x-ct).  ' (41)  This and Lemma 1 give rise to  {(!/)"«}, = (y3,rs={(ro,r«},.  (42)  This relation implies that: For the solitary wave solution (32), we hav^ the operator identity V = yDx, Consequently, the group' operation e' v has the effect of translation in x when it is operated on the solitary wave solution, fe'M.i.-c  = s(x- Cl + ay(c)).  (43)  Now let us assume that the solution/Or, f) splits into N well separated solitons as 1 — <*>, fix, / ) ~ E s,(x-c,t  + 6,)  (44)  as/-".  For such a wave profile, interactions between solitons are small, hence at least for small a we may assume {e° t '"}, l <«,.«,) as  {^""l/b,.  (45)  In view of (43), we can write this as {e  ot,  t(}/iT,i)~E s,(t - c , / + 6,-t-ay(c,)) a s / - « > .  (46)  F. Invariant solutions of the KdV equation One curious question would be whether there exists a solution which is mapped onto itself under the transformation e°". Speaking in a more general context, a solution, of a differential equation F = 0, which is mapped onto itself by the invariance group of the equation is called an invariant solution (or generalized similarity solution). 1 The necessary and sufficient condition for f to be the invariant solution of eaU is obviously f  = 0.  (47)  One of the best known invariant solutions will be the Green's function of the heat equation / „ - / , = 0, / = (417/)-1'2 e x p ( - * j / 4 / ) . Here the group involved is the dilation group generated by U = (xii, + 2tu, +11) 3„ (or equivalents U' = -xd x-2td,+u3 v). It is well known that the singlet solution of the KdV equation (32) is the invariant solution for 11 = U* - c"'t/ 3 (= 3, +c"'5,). The simplest generalization of this is to consider a group generated by V = + fiU* + qU 3, p, q constants. Then the condition (47) yields  iU . +ff  xxx  t" 2/„/„ + iff  x  + P(f  The idea here is precisely parallel to Lax's; Lax uses a condition T(f) = 0 to characterize the doublet solution whereas we use {Uu}, = 0; but they are related by (30). III. INVARIANCE GROUPS AND CONSERVATION ' LAWS OF NONLINEAR SCHRODINGER EQUATIONS The cubic Schrodinger equation - if„ - if 1/* + / , = 0 i s another well studied nonlinear equation. It i s known to share many common properties with the KdV equation. 8 , 1 8 , 1 9 In this section we study group theoretic aspects of conservation laws associated with a class of nonlinear Schrodinger equations. A. Conservation laws of nonlinear Schrodinger equations We consider a coupled nonlinear Schrodinger equation u„ + a(u, v;c) + iu, = 0,  vxx + a(u, ti;c)* - iv, - 0,  u=f{x,t),  «,=/,(*,/),•••,  v=g(x,  u,=f,(x,t), t),  vx =g x(x, t),  v,=g,(x,  (48  <),•••,  where a function a is subject to the condition  .v  Thus, two solutions fix, t) and/(x, t;a) = {e°"u} nx<„ of the KdV equation have the same asymptotic profile as I — <*> except that the phase of each soliton is shifted by the amount ay(c,).  {Vu}  group theoretic viewpoint, therefore, the doublet solution of the KdV equation is the invariant solution of the group  m  +//,) + qf x = 0.  An integration of this equation with respect to x, assuming/(± «>, 1) = 0, leads to the fourth order equation obtained by Kruskal and Zabusky, 1 2 and L a x . 1 5 The nature of the solution was carefully studied by Lax, and the solution was shown to be the doublet solution. From a  "„(/,«•;£) = k ( / , £ - ; c ) ] # ,  o„=e„o.  (49)  (See (14) for the notation #. ] Condition (50) amounts to requiring that the equation can be written as a Hamiltonian system, f  =-«r„  ff = * „  (so)  where bH/bg and 6 $ / 6 / are Frechet derivatives of ft - ! E(f, g) dx, £ = energy density. Equation (48) reduces to the cubic Schrodinger equation for the special case of a = u2v and g=f*. We assume that an initial value problem is well posed either for a periodic condition/(x, t)=f(x + x0, t), g(x, t) =g(x + xj, t) or for a boundary condition f(± ",t) = 0, g(± t) = 0. Let us suppose that the system described by (48) has a constant of motion I(f,g) = Ip(f,g)dx where the integration is over the period or from — * to + The following theorem establishes the relationship between the I and an invariance group of the equation. In the following, quantities 61/6u and 6l/6v represent { « / ¥ } , . * , . „ and {6f/6*}, Theorem 2: If bl/bf and 6I/6g are Frechet derivatives of a constant of motion Hf,g) = Ip(f,g)dx associated with Eq. (48), then the 5h) 3„ is a generator of an invariance group of the equation.  operator V = iifil/bv) ?„ - iifi  Proof-. We consider infinitesimal transformations of solutions f,g into solutions f+up, g+op. <f> and d must satisfy the equations A = +a,(.f,g; c)0 + av(f,g-, c)i; + 10, = 0, B = + n,{g,f; c*)4 + av(g,f;c*)4>-id,= 0. The effect of this transformation on I can be found easily; by integration by parts, we arrive at  Hf+t<t>,g+«l')  = l(f,g)+tf  el(f,g)  + e61.  0 + ~ 4') dx  Thus, d/dtbl  = 0. Next obviously,  The remaining three generators are of the generalized type, and there exists, at present, no analytic method of finding corresponding global transformations.  On integrating by parts this yields 0 = I(P<t> + <?#) dx . +. (d/dt) 5/ where P = - Mi>„ + a(u, v,c)* +  iw,]} /> f  The constants of motion associated with the generators (51) can be found by the simple integration process; they are / ' = J p' dx where pi=mv-it(u xv-tm x),  p2 = vv,  s  P = i(" xi' x - s w V ) ,  'laj/,g;c)-a u(g,/;c*)]j f,  p = i(u xxxv +  7  Q = - {t/[«  Because (d/dt)  =  2  p = u„v„ + j u V - 2(«,i> + uvx)  + "(«, v;c) + iti,h,c  p" = "xxxx," + 5 (ui'mV + wixvxx  61=0,  p' = iti,v,  e  -  h'U Iv 1), 3tt}v xuv,  + 2r,x„xxv  + ,m„vx  u2vx)  +  we obtain  (*) The operator U 2 i s not integrable. These constants of motion, except the first one, agree with the ones obOne can prescribe arbitrary admissible functions for tained by Zakharov and Shabat. 