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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 P w ^ . , 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. The equation (u -u =0 is studied and it is determined n K xJ xx yy 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 X X u {Vy-wx=0, vy-avw-bv-cw=0}, equations of a fluid flow {w^+vx=0, w -v~"'"wP=0} and the Liouville equation u =eu. x J H 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. I am particularly grateful to him for introducing me to the wondrous world of.symmetries.. TABLE OF CONTENTS page INTRODUCTION 1 I. CONTACT TRANSFORMATIONS. 15 1.1 Contact transformations. 15 1.2 Invertible contact transformations. 17 1.3 Infinitesimal contact transformations. 20 1.4 Properties of generators. 23 1.5 Invariance contact transformations. 26 1.6 Invariance contact transformations of 29 differential equations. 1.7 Invariance groups of linear equations. 30 1.8 Contact transformations of the spaces y,X. 37 II. INVERTIBLE MAPPINGS OF NONLINEAR SYSTEMS TO 41 LINEAR SYSTEMS. 2.1 Theorems on the existence of invertible '. ^ 42 mappings. 2.2 Remarks on the use of theorems. 50 2.3 Examples. 5 2 2 A. The equation z + %(z ) - z = 0. r2 xx x y J  ^  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 j 7 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. 71 B. An equation of a fluid flow. 75 3.3 The Liouville equation. 79 3.4 A series of Lie-Backlund generators and the 81 linearization. IV. 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 84 sequences. 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: 101 a determination of an invariance group of the 2 equation (z ) -4z z =0. n K xyJ x y 4. On the remarkable nonlinear diffusion equation. ,105 5. Invariance transformations, invariance group 110 transformations and invariance groups of the sine-Gordon equation. 6. Group theoretic aspects of conservation laws of 118 nonlinear dispersive waves: KdV type - equations and nonlinear Schroedinger equations, 7. On the relationship between conservation laws 127 and invariance groups of nonlinear field equations in Hamilton's canonical form. 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), z'=z' (x,z,z,...,z), z ' =z ' (x,z,z,...,z) 1 P 1 q . 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). If we set k z = u, z = u, k=l,2,3,..., (0-2) k k then the set w [u] consisting of points (x,z,z z,...) = (x,u,u,u, . . .) , x R (0.3) 1 2 1 2 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) 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 (0.4) for x and introducing it into the rest of (0.4) we obtain z', z1 as functions of k x* : z 1 \ w[u|]: (x,u,u) Z ' =U ' (X ' ) , Z ' = V C k ; ) (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=l,2,3,... , k k (0.6) 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','...) , x'tR. 1 2 1 2 It is intuitively clear that in contact transformations z' is related to x' and z' k because z' must behave as k the krth derivative of z' with respect to x' when (0.3) is introduced. Obtaining an explicit form of z' in terms k of x' and z' from the condition z 1 w[u] : (x,u,u) w[u'] : (x'.u'ju') Fig, 2 that (0.2) yields (0.6) is not only messy but also becomes • confusing for vector x and vector z. It is very convenient to replace lq.(0.2) and Eq.(0.6) by equivalent differential forms. Eq.(0.2) implies dz = udx, dz = udx, dz = udx, 1 1 2 2 3 (0.7) and if we use (0.2) it can be written as dz = zdx, dz = zdx, dz = zdx, 1 1 2 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, 1 1 1 1 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}. (0.9) k k+1 k k+1 + 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) , (0.10) k k l k 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 toa 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+...) - z'(x'dx+x'dz+x'dz+...) , 1 v x z zi J i x z z T J , . . . . 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 z 2 l x Z! z 2 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), z' = z'(x,z,z), z' = z ' (x, z,z,...,z) . (0.11) 1 1 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 trans-formations 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) (0.12) 1 5 z' = z + er(x,z,z,...,z ), k=l,2,3,... k k k 1 q v where 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 trans-formations have this property. will be called higher order infinitesimal contact trans-formations . 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 equation of a flow curve originating at a point (x,z,z,...) be z 1 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 , . . . j k k ' '1 , ... , z X Fig.3 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, (0.14) where T(a) is generally a nonlinear operator acting on to. 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, (0.16) where I represents the identity transformation. The explicit form of the operator T(a) is given by OO J| T(a) = e a £ = I * £n, (0.17) n=0 where I is defined by £ = C8x + ?3Z + •+ C3Z + . . . , (0.18) 1 1 2 2 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 trans-formation 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, (0.19) 1 n where f is an analytic function Rn+^->R. 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), x£R. (0.20) I n I n The set [u] defines a curve (generally a manifold) in the Sp£LC6 (x j Z } Z , , , , , z) , Let us demand that the curve fn) 1 n w [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). We now suppose that the Fie.4 z=u contact transformation (0.1) maps the hypersurface f=0 into a hypersurface f' ( x \ z \ z \ . 1 .,z') = 0, n (0.22) 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 . (0.24) . 1 n In other words, a contact trans-formation 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. Parti-cularly important invariance contact transformations are those which form groups in the Fig. 6 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 +uu +u =0, XXX X L 2 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 ] . These findings, lead to a general theorem [Appendix 7 ] that with any conservation law, admitted by a Hamiltonian system, + Appropriate references are given in the appendices. tt 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 of the two systems have the same structure. 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 sum-mation 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 , . . . ) 1 2 n where x = (x^ ,X2 , • • • e z = > > • • • > e a n d Z..R™ consists , of- coordinates zV . . with v = 1,2,...,N 1 2 n and i, = 1,2,...,M. For instance, r 1 1 N N. f 1 1 1 N . z i_z1,z2, . . . , z M_ 1, zmj , z L z -j. 1' z12 ' Z21' ' ' " ' ZMMJ • w'-11-' denotes the space with coordinates = (x , z,j , . . . ,z) , where = (x,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 x' = x ' (x, z , z , . . .) , z1 = z'(x, z , z , . . .) , 1 1 (1 .1) k' = ^ ' ' z ' |' ' ° ° ' k = 1> 2 > 3 , . . . where x-' e y, z , v e y, z-'v- . ey. We write (1.1) as l i 1i 2...i k 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 dzv - zVdxi = 0 j v v _ dz: . - z-1 ^  1 n t • ill 1 1 2 k (1.3) 1x 2...i kj d xj 1 y 3 y • • • is called a' contact condition. We express the set (1.3) simply by {dz - z-dx- = 0}. k k 1 1 Definition 1. The transformation (1,1) is a contact trans-formation only if it preserves the contact condition, i.e., {dz - z.dx. = 0} + {dz' - z!dx! = 0}, (1.4) k k 1 1 k k 1 1 Let u(x) be a function + We consider a space w[u]Cw consisting of points 03 = (x,U,U,U,,,,,U,,.,), X € R " 1 2 k where, as z, u is a vector with components k k uV . . = 3 3 3 uv(x) , 1112." ' ,:Lk x i i x i 2 ' ' ' x i k 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 o)'^^. = Given go'^, one can determine n>0, from the condition (1.4), A contact transformation is said to be invertible, or 1-1,~ if there exists a- space-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 w^^ , x' = x'(x,z,z), z' = z'(x,z,z), z' = z'(x,z,z), (1.5) 1 1 1 1 1 x'rw^ 1^ R M, z'rw^ 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 3 z' + z- 3 z1 = z!(3 x» + z-3 x!) (1.7) 'i x, i z 3 Z"J or from 3 z' = z!3 i! (1.8) i J z ± J and p(x,z,z) from p = 3zz' - z!3zx! = 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 - 1 0 ) 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), z' = z'(x,z), (1.11) R^, R^. The transformations for z, n>0, n are determined from (.1.4). 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 " J u' = aj'(w) = Too, to = co(co') = T_1co'. (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=0 Defining an operator «> n = I • ( 1 - 1 7 ) £ = + CV9 v + 'x. T ' u z v ' ,»i0zV + 1 I with ^ = (3 x!) n, CV = (3 z'V) = (3 £.'V) ^ a iJa = 0' s v a Ja = 0' v a l Ja = 0 , we write (1.17) as co' = (1 + a£)w + 0(a2) (1-18) 2 2 where 0(a ) represents terms of order a . 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 x k = 9 x k + zkazV-+ zik3zV + ••• • Now instead of starting from a transformation of the form (1.15), we start with an operator A = + ? V9 zv + ?i9zV + ••• > CI - 20) i i where E,1, £Vey, and <;V , . , . . . are determined recursively from i 13 (1.19). We consider a group transformation 00 ft a£ r ct e w = 2, ^T U ) n! (1.21) Transformation (1.21) always satisfies the contact condition (1.4). It was previously shown [Appendix 5] that a trans-formation 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 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 a) Point groups: = £x(x,z), £V = £V(x,z) b) Lie contact transformation groups: z is scalar and (1.22) there exists a function R such that C 1 = 3Z.G, C i (1.23) 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 1^1-' -»- 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 . .] zv J J 1... J (1.24) Let A be the space of all generators a. 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 = + £V3 v . X * z 1 The following property is very useful [in Appendix 6]: Proposition 2. Generators of the formZ = C v3 zv commute with the total derivative operator D i Proof. Noting that D eX, we have 1 U,D ] = U"l)3 + (lzV - D x C V)3 z V - (D Cv- D x Cv)3zv= 0. • i i i i i We introduce into the space X an equivalence relation by: Definition 2. Two generators I and are equivalent if and only if (SL - & 1 ) g | q = 0 for any gey. The equivalence is represented by Z = 8,'. The symbol I g = g indicates the evaluation of the quantity for those values of w =(x,z,z,...) which satisfy the equations g = 0, D x g = 0, D D g = 0, ... . (1.25) i i j The equivalence class N  of OtX consists of generators A = C XD X , 5 l e Y-i (1.26) Proposition 5. A generator £ = + £ v9 7v is equivalent to £ = U v - zV?1)3zV. Proof. Obviously, £ - £ = + zV^13 From (1.19) we find its full expression to be £ - £ = £X(3 + zYa V+ zV.3 + ...) = . • X- 1 Z 1 ZV J ^ X-1 J 1 As a result, elements of the quotient space XA. can be represented by generators of the form £ = 0 V ( w ) 3 z V , v ^ 6 ey. The generators X and £ satisfy the same commutation relation. Namely, /\ /N Proposition 4. = ^3 ^^ a n d o nly if Proof. From (1.24) we have [£1,£2] = (£152 " ( J ll C2 " A 2 ? 