9 The phase shift opera<t> and i at an initial time. Thus, the Eq. (») implies tor U 3, the x translation operator t' 4 , and the t translathat P and Q are identically zero. Furthermore, the tion operator ifi have given rise to the probability densecond terms of P and Q are zero because of condition sity p 3 , the momentum density p', and the energy den(49), hence, P = 0 and <? = 0 yield the equations to be sity p 5 . The first constant / ' also has a simple meaning proved. if we consider the cubic Schrodinger equation as the This theorem enables us to find constants of motion Schrodinger equation for a particle with negative mass if we know the invariance groups of Eq. (48); the proThe / ' represents the initial position of the particle, c e s s involves a straightforward integration process <* o) = 0r-/V> = jf = V = velocity. (61/Sf, il/6g)—l. However, we note that there may be a generator which i s not integrable to a constant of Let us define the Lie Hamiltonian by motion. This theorem can be extended to a general 3 ! e = C/5 Hamiltonian s y s t e m . ' 0 f (P<j> + Qii)d:r = 0.  ^'if) »" ( 'S) "  B. Invariance groups of the cubic Schrodinger equation and its conservation laws  Then, we find that the operator V' of (51) satisfies the relation [H, V'] + 3t/ ' / i t = 0 with  We look for the operator of the form (16) which satisfies the invariance condition (12) for F l =f xx+f 2f + if, = 0 and F 2 = (F 1)* = 0. Assuming the transformation to be the generalized type with i] = T[(x, t, u, v, ux, «am, vxxxxx), and carrying out Lie's algorithm, we arrive at the following eight generators [writing only the first term of (16)]: (/, = (- jixu + tu x) a„, l/j = (iti<„ + itu 2v + ixu x + in) 3„, U 3 = iud u, U, = uxd„  V i = i(u xx +  u2v)d„ (51)  V e = (« xxx + Ztivti x)d u, U 1 = i(u xxxx + u2v„ + 4ww/„ + 2uuxvx + 3 vu2x + | u V ) 3,, V e = hxxxxx  + 5(uvu xyx + im xvxx  +  2vuxuxx  t),  /=/(* + a,t), f =f(x, t + a).  Let us assume that Eq. (15) is a time evolution type: x° = time coordinate, a  = 0, (55)  To carry out a consistent analysis, we must take into account the relation (13),  + at, t),  = exp(ia)f(x,  IV. GENERAL PROPERTIES OF GENERATORS OF INVARIANCE GROUPS OF TIME EVOLUTION EQUATIONS  F (z;c) ~ [H(z;c)]^ + v 0 = 0.  TP = iud„ V* = id M, V s = -d t. The effects of the group transformation e""', a real, on & solution /Or, /) can be found easily for i < 6,  f = af(ax, <ft), f  We note that the second generator U 2 which is not r e lated to a constant of motion also satisfies the relation. A general analysis of this property of the generators will be given in the next section. Some of the other commutation relations among U l are [u', U y ] = 0 for 3  2  V 2 = - xd x — 2/3, + u3„,  / = exp[- i(ax + a>t/2)/2]f(x  SU  St  F 1(z;c) = H(z;c)+u  + uvj^n + u2v x) + f «Vw„] 3.. The first five generators can be cast into "genuine" Lie type operators by Lemma 2: T7 i = -ld x-$ixud„  ' W = energy = /5. (54)  +«o> = °,  «.  (56)  We define two operators associated with H 121 and U by H=H9,  (52)  k = \,2  dV  dP  + H*d„  (57)  (58)  - 124 As was mentioned in the first section, they must be interpreted as their infinite prolongation.  to associate a conserved quantity with an invariance group of the equation:  By the definition of a time evolution equation, H is not a function of the coordinates corresponding to ^ - d e r i v a tives such as i< ol , v v o . In such a case, we can always express any coordinate of V-derivatives in terms of other coordinates by making use of the relations (55) and (56). Thus, we assume, without a loss of generality, that 17 is free of these coordinates.  3: If the operator defined by (16) is a generator of an invariance group of the equation H + « 0 = 0, and if H satisfies all the conditions in Lemma 4, then the quantity (£/) defined by (60) is a constant of motion,  A key in the present analysis is to write Eq. (55) as  W + D„)v = 0.  Theorem  V  i. e., d/dx"(V) = 0.  We note that in proving this we did not assume the r 1 quantity / to be independent of time.  fix) fix) dx' •••dx "  Lemma 3 can be generalized to a set of nonlinear time evolution equations of the form (59) H ' + ul = 0,  ff'=  #<(*•,u,u,u, ...,u),  ( = 1,2,...,Mt  We first prove:  (63)  Lemma 3: If U is a generator of an invariance group where u = (u 1,u',.. ,,ti"), u = ( u ' j u 2 , . . . ,u") of the equation H + u0 = O, then under condition (56) we n fc k * have [U,H] + BV/Bx" = 0. <> u'v= /with ' ( * ) ,aetc. In this case we have Proof: We have [C/,//] + dU/ix = aiv + and a*B = UH -HJ) + 3 oT). It i s sufficient if we prove that a vanLemma 3': If V = is a generator of an invariance ishes under (56). Indeed, 0 = UiH +1<0) = UH + £>0 n = VH group of Eq. (63), then we have [u,H] + BU/Bx° = 0 i + + "on- + "el. + • • = VH + 3 0tj where - HiHu ••• = HH*3,, 2„t and- dV/dx° = i?x<,v') 3„i. = un-rS^-Hn. For Hamilton's equations of a field [t/1 = P=pix), 3 i/ = Q = ?(*)] Now, we define the following quantity: 2 , - g + « o = 0. + P 0 =O {V) = Re f ivUu)^ ftx ) dx' dx • • | • dx", Re = real part, (60) where the integration should be taken over the whole space of interest. Obviously \U) is a function of only. The following lemma describes how it develops in time for a class of nonlinear systems:  Lemma 4: It H  of the equation  equation  H  we obtain the familiar expression with  jca.  The theorem above can be specialized to a real differential equation: If  Hix. u,u.u...., u). 12 r  H + wo = 0, satisfies an equation +tt0= 0 satisfies the  H + uH„-D,(uH  U i) +  --- +(-  V./*tl)  an  and s  .,]..,<,, v'dn " * !./*(*> t  vanish for v = ( v , — , v") = normal vector on the boundary surface, then  )= 0  (64)  if* + vH. + uiHJ* - C,[t>WB| + h(H„()#] + •and• if•all the surface integrals /s["t7,...»W|,j ...Jiu/ii) dn for S = boundary, then the quantity (t j + ( - [ t t f . +•'(«„ )#] = 0 (61)vanish = ittiUi(]„,, dx 1 • ••dx* is a constant of motion. and if all the boundary integrals  in the equation  Here,  v — the whole space inside S.  The following equations which have been attracting considerable attention in the study of propagation of nonlinear waves satisfy the condition (61) or (64): generalized Korteweg—de Vries equation  (K) 2"if+f"B,f+S,f=0,  cubic Schrodinger equation in « dimensions d 00 =([£/,//]-3 U (62) d? - t [ E ( ? , , ) V + / ' / * ] + » . / = 0. Y7, Proof: For brevity, we write (60) as (l/) = ReJ vUudx. Hirota equation 8 Then, we have d/dx°(U) = Re !(v 0Uu + vD t Uu)dx 0 = Re/[-W*ri + «(3l//3x -//ty)«)dx. Here, we have used a(3,)'/+ib(Bx)\f + cff'BJ + idf*f' + 3 , / = 0. the relations (5S) and (56). On the other hand, we have <U//) = Re Jt)t//jWx = R e / w j ( . d x . Applying Green's theorem repeatedly, and using the hypotheses, we find (t///) = Re /(-#*!))<fx. Putting these two together, we obtain  However, the heat e q u a t i o n / „ - / , = 0 and Burgers equation/„ = 0, both of which represent a disx sipative system, do not satisfy Eq. (64).  +ff -f, id/dx"){V) = <[£/,//] + BU/Bx"). The application of TTieorem 3 to the KdV equation  The combination of Lemma 3 and 4 leads to a method  and to the cubic Schrodinger equation has turned out to  produce only a few constants of motion,  The first term is  KdV equation:  m = -f«dx, <U 2) = fi« 2dx, {U i ) = 0,  + ? M „ S + v^A^)} + EoU0„o +E + =Zb, A  fori>2.  cubic Schrodinger equation:  (V 2) = fi mi* dx,  <t/<) = 0 for f > 2.  +  £oA«.J+?t'o•EhA-.'e+ZA-.J ) = UA 0 +Lu„ UA^ UA V m.  V. CONCLUDING REMARKS We have shown that provided one considers the group transformation which is more general than the one considered by Lie, one can associate one invariance group with each constant of motion of a class of physical systems. Thus for such a system one can derive the constants of motion by finding the invariance groups of the equation. One of the best known methods of finding conservation laws is to use Noether's theorem. The difference between the two is that the groups in the present approach leave the differential equation invariant whereas the groups in Noether's theorem leave an action integral invariant.  Hence, (») gives D t UA = t/[>l0  = UD aA.  APPENDIX B: A COMMUTATOR OF GENERALIZED LIE TYPE OPERATORS We consider two operators of the form (19),  l/1=T)Ie. + £13„, U 2=^d, + (%.  We must interpret these as simplified representations of (9). The commutator of the two is defined as  [l/», U 2] = [(£/y) - (t/V)] 3„ + [(t/>£2) - (£/'£>)] 3„+ • • • +[(t^rii...,) - (t/!i!...,)]  In the following communication, a generalization of Theorems 1 and 2 will be discussed. ACKNOWLEDGMENTS  (" 0o i4„ o + u 0 o A„ o )]  We write this as  V = [t/ 1, U 2] = t)3. + £3V + • • • + T,,..., 3„(ii  I am sincerely grateful to Professor Carl E. Wulfman for many helpful discussions and for a number of valuable comments on the manuscripts. I also thank the Research Corporation for supporting this research. APPENDIX A: PROOF OF LEMMA 1  We prove that this satisfies the condition imposed on (9), i . e . , the condition (6). In fact, by applying Lemma 1,  D UA = UD A.  2 It is sufficient if we prove To avoid 0 0 V t complex indices, we represent a set of indices i •• • k appearing in the expressions (7) and (9) of D 0 and U by t ( a circle " or by a dot •, and write D 0 and V as Similarly £ , . „ , = D , . . . , £ . Therefore, the operator obtained from the commutator of two operators of the ^0 = 3,0+? ("o^ + fo'Vform (9) also assumes the same form.  T),..., = U \... = I/'B (...,r, ! - U 2D = D,... (UW -t/V)=.D ...»1-  where the sign indicates a summation over all the parenthesized quantities in (7) and (9). Then, by the definitions of D 0 and U,  =£ bDaA^+ZD^) Using  • T h i s study has been supported by a R e s e a r c h Corporation grant. ' R . L . Anderson, S. Kumei. and C . E . Wulfman, P h y s . Rev. L e t t . 28, 988 (1972). I have learned that the idea of the 0 o derivative dependent infinitesimal transformation is not new. F o r instance, H. Johnson, P r o c . A m . Math, Soc. 15, 675 (1964). I am grateful to P r o f e s s o r A. Kumpera of University 0o 0 0 Oo 0o of Montreal for bringing this p a p e r to my attention. 2 S. Lie, (Chelsea, New York, 1970), 3 Vols. (Reprints of 1888, 1890, 1893 e d s . ) ; +E[(t/«0„)A.o+(I/t>0„)AvJ. (Chelsea, New York, 1967) (Reprint of 1897 ed.); (Chelsea, New York, 1967) (Reprint (») of 1893 e d . ) .  +E[U>1o) A„ + (DoS.Mj.  D 0rj a = j]0o = Uu , D £ = [ = Uv  D 0UA=E (tJoA^+E.-DoAJ  , Transformationgruppen Differentialgreichungen CoTitinuierliche Gruppen  5  I . . V. Ovsjannikov, Group Theory of Differential Equation (Siberian Section of the Academy of Sciences, Novosibirsk, USSR, 1962). [This book has been t r a n s l a t e d into English by G.W. Bluman (unpublished).] *G. W. Bluman and J . D . Cole, Similarity Methods for Differential Equations (Springer, New York, 1974). ' R . L . Anderson, S. Kumei, and C . E . Wulfman, Rev. Mex. F i s . 21, 1, 35 (1972); J . Math. P h y s . 14, 1527 0 9 7 3 ) . e N . H . Ibragimov and R . L . Anderson, "Lie—Biicklund Tangent T r a n s f o r m a t i o n s . " J . Math. Anal. Appl. (accepted for publication). (See Sov. Math. Dokl. 17, 437 (1976) for e x c e r p t i o n . ) J S . Kumei, J . Math. P h y s . 16, 2461 (1975). ' A . C . Scott, F . Y . F . Chu, and D. W. MacLaughlin, P r o c . I E E E 61, 1443 (1973). ' V . E . Zakharov and A . B , Shabat, Zh. Eksp. T e o r . F i z . 61, 118 (1971) [Sov. P h y s . - J E T P 34, 62 (1972)]. 10 T h e notation v is from Ovsjannikov (Ref. 3).  1<  S . Kumei (unpublished). Also, L . V . Ovsjannikov (Ref. 3). N . J . Zabusky, in Nonlinear Partial Differential Equations, edited try W. A m e s (Academic, New York, 1967). " R . M . M i u r a , C . S . G a r d n e r , and M . D . Kruskal, J . Math. P h y s . 