1 ) 3 Z v = V Therefore, = { ( l ^ - £2C^) - z ^ O ^ - £2^J)}3 On, the other hand, we have z^ A /S j J 3 zV l l J J The converse is obvious. • 1.5 Invariance contact transformations. A set ,Q,f K C. functions f V (to ) = fV(x,z,z, . . .z) , v = l,2,...,K, (1.27) I n 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 1 2 K 1 2 K y ,y ,...,y , for which the Jacobian of (f ,f ,...,f ) is nonzero: ' ' ' Q in D (w) . (1.28) D C y 1 ^ 2 , ... . ,yK) We denote by D(w;f=0) the set of points to^ n-' satisfying the equations f^(to n^-^)=0, v = l,2,...,K. K equations . ... ..••'.. fV(to(n)) = 0, 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 {fV} 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^ ,y2 , . . . ,yK, with the property that the functions f V all vanish with the substitutions y V = 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 y v = + 4>V (x) , v = l, 2 , . . . ,K, (1.31) where Av is a linear operator defined by y A?zv = (a^(x) - aix;i(x)DY +....+ a ^ Z - ^ f x J D , D y . . .D }zy, i n (1.32) tor. .matrix A = 1^1' w e sometimes write (1.31) in the form y = Az + cf> (x) (1. 33) y y y x. y x. x. x. i ii i 2 i n a^ J :R (xj+R, <j> :R (x).->-Rt Defining an operator. ,matrijc where y,z and (p are column vectors. We consider a system of equations f = 0, D f = 0, D D f = 0, ... (1.34) i i j where f=0 is the independent system (1.29)'. 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 {fv(w)=0} and its explicit form {yv=<j>v (co)} admit the same invariance group, generators. Lie gave' an algorithm [-6,7,23] to determine invariance group generators = + £V(co'-n-')3 v satisfying the x i z condition (1.36) for given m and n. Clearly we have Proposition 6. If and I is an invariance group generator of the system f~0, then 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 ) 3 z 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 M M N co - (x ,u ,u ,u, . . . ) , x eR , u:R (x)-tR . When u is a solution of a system of differential equations, - id. -£ (x ,u ,u ,u, . . . ,u) = 0, (1.37) 1 2 n 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) , (1.39) 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 trans-formations of linear equations bear special properties. We consider two cases separately: i) SL = 0 V(x)3 z V, 0V:RM(x)+R ii) £ = e v ( w ^ ) 3 z V , e V: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=0V(X)3£V. Namely, Proposition 7. A system of linear equations ^Z^- $V(.X) = 0, v=l,2,...,K, K<N, $V:R (X)->-R, with linear operator A. defined by . j4 = {AV (X) • + AV:1.-(X)DV + . . . + AV1l12:,-Jin(x.).Dv Dv; • • • }Z;J y y - y x^ y A i n ' (1.40) Avi.••j:RM(x)^R, admits a generator L=Uy(X)3zy depending upon an arbitrary solution {Uv(X);v=l,2,...,N} of the system of linear differential equations ^Uy(X)=0, i.e., AV(X)Uy + Avi(X)uV + ... + A v il i2- •-^(X)^ . • = 0, y y 1 y 1 1 ii^2 • • ^ n (1.41) v = 1, 2 , . . . , K. Here and in the following, U y • , = 9 3 ...3 Uy(X) . 13 • • .K X-j^  Xj Xj, Let Z=(l+cL)Z, c constant and L=Uy (X) S^y def ined.-.above . It is easy to see that if 4 vZ y-$ v(X) =0, then the Z, a super-position of cU(X)' with Z, satisfies the equation 4^zy-§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 vU y (X) = 0. U J It is clear from Proposition 5 that: Proposition 8. A linear system Fv(ft^)=0, v=l,2,...,K, K<N, admits a superposition generator L=Uy(X)32y of its explicit form Y v = A^ ZW + $V(X) . (1.42) ii) The second case. In this subsection, we restrict our-selves to a linear system Fv(f2^n^)=0 whose explicit form is linear homogeneous, i.e. 4^Zy=0. In this case, two types of generators can arise: • "" . . a) 0 V is linear in Z,Z,...,Z, i.e., 0 V=B vZ y+Y V(X) 1 m y for some linear operator B . b) 0 V is nonlinear in Z,Z,...,Z. 1 m The generators will be called linear generators and non-linear 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. For instance, the wave equation Z ^ - 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, ^ defined by (1.40), admits a nonlinear generator ' 0 V (ft ) 3 iff the system admits a linear generator of the form L = {BV(X)Zy + BV1(X)ZV + u K J u v 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 = {3ZV .0v^(m))>lz=u,z=u,...,z=u' t 1 - 4 4 ) 1 • • • J 1.1 m m and U:R^(X)->-R^(Z) is an arbitrary solution.of a system of differential equations 4 vU 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 homo-geneous system, we may restrict ourselves only to linear generators of the form L = (B^Zy + yV)8zv (1.45) where B* = B^(X) + B^1(X)D + ... + B^ 1l i2--- im ( X)D D ...D , y y y y Ai i Ai i X1 ±2 .J-m BV1. . • j : R M( X)+R, and ¥V:RM(X)-*R. To illustrate a use of the theorem we consider the following example. (1.46) Example. Consider the equation Z XX z YY 0 . (1.47) This equation admits a linear generator L ='{g(Ux+UY)-(Zx+ZY)}3z X Y (1.48) 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(Ux+UY), BY(X,Y) = g(Ux+Uy). (1.49) Comparing (1.49) with (1.44), we look for a function 0 satisfying Clearly, we have 0=G(Zx+Zy), G being an arbitrary function of Zx+Zy, and consequently Eq(1.47) admits a nonlinear generator L=G(ZY+ZY)37. 0 Z ez = g(zx+zY). (1.50) We state a few basic properties of the linear generators. Their proofs are given in Appendix 2. Let B be an operator matrix B . Proposition 9. If L=(£^Zy)3zV, L=(B^Zy)3zv and £=(B^Zy)3zV satisfy the commutation relation [L,L]=l!, then [B, Proposition 10. If L=(B^Zy)9zvand L=(B^Zy)3zV are generators of invariance groups of a linear homogeneous system 4^Zy=0, then so is L= ( S V3 KZ y) 3 7v • Let A be an operator matrix A =\|. Proposition 11. L=(5^Zy)3zV 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=(£VZy)37v 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)kU(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 VZ 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, . . .) . (1.51) 1 2 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 T f O ^ ) = f(T"W n ; )). (1.52) The inverse transformation r^ -y is defined by T_1F(ft(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 = (£Xi)9x_ + (£ZV)9ZV, (1-54) where X,Z are X,Z components of (1.51). Since the trans-formation (1.51) is a contact transformation, if leX, then T£T In view of Poposition 3, we have T£T = (JtZV - zVj?,Xi)9zv. • (1.55) 1 • i x- i Similarly, T _ 1LT = (LzV - zVLxi)3zV. (1.56) We let £ = 6V(to(-m^) 9 v and L = v and consider two Z lu cases, z scalar and z vector, separately. 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) , Z = Z (x ,z,z) , Z. = Z. (x , z,z) , ... . x 1 1 i x x 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 T"1 ; ^rT- 1oC m)^rT" 1o( 1) and T0 (oo ^) 9 T = {etT^ft^^pfT"1^1-1^ } 9 7 , (1.58) Z Li T 10(fi(m;)) 9 7T = { e C T ^ ^ a ^ 1 ) ) }9 , (1.59) where p(T _ 1ft C 1 )) = p(w ( 1 )) l u = - ( n ) and a((.(1)) = { p ( W ( 1 ) ) } _ 1 (1.60) with pfw^1^) = 9 Z - Z-9 X.. v J z 1 z 1 z vector. In this case the most general 1-1 contact trans-formation is the extended point transformation: = Xi(x,z) , Z v = Z v (x, z) , . . . , (1.61) yielding TeV((0(m))9 V T - 1 = {6 y(T" 1n ( m })p V(T- 1ft ( 1))}9 7 V, (1.62) Z ji Lt where = {9 zpZ V - (1.63) and T - V ( ^ m ) ) 8 z V T = {© y(Ta) ( m ))a^ ( 1 ))}^v> U-64) where y V ^ - - ZV a ^ h = {3zyzv - z ^ y V l ^ ^ (1.6S) From these results follows that: Proposition 13. An operator L = 0 V (fl ^  ) 9 z V is an invariance group generator of the equation F(fl^n))=0 iff T~ 1F=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 trans-formations. 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 trans-formations 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 - f (x , z , z , z , . . . , z) = 0, xeRM, zeR, (2.1) 1 2 n 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 } ) ) } 9 z , (2.2) where M 1) U(X') :R.->R is an arbitrary solution of some n-th order linear differential equation I4U-- = '"{A (X') ' + A 1 (X) D V + ... + A1-1 • • •LN(.X).I)V ..P Y }U = . 0 , xi • Xi1-'- x i n and (2.3) 2) X(O)(1')) iw^+R1^ is a component of a Lie contact transformation X = X(EO ( 1)), Z = Z(U) ( I ; )), Z = Z ^ 1 ) ) (2.4) and a(u>C1)) = {p(a)(1))}"1 = (BzZ - Z i9 zX i)' 1. (2.5) The transformation (2.4) maps Eq.(2.1) to a linear equation which has an explicit form j4Z-$(X)=0 with A defined by (2.3). Proof. Suppose that Eq.(2.1) is transformable to a linear equation by an extended Lie contact transformation.w+fl. By Proposition 8, this linear equation admits the superposition generator L=U(X)8Z of the equation 4Z-$(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. In view of Proposition 13, F=0 admits 1 n the generator L=U(X)9Z. Thus, by the invariance condition (1.36), we have LF = F U + F z - U i + ... + F z _ Ui i i ?...i = 0 (2.6) i 1i 12'"" xn 1 2 n for any 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 - A 1F z)U i + (AF - A1^F„)U.. + ... . (2.7) i i j 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 i I j v J-vanish. This is possible only if F has the form G(4Z,X), 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 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 f(x,u,u,...,u) = 0, x e R , u:R (x)+R 1 n ,M M (2 .8) 4U - . $(X) = 0. (2.9) We now turn to a system of nonlinear equations. We have: Theorem 5. A system of K independent n-th order nonlinear equations fV(>(n)) = fV(x,z,z,z,...,z) = 0, v=l,2,...,K, K<N, 1 2 n (2.10) xeRM, 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 = {Uy (X(x,z))crV(x,z,z)}3 , y — zv where 1) UV(X), v=l,2,...,N, is an arbitrary solution of some system of n-th order linear differential equations (2.11) LJP = {AV (-'X) -+ A V 1(X)D Y + .... + A V l 1 - • •1n(X)DY . . . D Y . • }UU .= 0, 11 11 * • y M i X... 1 -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), Z = Z(x,z) (2.13) with inverse transformation x=x(X,Z), z=z(X,Z) and v <J*(x,z,z) = O z y Z V - z V 3 z y x i > 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 A vZ y - 3>V(X) = 0, v = l, 2 , . . . ,K. (2.15) Proof. 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), ... (2.16) 1 1 1 mapping Eq.(2:,l,0) to a linear system (2.15). By Proposition 8 this linear system admits the superposition generator L = U V ( X ) 8 7 V . 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 = Uv(X)9zv (2.17) and Eq.(2.10) into, say, Fv(X,Z,Z,...,Z)=0, v=l,2,...,K. 1 n The system {Fv=0} is solvable in explicit forms for K components of the Z,,Z., . . . ,Z.. Without loss of generality, 1 2 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, v = l, 2 , . . . , K, (2.18) 1 I n where cj>v are independent of Z y, y = l, 2,...,K. According to Proposition 13, the system {F — 0} admits the generator (2.17), and hence so does the system (2.18). Thus, + + *zyui + ••• + ^zv . u? .i = °> (2.19) i 1 - . . . . 1 I n v J 1 n where v=l,2,...,K and cj> ^ y= ^  ^l-1 ^ V ' e t C - - 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 Uy,UV,... appearing in Eq.(2.12). Eq.