9, 1204 (1968). " M . D . Kruskal, R . M . M i u r a , C . S . G a r d n e r , and K . J . Zabusky, J . Math. P h y s . 11, 952 (1970). 1S P . D . Lax, Commun. P u r e Appl. Math. 21, 467 (1968); 28, 141 C1975). " C . S . G a r d n e r , J . Math. P h y s . 12, 1548 (1971). " H . Shen and W. F . Ames, P h y s . Lett. A 49, 313 (1974). I owe t h i s reference to P r o f e s s o r R . L . Anderson. " V . E . Zakharov and S . V . Manakov, T e o r . Mat. F i z . 19, 332 0 974). 19 Y . Kodama, P r o g . T h e o r . P h y s . 54, 669 (1975). « S . Kumei (unpublished). n H and the e n e r g y H in Sec. m should not be confused. 12  On the relationship between conservation laws and invariance groups of nonlinear field equations in Hamilton's canonical form a ) Sukeyuki Kumei  bl  Department of Physics. University of the Pacific. Stockton. California  95204  (Received 28 January 1977) It is shown'that whenever fields governed by the equations 3/dlp0 = — 5H/5q a.  d/dtq  B  =  SH/bp  a  allow a conservation law of the formflp/fl;+divj= 0. there exists a corresponding Lie-Biicklund infinitesimal contact transformation which leaves the Hamiltonian equations invariant. A condition thai an invariant Lie-Backlund infiniiesimal contact transformation givesriseto a conservation law is established. Each such transformation, which may involve derivatives of arbitrary order, yields a one-parameter local Lie group of invariance transformations. The results are established with the aid of a Lie bracket formalism for Hamiltonian fields. They account for a number of recently discovered conservation laws associated with nonlinear time evolution equations.  INTRODUCTION 1,2  In p r e v i o u s p a p e r s , we have studied invariance p r o p e r t i e s of v a r i o u s n o n l i n e a r t i m e e v o l u t i o n e q u a t i o n s by a p p l y i n g t h e t h e o r y of g r o u p s of L i e — B a c k l u n d t a n gent t r a n s f o r m a t i o n s 3 (not t o b e c o n f u s e d with t h e B a c k lund t r a n s f o r m a t i o n s of r e c e n t l i t e r a t u r e " 1 ) and we h a v e shown t h a t e a c h of t h e w e l l - k n o w n s e r i e s of c o n s e r v a t i o n l a w s a s s o i c a t e d with t h e s i n e - G o r d o n e q u a t i o n , t h e K o r t e w e g — d e V r i e s e q u a t i o n , and t h e n o n l i n e a r S c h r o d i n g e r e q u a t i o n i s r e l a t e d to a different o n e - p a r a m e t e r g r o u p which l e a v e s t h e c o r r e s p o n d i n g differential e q u a tion invariant. T h e - g r o u p g e n e r a t o r s o b t a i n e d in t h e s e p a p e r s d e p e n d upon d e r i v a t i v e s of a r b i t r a r y o r d e r , s o t h a t t h e y a r e not of t h e t y p e c o n s i d e r e d in L i e ' s g e n e r a l t h e o r y of c o n t i n u o u s g r o u p s of t r a n s f o r m a t i o n s . T h e q u e s t i o n naturally a r i s e s : T o what extent can the p r e v i o u s r e s u l t s be generalized? In t h e p r e s e n t p a p e r , w e s t u d y i n v a r i a n c e p r o p e r t i e s of H a m i l t o n ' s e q u a t i o n s g o v e r n i n g t h e t i m e e v o l u t i o n of m u l t i c o m p o n e n t fields p„i\), qa(x), p  a  =-6H'&,  u  ,  «  0  = 6//.'6/>0,  a = l, 2, ... , A \  (1)  w h e r e x = (v",.v',.v 2 ,.v 3 ) and />„ = ?,o/>„, q a = ? / / / „ • W e a s s u m e that an e n e r g y d e n s i t y H a s s o c i a t e d with H c a n d e p e n d on c o o r d i n a t e s .v ( i n c l u d i n g .v n ), pa, and qQ, and t h e i r s p a t i a l d e r i v a t i v e s of a r b i t r a r y o r d e r . 5 T h e m a i n i n t e r e s t of t h e study i s : t o e x a m i n e t h e r e l a t i o n s h i p b e t w e e n i n v a r i a n c e g r o u p s a d m i t t e d by E q . (1) and c o n s e r v a t i o n l a w s o b e y e d by t h e f i e l d s . We w i l l p r o v e  that: Tltc existence of A' independent conservation laws associated with Ihc fields of Eq. (1) necessarily requires the exislcnce of A" one-parameter groups which leave Eq. (1) invariant. T h e p r e c i s e r e s u l t will b e s t a t e d h e r e a s a t h e o r e m . T h e n o t a t i o n s in t h e t h e o r e m a r e th^ following: A and J ' a r e q u a n t i t i e s a s s o c i a t e d with t h e fields and a r e f u n c t i o n s of x , pa, and q a , and of t h e i r s p a t i a l d e r i v a t i v e s of a r b i t r a r y  o r d e r ; £>, r e p r e s e n t s a d i f f e r e n t i a t i o n with r e s p e c t t o x ' , and t h e q u a n t i t y bA/if (f=q„ o r pa) i s defined by  with  Theorem: If, when pa and qa a r e s o l u t i o n s of t h e H a m i l t o n i a n e q u a t i o n s (1), t h e f u n c t i o n s AU,p a,q0,"r) and J ' ( . v , p a , q a , " ' ) obey t h e c o n s e r v a t i o n law D„A + = 0 , t h e n t h e p r o l o n g a t i o n of t h e o p e r a t o r A = (6A/6p a)d tl -(6A/6q l>)ct i s a g e n e r a t o r of an i n v a r i a n c e g r o u p of t h e H a m i l f o n i a n e q u a t i o n s . C o n v e r s e l y , for any o p e r a t o r of t h e f o r m A' = (6A'/6p a) ?flo - (M',< whose prolongation becomes a generator, of an i n v a r i a n c e g r o u p of t h e H a m i l t o n i a n e q u a t i o n s , t h e r e e x i s t s a flux _?' w h i c h t o g e t h e r with a d e n s i t y A' f o r m s a c o n s e r v a t i o n law D^A J ' = 0. T h e c o r r e s p o n d i n g r e s u l t for H a m i l t o n i a n s y s t e m s with finite d e g r e e s of f r e e d o m g o v e r n e d by t h e e q u a t i o n s — dH/?p a, pa = - cH/cq a h a s b e e n o b t a i n e d by Peterson.6 W e will p r o v e t h e t h e o r e m by u s i n g a L i e b r a c k e t f o r m a l i s m , i n s t e a d of a P o i s s o n b r a c k e t f o r m a l i s m , for E q . (1). T o e s t a b l i s h t h e L i e b r a c k e t f o r m u l a t i o n , o n e n e e d s to a s s o c i a t e a p p r o p r i a t e o p e r a t o r s with p h y s i c a l q u a n t i t i e s of t h e s y s t e m . Such a f o r m a l i s m is known for H a m i l t o n i a n s y s t e m s with finite d e g r e e s of f r e e d o m . 