(2.12) is solvable for uV, y=l,2,...,K. This is seen as follows. Suppose this be not the case, i.e. rank |A^|<K, y,v<K, .We fix X at X=X° and assign to UV(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 + A^uV + ... + A ^ l - ' - W = 0, v = l,2, . . . ,K. (2.20) I ^ /A. U U • ^ ... T /V "U • l y y i y .. . 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 VZ y + AvizV + ... + A^l-^nzV . + (X) y y i y xi•••xn /s 1 where A^ =0 for y<K, and consequently the explicit form (2.18)of the transformed system {Fv=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 fv(x,u,u,...,u) = 0, v=l,2,...,K, K<N, (2.22) 1 n - so. x 6R 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 AVU y - $V(X) = 0, v=l,2,...,K. (2.23) 2.2 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 . (2.24) 1 z 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. At this point, Z still admits functional arbitrariness. From Z and X we determine Z • using conditions (1.7) or (1.8).- "Next-we. use ^ -1 (2.5) for the Known p=a to limit the arbitrariness m Z. The resulting transformation X=X, Z=Z, Z=Z maps the non-linear 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 £ = U V(x,z) - z^ i(x,z)}8 z V. (2 By Proposition 3, (2.25) is equivalent to a generator of a point group £ - 5^x^)3 + CV(x,z)3 v . i z If there exists a mapping to a linear system, we can find the functions X(x,z), av(x,z,z) by comparing the resulting ^ 1 invariance group generators (2.25) with (2.11). The functi ZV(x,z) are to be determined from these functions using equations (2.14). Eq.(2.12) is found on determining the invariance group. 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. 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 fv(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 To illustrate the use of our theorems, we consider z =p, z =q, z =r, z =s, z =t. x ^' yi xx ' xy ' yy 2 A. The equation z + h{z ) z y o. We consider the equation t f z y o. (2.26) t This is an integrated form of Burgers' equation z +zz -z =0 which will be discussed in 3.1. xx x y The generator (2.24) is now i=Q(x,y,z,z^,z )3z. Applying Lie's algorithm, we find that Eq.(2.26) admits 2 2 = (y z + yxz + %x +y) 3 , = (yz + Jgxz )3 , 1 y 7 x 3 J z' 2 w y • x' z' JU = z 3 , = z 3 , £ = 3 , (2.27) 3 y z ' 4 x z ' 5 z' ^ ^ - (yzx + x)3z, ij = U(x,y)e_JsZ3z, where,:.in , U(x,y) is an arbitrary solution of the heat equation U ^ •• IP =. 0 . This indicates that Eq.(2.26) is equivalent to a linear equation. To find the mapping, we - 3"Z compare 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). Thus, the mapping is a point transformation. From (2.5) and v p=(a) =e2 , we find that 3 zZ=e 2 . The mapping is then X = x, Y = y, Z = 2e + h(x,y), (2.28) where h(x,y) is an arbitrary function of x and y. 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 = alx(w,v)wx+ aiy(w,v)wy+ blx(w,v)vx+ biy(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 vy~ vx wy^®, the system (2.33) admits a generator of the form I = -{U1(w,v)wx+ U2(w,v)wy}3^ - {U1(w,v)vx+ U2(w,v)v^}3v 1 2 where {U (w,v),U (w,v)} is an arbitrary solution of the system of linear differential equations fi.r t r T _V\ fu lrlll 4- n ^ ^  \j j kj — \j — (2. biy(w,v)U^ - aly(w,v)U^ - blx(w,v)U2 + aix(w,v)U2 = 0, i = l,2 where UX=3 U 1, U 1 = 3 U 1. w w ' V V Proof. Since D fx=D f 1 =0, we find that x y ' - Af x = w U 1fa 1 Xw + a i y w ) + v U 2(b 1 Xv + b l y v ) x w ^  x y y v v x yJ + w U 2(a l xw + a l y w ) + v U 1(b l xv + b i y v ) y w v x yJ x v v x yJ+ v U1(blxw. + b i y w ) + w U 2(a l xv + a i y v ) x w^ x yJ y v x yJ + v U 2(b 1 Xw + b i y w ) + w U^(alxv + a l y v ) y w^ x yJ X' v v x yJ Using Eq.(2.33) in the first two rows of this expression, we find that „ ri i T r,iyTtl iyT,l ,ixTT2 , ixTT2. if .c r n = J-(b - a 7U - b U + a U ) I£^=£2=0 v W V w VJwhich 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 . Clearly X=w, Y=v, af=-w,;, ai=-w , a 2 = -v , a 2 = -v . The 3 ' 1- x' 2 y 1 x' 2 y definition of a v leads to 9ljrx=l, 9,,x=0, 9,^=0, 9„y=l. Thus y W ' ¥ ' m3 ' V' we have a solution x=W, y=V. 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 / Z y = biy(X,Y)Wx - aly(X,Y)WY - blx(X,Y)Vx + alx(X,Y)VY.,= 0, (2.37) where Z 1 =W, Z2=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, (2.38) where p,q,r,s and t denote the variables defined at the 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)Uqq - 2b(p,q)Upq + c(p,q)Upp = 0, (2.39) where U =(9 ) 2U, U =9 9 U, U =(9 )2U. pp p pq p q qq q Proof. Introducing w=p, v=q, , v =t, w^=vx=s, we write Eq.(2.38) as a system a(w,v)wx + b(w,v)(w + v x) + c(w,v)vy = 0 (2.40) w - v = 0. (2.41) y x 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 (2.42) " U w = (2-43) Eq.(_2.43) allows us to introduce a function U(w,v) with a property U 1 = 8 U E U , U 2 = 3 U E U , (2.44) iir T»r  7 "\t "\r ' v J and then Eq.(2.42) takes the form aU - 2bU + cU =0. (2.45) VV WV WW J Introducing (2.44) into (2.34) and using (2.41), we find that i = (DxU)8w + (DyU)3v. Recalling that w=p=zx 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., Z + pZ = 0, Z + qZ = 0. x r z ' y n z The solution is Z=Z(p,q,a), a=-z+px+qy. From (1.8), we get Z + xZ = P, Z + yZ = Q p a ' q 7 a xwith P=ZX and Q=Zy 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] : X, Q = y. (2.46) 0 (2.47) 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 (Dxa)(Dy3) - (D a)(Dx3) = f. (2.50) X = p, Y = q, Z = -z + px + q'y, P This transformation maps Eq.