7 In t h e following we will d e v e l o p a s i m i l a r f o r m a l i s m for t h e field e q u a t i o n s (1) by a p p l y i n g t h e t h e o r y of L i e Backlund tangent t r a n s f o r m a t i o n s . T h e f o r m a l i s m t u r n s out to b e v e r y a p p r o p r i a t e in s t u d y i n g t h e c o n n e c t i o n of i n v a r i a n c e g r o u p s of E q . (1) to c o n s e r v a t i o n l a w s . In t h i s a p p r o a c h no r e f e r e n c e is m a d e to i n v a r i a n c e p r o p e r t i e s of an a c t i o n i n t e g r a l ]L tlx: W e d e a l d i r e c t l y with i n v a r i a n c e p r o p e r t i e s of d i f f e r e n t i a l e q u a t i o n s .  a)  T h i s work has been supported by a Research Corporation grant. " P r e s e n t address: 151U-G9 Sekido, Tama-shi, Tokyo 192-02, Japan.  All t h e r e s u l t s in t h e following s e c t i o n s r e m a i n v a l i d for a g e n e r a l c a s e of n s p a t i a l v a r i a b l e s .  I. LIE BRACKET FORMALISM  q a(x) — <7„(.v) + e0„(.v), ¥-=9 -D.Q ,+DMj  Similarly, for a variation  We consider groups of Lie-Backlund tangent transformations generated by the operators®  U=  - c.a,. + U),F m)amaii - iD.c.te,..,  _ { ( - D where  (  ) . . . ( -  D y  ) G  a  }  + ( _ D ( ) . . . ( - D J ) < ( ? a a i . ..., + ••• .  (2)  +  represents a total derivative operator  j  = aTi  (9)  6J/6p  We adopt (8) and (9) as the defining equations of a and 6,7/bq a . We call ty a density of J. With the functional J we associate an operator J which is obtained from (2) by substituting a and a for F a and G„: In simplified notation  6J/6p  bj/bq  (10)  M S  (3)  /><,,i..., i 9<,,i...i represent coordinates associated We designate the operators of this particular form by with derivatives x boldface letters. Then, with the energy functional H, a a Throughout the paper we adopt a summation convention the following Lie—Backlund Hamiltonian operator will for repeated indices: a greek index runs from 1 to AT be associated: and a Roman from 0 to 3. In contrast to conventional contact transformations, we allow F and G to be func? tions of .v and a and any of their derivatives a of order. In the study of Eq. (1) which is a perator corresponding bto a functional The operator time evolution type we can assume without a loss of nd to be equivalent to is found generality that the F a and C a are not functions of time (12) derivatives of and This will be assumed in a a the following for all the operators of the form (2). To Let us denote the set of all the operators of the form avoid a complex expression we write the operator (2), (10) by SJ. We can prove that n closes under the comwhich we call a Lie—Backlund operator, as mutation operation defined in (a) above: U = FJ -G„?. (4) a 9ft a Pa' If two operators A and B belong to f2, We must always consider this to be the infinite series the commutator C = [B,A] also belongs to n , and its given by (2). We denote a set of operators of the form density C is given by any one of the following: (2) by A. It is known that the U have the properties 3 2 1 (a) If U\ U'eA, then U = [U\ t / ] e A with F' a = U F\  ?i • • • p (x),  •'" ®(.v).  («/>„) C</ft)'°'  p lx), q (x), arbitrary p (x)  q (x).  an  •'-MSK-MSK  Proposition-.  - U*Fl  and C a = U'G* a - U'G' a.  2 s  -  C 2 = B/4, C3 = -aS.  (b) If V 1 , V , [ ) ! € A , the Jacobi identity holds 9 :  [[U1, W],U3]+[[U2,£/3], £/']+[[t/s.f/1],^]^. (c) Members of A commute with the total derivative operator Dr. [U, D i ] = 0 . 2 - 9 ' 1 0 This last property will be used frequently in the following without comment. We define the time derivative of the U, which we denote by U^, by  <5> A g a i n , t h i s i s a s i m p l i f i e d e x p r e s s i o n ; t h e full e x p r e s s i o n i s o b t a i n e d b y r e p l a c i n g F a a n d G a in (2) b y S r 0 F a and a^G,,.  Now, let us consider a variational problem of a functional  The proof will be given in the Appendix. Following the usual definition of a Poisson bracket for fields, we have C = J(Tidx' = {£,A}. Thus, we might state this as: The commutator of the operators associated with the functionals A and B is equal to the operator assoicated with the functional We note that the canonical commutation relations among pa and q a are not carried over to the operator formalism: The operators corresponding to pa and q a are P a = d q and Q a — - S p and they all commute.  [A,B\.  II. INVARIANCE GROUPS OF HAMILTON'S EQUATIONS AND CONSERVATION LAWS We now turn our attention to the theorem stated  1 3 J\p,<!,x°) = jj(x,p,q)dx', dx' dxwell-known dx 2dxequation . which (6) describes earlier.=The evolution of a functional A = IA dx' is The density $(x,p,q) depends on x and pa, q a time and their derivatives of arbitrary order except ^A = {H,A} + fB xo/ldx'. ones involving time derivatives. For the variation paix) -/>„(*)+ £!>.(*) we have - 6J = e / c  iha dx' + surface integral  (13)  (7)  the (14)  We associate an operator K = [H,A] +A 0 with the quantity on the right-hand side. In view of (12) and (13), it is obvious that: The density K corresponding to the operator K=[H,A] + A o is any of the following:  with  (8)  ^fMHfm^  we have  or  (*»)  In the following, we prove the theorem by showing basically the following equivalences: A is a generator of an invariance group of Eq. (1). A satisfies [ H , A ] + A ^ , = 0  A satisfies DJ\ + £XDiJ< = 0.  (•i  According to the theory of groups of differential equations," the operator V of (2) becomes a generator of an invariance group of Eqs. (1) if and only if U satisfies the equations  Here, the symbol (•• •)!,, means: Evaluate the quantities under conditions (1) and the conditions implied by them. We note that there exist generators which do not take the special form given by (10). We start from the following properties of a generator of an invariance group of Eq. (1):  even pointwise: For any given values of x, pa, qa, Pa it la i>**"> the operator vanishes. This should be true as long a s there e x i s t s a solution which takes the designated values at the given point x. It is also clear that we can allow f a and g a to be functions of x instead of x' because x", if it appears in f a and g a, acts simply as a parameter and has no consequence for the proof given here. Now, we combine the results obtained above to prove the theorem stated at the beginning: Proof of theorem-. In the following, we a s s u m e , that the index i runs from 1 to 3. First we show DrjA + Dj' = 0 —A is a generator. Under condition (1), we have tlA + B j t A ^ D c A , hence, by the hypothesis  H/ +  =  (T)  Because we may assume that neither A nor J ' contains time derivatives of pa and qa, this relation must hold at initial time x° = f where arbitrary initial values may be imposed on pa and qa. Consequently, the equation (f) holds not only for solutions pa and qa but also for arbitrary functions f a(x') and g a{x'). Thus, noticing that the Lemma 1: The Lie-Backlund operator V defined by left-hand side of the equation Of) i s the /C2 of (*»), we (2) satisfies the equations have K, = -D{J'. This implies that 6K/6p a and 6K/6q a ([H, U] + U :f i)pa | „ = 0 , ([H, U] + U xo)qa | w = 0 , (16) vanish identically, and, as a result, [H,A] +A^>s0 by (»*). In view of Lemmas 1 and 2, we s e e that this is if and only if U is a generator of an invariance group the necessary and sufficient condition for A to be a of Hamilton's equation (1). generator of an invariance group of Eq. (1). Conversely, Proof: In view of the definition of H, under the conif A i s a generator of an invariance group, in view of dition (• • • )l w we have an identity D0= 3,0 + H. Using Lemma 1 we obtain two equations 6K/6p a = 0 this relation, we obtain ([H, £/] + U x o)p a \ w = { - H G „ and 6K/6q a = 0. According to Lemma 2, these vanish + £7(6/7/6q„)- 3, 0 G J l „ = {l/(6fl/6 9 < 1 ) - D 0 G „ } | r = V(6H/tq, identically. This implies that the density K in (**) must + pa)\w. Similarly, ([H,C/]+ V^)q a]w=U(-6H/6p a have a divergent form; for instance, K 2 — (H + d ^ A + q a ) l „,. These relations obviously prove the statement. = -Dj' with J ' = 3'(x,p,q). Now if we let />„ and qa be solutions of Eq. (1), the quantity in the middle of In the following analysis, it is often helpful to conthis equation becomes equal to D^A, and the equation sider an initial value problem of Eq. (1). We say that leads to the desired result DqA + D , J ' = 0. functions /„(*•') and g a(x'), x' = (*', x2, x 3 ) , are admissible if the initial value problem pa I =/„, qa I =g a III. INTEGRABILITY OF GENERATORS TO has a solution. A set of all such admissible functions CONSERVED DENSITIES will be denoted by I. The following lemma states that Eq. (16) holds without the condition I v . We have proved that with every conservation law Lemma 2: If V is a generator of an invariance group of Hamilton's equations (1), the operator [H,£/] + U,o vanishes identically for arbitrary functions /„(*') and g a(x') which belong to I. Proof .'We have, by definition, [H, U] + f -N aBt, w h e r e MJx,p,q) N a(x;p,q) = -RGa-V(t,H/bq  =  = H F a - U(bH/bp a) + 3xoF a, + B^G. B y L e m m a 1, if 0L)  Pa' 9<x a r e solutions of Hamilton's equation, then M a = N a = Q. We let x°-1 = initial time. At t, both M 0 and N a are well defined (note that F a and C a do not depend on any x° derivatives of pa and qa), hence, M m=N a = 0 at t. Suppose that inital conditions were />„=/„(*'), ? „ = « „ ( * ' ) . Then, MJx,f,g) = NJx,f,g) = 0 with x = (t, i 1 , x2, x3). Because I i s a parameter of arbitrary 2 value, we may replace x by (x°, x , x 3 ) to obtain the desired result. Remark: If / „ , g a or any of their derivatives were not defined at some point, the function M a and N a , hence the operator [H, V] + U to, would not be defined at the point. We note that the relation [H, U] + U ^ e O holds  obeyed by the Hamiltonian fields one invariance group i s always associated. In the present formulation, the converse of this i s true only if the coefficients of the generator V take the special form F a = bA/5p a, Ca = 6A/6q a. Because there exists a systematic algorithm for finding generators of invariance groups, it is Important to know whether the generators found are lntegrable to conserved densities. For simplicity, we adopt the following notation:  /.(*) = ?«(*). f.J*)=P*M  with  01 =  1,2,...,A', (17)  s„(A)=f„ UA)=c. In this notation, our problem i s to tell whether a given set of S„ have the property S„ = iA/6f a for some functional A [ / J = SA(x,f y)dx'. As a general property of a functional, we have  where £ = ( f u t 2 , • • • ,£ a „). If S„ has the desired property,  -  130  then, because of the definition {(r//rfe„)/»[/, + t,0 1 J},.„ = 6 ( y fixed), this relation is written (or fixed a and j as,  (18)  This is the integrability condition of the set S„ to a conserved density A. It is not difficult to obtain from this a condition which does not involve integration: Using the fact that the functions <pa and 6 S are arbitrary, we can reduce (18)to £d:  S.) = S<!>  .(6?,  ?,  S„.  (19)  0(.v) — arbitrary function where ZT = ( - D , ) • • • ( = ?,«••• ®(v), and the notation y,. , a. . = a + a. + n, + • • •. All the ^  u >  I***;  i  ij  Roman indices run from 1 to 3. Because 6 is arbitrary, the coefficients of each A,,..., on both sides of (20) must match. Example. sine-Gordon equation To illustrate the results obtained above, we study the sine-Gordon equation, using / = .v°, .v — .v1, h „ - « „ + sin» = 0.  (20)  A canonical form qt-6H/ 6/j, pt-= - bH'bq for this equation is obtained by letting q = », H = \p2 + j®2 - cosq: <7i=/>, / > , - - 9 „ + s i n < / .  (21)  In the previous paper, 1 we have shown the equation u x t = sin» admits an infinite number of invariance groups, and it is straightforward to adapt these results to Eq. (21); four of the generators of invariance groups of Eq. (21) are v  i =  u  v  i = l 4 » n , " Sfl.cos? +  2 = M , -<-<7„ + si-V/)?,. + h j > 2 ) \ ~ ( - 4/>„,  + 1px cos® - f ® ^ v  - 3qxqxJ>V e, 2  4 = (4/>„ " /> cos9 + \q J> + - (+ 5®xx cos® - §q\sin® - sin® cosq ip'sinq-ZqJ>J>-\p  4qxxx  2  qxx)* t-  Using a theorem given in the previous paper, 8 we see that C<'i and V2 are equivalent to the space and time translation operators ? and c t . To find conserved densities from these operators we must check condition (19). All of them satisfy the equation, and the conserved density A associated with each of the generators is found to be: A i = p q x = momentum density, A 2 = jp2 +  - cos® = energy density,  j -A, = 4pqxxx - 3pqxcosq + iq\p +  iqj,\  A,= -1 p\- 2 q\x - ip 2 cos q + \q2J> 2 +  ip" ~11\cosq + i cos2® + iq*.  These conserved densities are related to those obtained  by Lamb,' 2 and their group theoretic aspects have been studied by the author1 and by Steudel. 1 3 In the previous paper, 2 we also have shown that a series of conservation laws admitted by the nonlinear Shrodinger equation are related to invariance groups of the equation, where we have made use of a special property of the equation. The present results provide a unified view to the previous results. CONCLUSION In this paper, we have developed a new group theoretic way of looking at conservation laws associated with field equations in Hamilton's canonical form, and we have proved that the existence of -V independent conservation laws necessarily implies the existence of at least ,V local one-parameter Lie groups which leave the field equations invariant. The condition that a given invariance group is integrable to a conserved density also has been given. Because there exists a well established algorithm for finding generators of invariance groups of differential equations, and because many Euler —Lagrange equations can be put into Hamilton's canonical form, the present results should be useful in finding conservation laws for a variety of s y s t e m s . 1 4 Clearly, the present approach to conservation laws via a Lie bracket formalism is quite different from conventional approachs which make use of Noether's theorem: Noether's theorem as originally derived is too restrictive to give rise to conservation laws such as those dealt with here. However, as this work was being completed, the author learned in a personal communication from N. H. Ibragimov that he has been able to generalized Noether's theorem and with the aid of L i e Backlund contact transformations he has obtained results similar in part to those obtained h e r e . 1 0 ACKNOWLEDGMENTS I am grateful to Professor Carl Wulfman for his continuous interest and encouragement in the present work, and greatly appreciate his helpful discussions and valuable comments on the manuscript. I also would like to express my thanks to the Research Corporation for supporting the research and to Professor N.H. Ibragimov for a personal communication. APPENDIX: PROOF OF THE PROPOSITION To simplify expressions, we use notations D.... and ( - D 1 ) - " ( - D / ) = I>:.j; and = < 7oi-j * We represent a sum of the f o r m / + / ( + / l t + • • • by one term f lmJ. For instance, Eq. (2) and Eq. (8) become [/ = ( D . ^ F J ? , ^ - (Bj.jG.)? and &J/6p a We first probe the following relations: =  (Al) (A2) This relationship is entirely independent from Eq. (1).  We p r o v e the f i r s t r e l a t i o n : the s e c o n d f o l l o w s s i m i l a r l y . To p r o v e ( A l ) , w e a s s u m e that functions p a and g a decay sufficiently fast a s [ ( * ' ) ' + ( x 2 ) 2 + ( x * ) 2 ] " 2 so that all the s u r f a c e i n t e g r a l s w h i c h a p p e a r in t h e p r o c e s s v a n i s h . L e t u s c o n s i d e r a n i n t e g r a l Jv(x')l/(6M/6<i s) dx' w h e r e v i s s o m e a r b i t r a r y function e x c e p t that it does not d i v e r g e at infinity. If w e w r i t e 3,i • • ' 3 , / f = f ( - J I then i n t e g r a t i n g o v e r t h e w h o l e s p a c e  = JvUD-^/H^dx-  = JvD' imlV/n,  =!v<. lvK<-t  -  dx'  ti j  dx  '  Gav,.,d tll.P~r.i<Mp„f.,.)dx' ( u s i n g [Z)„ < v A „ J = °>  Now, w e h a v e the right hand quantity of E q . ( A l ) in the l a s t i n t e g r a n d . B o t h in the s t a r t i n g and in this final f o r m of the i n t e g r a l the function v a p p e a r s a s a f a c t o r . B e c a u s e v is arbitrary, this equation n e c e s s a r i l y implies (Al). Next, by definition, [B, A ] = C « ° 3 , o - C ' « 3, with  In v i e w of the e q u a l i t i e s (A2), w e o b t a i n  +  (4)  =D  '->  ~fer) fc)»., J  [ © (S) _  ©]  S i m i l a r l y , we obtain C ' s D J . ^ C , ] , . T h u s , we have p r o v e d the a s s e r t i o n for C T o provs" It for C, and C„ w e s i m p l y n o t e that t h e y a r e r e l a t e d t o C :t>yC,=C, + Df and + w h e r e / ' and a r e functions of  C,=Ci D^  g*  pu, q a , and of t h e i r d e r i v a t i v e s and t h e i n d e x i r u n s f r o m 1 t o 3 . T h e fact that functional d e r i v a t i v e s of t h e functional / d i v h ( x , p , q ) d x ' a l w a y s v a n i s h l e a d s to the desired results.  ' S . Kumei, J . Math, P h y s . 16, 2461 0 9 7 5 ) . *S. Kumei, J . Math. P h y s . 18, 256 0 9 7 7 ) . ' N . H . Ibragimov and R . L . Anderson, Soviet Math. Dokl. 17, 437 0 976); N . H . Ibragimov and R . L . A n d e r s o n , "Lie— Backlund Tangent T r a n s f o r m a t i o n s , " to a p p e a r Id J . Math. Anal. Appl. As t h e s e authors have c l e a r l y shown, the t r a n s formations which have been considered to p r e v i o u s p a p e r s LR.L. A n d e r s o n , S. Kumei, and C . E . Wulfman, P h y s . Rev. L e t t . 28, 988 0 9 7 2 ) , and Ref. 1 and 2 above) form groups of infinite o r d e r contact transformations; for instance. If we take our p r e s e n t problem as an e x a m p l e , the o p e r a t o r U d e fined by E q . (2) Is a tangent v e c t o r and g e n e r a t e s a t r a n s f o r mation which i s a local o n e - p a r a m e t e r group of contact t r a n s f o r m a t i o n s defined in a v e c t o r s p a c e of Infinite d i m e n 111 sion with coordinates ( p a , q a , p „ , i , g a , i , p a , n , 1 ^ i i • ' " t h i s s p a c e , no finite dimensional subspace e x i s t s which c l o s e s u n d e r the transformation except for special c a s e s w h e r e the transformation b e c o m e s an o r d i n a r y point transformation o r a first o r d e r contact transformation. The b a s i c Idea of the method of calculating group g e n e r a t o r s i s the s a m e a s the one due to L i e ; for instance, s e e G . W . Bluman and J . D. Cole, Similarity Methods for Differential Equations (Springer, New York, 1974). *For Instance, s e e R . M . M i u r a , E d . t Backlund Transformations, The Inverse Scattering Methods, Solutions, and Their Applications (Springer, New York, 1976). *A method of c a s t i n g Euler— Lagrange equations into H a m i l t o n ' s canonical form Is well known for the c a s e w h e r e Lagranglan d e n s i t i e s involve no d e r i v a t i v e s of fields whose o r d e r s a r e higher than one. T h e c a s e w h e r e Lagrangian d e n s i t i e s depend on higher d e r i v a t i v e s has been studied by T . S. Chang; P r o c . Cambridge PhUos. Soc. 42, 132 0 945); 44, 76 0948). • D . R . P e t e r s o n , M . S . t h e s i s . U n i v e r s i t y of the Pacific, 1976 (unpublished). ' R . Abraham Bad J . E . M a r s d e n , Foundations of Mechanics (Benjamin, New York, 1967). • T h e s e o p e r a t o r s , a s they a p p e a r , do not g e n e r a t e t r a n s f o r m a t i o n s In Independent v a r i a b l e s x . However, such t r a n s f o r mations a r e contained in them in dtsgutse as stated in the p r e v i o u s p a p e r ( L e m m a 2 in Ref. 2 above], and we a r e not excluding any of such t r a n s f o r m a t i o n s . See the e x a m p l e in S e c . Ill for instance. *H.H. Johnson, P r o c . A m . Math. Soc. 15, 433, 675 0 9 6 4 ) . " N . H . Ibragimov, Dokl. Akad. Nauk SSSR 230, 26 0 976). u F o r Instance, s e e L . V . Ovsjanlkov, " G r o u p p r o p e r t i e s of differential e q u a t i o n s , " Ozdat. S i b i r s k . Otdel. Akad. Nauk SSSR, Novostvlrsk, 1962 (In R u s s i a n ) . " G . W . L a m b , J r . , P h y s . Lett. A S2 , 251 0 9 7 0 ) . " H . Steudel, Ann. d e r P h y s . 32, 205 0 9 7 5 ) . u Note added in proof: Recently, It has been shown that many of the t i m e evolution equations which a r e solvable by an Inv e r s e s c a t t e r i n g method a r e w r i t t e n In Hamilton's canonical form; Y. Kodama, P r o g . T h e o r . P h y s . 54, 669 0 975); H . Flaschka and A . C , Newell, In Dynamical Systems: Theory and Applications, edited by J . M o s e r S p r i n g e r , New York, 1975).  x,  LIST OF PUBLICATIONS 1.  C. Wulfman and S. Kumei, "Highly Polarizable Singlet Excited States of Alkenes", Science 172_ 1 9 6 1  2.  (1971)  R.L. Anderson, S. Kumei and C.E. Wulfman, "Generalization of the Concept of Invariance of Differential Equations:  Results of  Application to some Schrodinger Equations", Phy. Rev. Lett. 28^ 988 (1972) 3.  R.L. Anderson, S. Kumei and C.E. Wulfman, "Invariants of equations of wave mechanics I", Rev. Mex. Fis. 21_ 1 (1972)  4.  R.L. Anderson, S. Kumei and C.E. Wulfman, "Invariants of equations of wave mechanics II", Rev. Mex. Fis. 21_ 35 (1972)  5.  R.L. Anderson, S. Kumei and C.E. Wulfman, "Invariants of Equations of wave mechanics III", J. Math. Phys. 14 1527 (1973)  6.  C.E. Wulfman and S. Kumei, "A simple 0 (4,2) approximation for hydrogenic Coulomb integrals", Chem. Phys. Lett. 23 367 (1973)  7.  S. Kumei, "Atomic radial integrals and a use of 0(2,1) transformations", Phys. Rev. A £ 2309 (1974)  8.  S. Kumei, "Invariance transformations, invariance group transformations and invariance groups of the sine-Gordon equation", J. Math. Phys. 16^ 2451 (1975)  9.  S. Kumei, "Group theoretic aspects of conservation laws of nonlinear time evolution equations", in Group Theoretical Methods in Physics, Proceedings of Fifth International Colloquim, Ed. B. Kolman and R.T.  i  Jjharp, Academic Press (1977)  10.  S. Kumei, "Group theoretic aspects of conservation laws of nonlinear dispersive waves: KdV type equations and nonlinear Schrodinger equations", J. Math.  11.  Phys. 18_ 256 (1977)  S. Kumei, "On the relationship between conservation laws and invariance groups of nonlinear field equations in Hamilton's canonical form", J. Math. Phys. j_9 195 (1978)  12.  G. Bluman and S. Kumei, "On the remarkable nonlinear differential equation 2-fa(u + b)" 2 u1 - u = 0", J. Math. Phys. 21 1019 (1980) ®X LX t  13.  S. Kumei and G. Bluman,"When a system of nonlinear differential is equivalent to a system of linear differential equations", Technical Report, Institute of Applied Mathematics and Statistics, University of British Columbia (March 1?81)_  

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