(2.38) to AZ E a(X,Y)T - 2b(X,Y)S + c(X,Y)R = where T = ZyY> ^ = a n <^ D. Lie's Theorem on the Monge-Ampere equation, 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 = a 1, • ' Y = a 2, P = 3 1, 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 I1(a1,31)=0, i=l,2, admits two invariance group generators = a(co(:i))I1(a1,31)3z, i = l,2, (2.52) where a ^ = p = = [a2,3^]-Proof. It is easy to check that the equation Z^Y^ admits 1 2 generators (X,P)9Z and L£ = I (Y,Q)9Z with arbitrary 1 2 functions I and I . On the other hand, according to Lie's theorem above, there exists a Lie contact transformation mapping the Monge-Amp&re equation in question to ZXy=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). However, they contain (2.2) as a 1 1 1 special case. To see this we choose special forms I -I (a ), I 2=I 2(a 2) and let i=Z1+l2. Then from (2.52) we obtain I = a{I1(a1) + I 2(a 2)}9 z = a-U(a1,a2)8 . (2.53) Observing that U(X,Y)=11(X)+12(Y) is the general solution of the equation UXy=0> w e s e e that indeed the Monge-Amp&re equation in question admits a generator of the form (2.2) 1 fl~) - 2 fll with X=a (eov J ) and Y=a (to ^  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 (z ) az - z =0. - A xx YL A special Monge-Ampere equation of the form g(z )z - z = 0 , g:R+R, (2.54) xJ xx yy ' s ' K Jarises 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 = (z ) az - z = 0, a real, (2.55) v x^ xx yy ' ' v 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)yq2 - 4y/ p a + 1dp}9 z = { (a + 4)xp + (3a + 4)yq - az}8z, (2.56) = (z + %ayq)9z, = y9z, £ 5 = U(p,q)3z. (2) a = -2: l^, ^3' £4> a s above and I 6 = (2.57) (3) a = -4: J^  = (xp -yq)3z>. = U(P^) 9 Z' £3 = {pl1(p"1-q,(p"1-q)y+z)}9z, (2.58) = {pl2(p"1+q,(p"1+q)y-z))9z. (4) a = 0: £ = z3 I = h(x,y)9 1 z z z (2.59) £ 3 = I1(x+y,p+q)3z, £ 4 = I2(x-y,p-q)3z. In all cases, the function U(p,q) represents an arbitrary solution of the differential equation U - (p)aU = 0 (2.60) pp qq v 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 )"4z - z = 0. (2.61) v x^ xx yy Comparing (2.52) and the generators 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 = -2p_1(z - px -qy), (2.62) P = -(p - 1 -q)y -z, Q = (p _ 1 + q)y - z. The inverse transformation is x = %Z - % (X + Y) (P + Q) , y = -(X + Y) - 1(P - Q) z = -(X+Y)_1(YP+XQ), p = 2(X+Y)_1, q = %(-X+Y). (2.63) If we introduce the general solution of U X y = 0 , U=F(X)+G(Y), where F and G are arbitrary functions, into (2.63) by Z=U, P=U and Q=Uy, then we obtain a parametric representation X 1 of the general solution of the differential equation (u ) u - u = 0. x^ xx yy (2.64) 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{YFt(X)+XG'(Y)}, (2.65) where F' and G' denote derivatives. We note in passing that Eq.(2.61) is also trans-formable by the Legendre transformation (2.46) to ZXX " 00 " \ Y = ( 2 - 6 6 ) and in turn Eq.(2.66) can be mapped into by a composition of the transformations (2.46) and (2.62). Explicitly, the transformation x = 2(X+Y)_1, y = %(-X+Y), z = (X+Y)"1Z - 4 maps the equation - (x) z ^ = 0 to the equation Remark 2. Let Eq.(2.55) be written as D (z ) a + 1 - D (a+1)z = 0. (2.67) x\ xJ y y r\j If we introduce a potential z by ( z x ) a + 1 = z y, (a+1)Zy = z x, (2.68) z satisfies the equation (z f z -z = 0, 3=-a(a+l)_1. (2.69) K yJ yy xx ' K  J  K JThe * transformation (2.68) may be viewed as a Backlund + ) transformation between Eq.(2.67) and Eq.(2.69). For a=-2, the transformation (2.68) becomes an auto-Backlund transformation: (z )"2Z - z = 0 , ( 2 - 6 8 ) , (2 ) ~ 2z = 0 . (2.70) v x^ xx yy K yJ yy xx K J For a=-4, Eq. (2.69)takes the form z - z = 0 (2.71) v Y yy xx 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 7 g = yg""" + vgT , y,v constants. X The case y=0 yields Eq.(2.61) and the case v=0 leads to Eq.(2.71) t) Consider a system (g =0}. If this k £ has the property that for any solution of a system {Fy(x,z,...,z)=0} the corresponding z,z,z,... satisfying m 1 2 {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 + z z - z = 0. (3.1) xx x y 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] ~ ' -2'Zy x = X, v - Y, z = , (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(ZDX - 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 Zxx + - z = 0. (3.5) One then sa-ys'"by inspection" that the transformation 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 trans-formation (3.6) with c=% by applying Theorem 2. The second example is a nonlinear diffusion equation f = D (z~2z ) - z =0. (3.7) This equation admits a four parameter point Lie group [ 6] with no generator of the form (2.2). As in the above we let z=z and instead of Eq.(3.7) we consider an X integrated form f = (z )~2z - z = 0. (3.8) v x^ xx y ^ J This equation admits [ 23] seven generators of point trans-formations including the generator S. = U(z,y)zx9z, (3.9) involving an arbitrary solution of U z z-U =0. Comparing (3.9) with (2.2), we find a point transformation X = z , Y = y, Z = .x (3.10) which maps Eq.(3.8) to the heat equation AZ = Z^-Zy = 0 From (3.10) it follows that z=z =(ZV) ^ and one can X A verify that the transformation x = Z, y = Y, z = (Z x) - 1 (3.11) transforms Eq.(3.7) to f = Zx 3C ZX DX " 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 , . . . ,zx,z) - D yz = 0. (3.13) For such an equation we can always introduce a potential z by 2 = z x, g = z . (3.14) The equation governing z is ^ zx...xx"--' zxx' z^ - Zy = C 3 - 1 5 ) 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. We consider a system v - w = 0 7 X (3.16) v = avw + bv + cw, y where a,b and c are constants. Equations of this form arise in physical problems. For instance an equation governing a fluid flow through a reacting medium I 46,45] w + v + c w = 0 y y x (3.17) Vy = k^(a - w)v - k2w(b - v) and an equation describing a two wave interaction [47-49], v + C-.V = -avw - bv - cw y i x w + 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 . v, cv v bw. , 7 (x,y,v,w) v- (-' {-' — ) , (3.19) F a a we obtain v - w = 0 (3.20a) y x ^ • J v y = v w + v + w . (3.20b) This equation admits only a trivial invariance point group generated by A- = 3 , £ 0 = 3 , £ _ = x3 -y3 - (v+l)3 + (w+1)3 . (3.21) 1 x' 2 y' 3 x 7 y K v ^ ^ w ^ J However, if we introduce a potential z by v = z , w = z , (3.22) x' y' y. J then the corresponding equation for z, i.e. z - z z - z - z = 0 (3.23) xy x y x y 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. = 3 , £ 0 = 3 , 2, = x3 - yd - (x-y) 3 , 1 x' 2 y' 3 x J y v- 1J z ' (3.24) 3 4 = 3 z > = U(x,y)eZ3z, where U(x,y) is an arbitrary solution of the differential equation uxy - U x - U y = 0. (3.25) Now applying Theorem 2 to we find a transformation X = x, Y = y , ^ Z = e"Z, (3.26) + ) which maps Eq.(3.23) to J ZXY " Z x - Z y = 0. (3.27) The transformation connecting Eq.(3.27) to Eq.(3.20) is then Z z x = X, y = Y, v = - -JL t w = - I. (3.28) It is known that Eq.(3.17) and Eq.(3.18) admit trans-formations 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 zXY=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 and a r e unchanged. The first extension of takes the form *3 = X 3 x ' y 3 y " Cx-y)3? - + t V 1 ) 3 z y - C 3 ' 2 9 ) Clearly, this corresponds to (3.21). The first extension of Z^ is £4 = 3 Z + 0-9z + O O z , (3.30) x y and hence = 0 in the (x,y,v,w,) system. For the generator we have lr = UeZ3 + (U +Uz )eZ3 + (U +Uz )ez3 . (3.31) D z x x z x y y Zy Because of the appearance of z in the coefficients of 3 S3 and 3 =3 , the first extended part of (3.31) can V 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 = / zxdx + Zydy =. / vdx + wdy. (3.3 2) Then the first extended part of (3.31) can be written as i s = {(Ux+Uv)e x /vdx+wdy}3 + { ( u + U w ) e / v d x + w d y } 9 _ v 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. The same can be said for the preceding two examples. For instance the generator £=U-(;x ,y).e 2 d • o t". I • q .(3.5) b e c o me s , via z = an integral dependent generator, 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 £ = {(U -%Uz)e-is/2dx}9 A z (3.34) a,3 real (3.35) w v aw' 6 = 0 X which describes a fluid flow through a long pipe-line. The system (3.35) was found to admit an extra generator when a =-1: w + v = 0 y x -1 "3 n w - v w = 0 . X (3.36) The extra generator is A = gO,y)9 x + {w"33wg(w,y)}3v, (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 = (w g)3 + (v g - w 3 g)3 . (3.39) v x°J  w X s w6-^ V 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 U1=U1(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 = z x, v = -zy, (3.40) we write the second equation of (3.36), as (z ) 3z + (z ) _ 1 = 0 (3.41) v xJ  XX v Y The generator (3.39) can be written in terms of x,y and z as 1 = ( D x U ) 8 w " ( D y U p V = ^ D x U ) 3 z x + ( D y U ) 9 z y ' ( 3 ' 4 2 ) where U=U(z ,y) is ••••def ined~ by X U(zx,y) = / * g(s,y)ds. (3.43) From (3.38) and (3.43), we obtain an equation for U: (zx)"'eOz ) 2U - 3 U = 0. (3.44) x y Noticing that (3.42) is the first extended part of the generator £ = U(zx,y)3z, (3.45) we expect that Eq.(3.41) admits the generator (3.45) subject to the _ condition (3.44). A direct calculation verifies 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 -XZX, z x - X, z y Zy> (3.46) which transforms Eq; (3 . 41) * to a : linear equation • X ZXX " ZY = (3.47) Let U(X,Y) be a solution of the differential equation x;3Uxx - UY = 0. (3.48) Then, from the first three relations in (3.46), we obtain an implicit solution of Eq.(3.41): x = -UX(X,Y), y = Y, z = U(X,Y) - XUX(X,Y), (3.49) and the rest of (3.46) leads to w = X, v = -Uy(X,Y), (3.50) which together Eq. (3.36). To with (3.49) defines an implicit obtain an explicit solution, we solution of 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 z x y - e Z = 0. (3.52) One of the generators admitted by this equation is £ = {f(x)zx + g(y)zy + fx(x) + gy(y)}9Z, (3.53) where f(x) and g(y) are arbitrary functions and f and g x y are their derivatives. 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)ux + g (y)Uy + fx(x) + gy(y) = 0 u u - e = 0. xy (3.54) The solution is found to be u = £n 2<j> il) O + M (3.55) -1 -1 where (J) (x) = Jf dx and ip (y) = Jg dy. This is the general solution of the Liouville equation [37]. Introducing. U=cp+ip, we write (3.55) as u = £nI2U U U 1 x y (3.56) Recognizing U as the general solution of the equation U x y=0, we conclude that the transformation z = ,£n|2Z Z Z 1 x y (3.57) maps a solution of Z =0 to a solution of z - e = 0. r xy xy One can also see this from the equality z - e Z = (Z_1D + Z _ 1 D - Z _ 2 Z - Z~2Z - 2Z_1)Z , xy v x x y y x xx y yy xy' (3.58) hence Z =0 z -e =0. xy xy 3.4 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 form L = (Z?vZy) 9 y V admits an infinite sequence of L-B !fl L 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 { (zTz > - z. = 0 and it is _ shown that only for a=-2 X X L the equation admits an infinite sequence of L-B generators. The analysis of these generators in turn leads to a trans-formation 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 trans-formation 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,ux,uxx,... 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 = 0. (4.1) xx x t A generator of the L-B invariance group of Eq.(4.1), I = e(x,t,u,ux,...,ux>> x ) 3 u , (4.2) must satisfy the determining equation (Dx)2e + uDxe + uxe - Dte = o (4.3) 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. (4.4) xx x x t y. j 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. 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 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 i = (Bv)3 = 6 3 (4.5 v ( n )(x,t) ( B) v (x , t) , n=l, 2 , 3 , . n (4.6 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 0 ( n ) (x,t,u,ux,uxx, . . .) = (B)n6(x,t,u,ux, . . .) , (4.7) which solve Eq.(4.4), and hence satisfy Eq.(4.3). In other . words, once we find a generator £ =(#v)3^ of the linearized equation, , we can construct a sequence of L-B generators £ = {(3)ne}.3u, n=l, 2,3,..., (4.8) for the original nonlinear equation from any known generator £=03^. 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 = {bX(x,t)vx + bt(x,t)vt + b(x,t)v}3y (4.9) admitted by Eq.(4.4) is £=v3v, 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 £ = {(£)n u x}9 u, n=l, 2 , 3 , . . . with D = D + hu + hu D"1, X X X ' where D ^ denotes an integral operator with the property X D D - 1 = D - 1 D =1. Comparing (4.10) with (4.8), we let s =D X X X X and consider an operator corresponding to (4.5): 69 - (5v)3 = (v + %uv + hu D _ 1v)9 . (4.12) v v 7 v v x x x ^ v 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. This leads us to X X consider a generalization of (4.9) by including a D term: £ = (Bv)9v= {bX(x,t)vx + bt(x,t)vt + b(x,t)v + b"x(x,t)Dx1v}9v (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: (4.10) (4.11) Bf = D t + (%ux + %u2) + hutD x t X -1 5" = tJ5' + %xD + %xu + %(u+xu )D _ 1 X (4.14a-c) to • + '-iX + -2 D -1 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 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 u (4.15) G( U' U X' U X X'••'•' u x... x) " u t = 0 C4.16D with invariance group generator £=09^. We want to allow to depend on integrals such as D _ 1 u = / u dx, ( D _ 1 ) 2 U = // U dxdx, (4.17) X X 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 D x 1(u) 2, D x 1(u x(D x 1u)), 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 v - v = 0. (4.19) u u x u x...x t J X X • • • X Let 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 . The form of B, however, is still quite arbitrary. For instance, Bv could contain terms such as D^Cfv), D ^ g D ' V v ) ) , ... (4.20) where £ and g are functions of x,t. 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 + I b ( - k ) ( D " 1 ) k } v = 5V, (4.21) k=0 x k=l where b^-' and b^ - are functions of x,t. For example, by repeated integration by parts, we have for the first expression of (4.20), D x 1(fv) = Jfvdx = fD^v - (Dxf)(Dx1)2v + ((Dx)2f) (D; 1)^ + .... (4.22) For the second expression of (4.20), we apply this procedure to each integration operator D ^. Thus, we should start with a generator t= (Bv)3v 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 - 2 3 ) 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 £=03u with the following 9: (a) ' (b) _ (c) ' , „ (d) . 1 2 u x > 9 = _ u t ' 0 = t u x " ' ; " t ut" 3 X Ux" 3 U' (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. With 0 = ( B ) n 0 ^ ^ and.notations u =un, u =u9,..., one sequence is: X -L X X Z d(0) Q (a) D(l) A Q(b) 0V  J  _ .Q «., ./_ u J  = Uj + uu^ = 0V J , (3) 5 10 5 2 = u r + T u u 7 + -=- U..U- + u u , . . . . , 5 3 3 3 1 2 6 x and the other is e C ° ) = 0 ( c ) = t i a 1 - l , 0 ( 1 ) = t ( u 3 + u u 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.:- -.... ' 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. 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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 Inter-science, 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 and M. Yamaguti, On further properties of solutions to a certain semi-li-near system of parital differential equations, Publ. Res. Inst. Math. Sci., Kyoto Univ., 9 (1974) 577-595. 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. APPENDIX 1,.. PROOF OF THEOREM 3.. , Proof. For brevity we write 0V(X,Z,Z,...,Z) = 0V(X,Z). 1 m Suppose the system ^ vZ y= 0\admits L = '0V(X,Z)9 v . 1'hcn, . y z, 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 0v(X,eZ+U)3zV = h v ( 0 ; ) 9 z v + e h v ^ h z V + e 2 h v ( 2 ) 9 z V + ... . (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 admit-(1.43). The function 0v(X,Z) yielding (1.44) is not unique, but if we restrict to 0V(X,Z) which depends on X and only on those Z's which appear in the coefficients of 9zv i-n (1.43), then 0 V is determined within an arbitrary additive function <}>V(X). We let Z->-eZ in such 0V(X,Z) and expand 0V(X,eZ) in the series in ©V(X,eZ) = h v( 0) - e h v ( 1 ) + e 2 h v ( 2 ) + ... , (A2) where h^^ = 0V(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 our disposal. With such a choice of (|)V(X), the operator 0V(X,Z)3zV becomes an invariance group generator. To see this we let U+Z in (1.43), then let Z+eZ. The resulting operator is still an invariance group generator of the same system. We write its power series expansion in e as L = e g v ( 1 ) 8 z V + £ 2 g V ( 2 ) 3 z v + ... . (A3) By the construction of 0v(X,Z) above, we have Since every i n (A3) is an invariance group generator of we see that the operator 0V(X,Z)3 v = h v C 1 ) 3 v + h v ( 2 ; ) 3 v + ... is an invariance group generator of 4VZ^=0. APPENDIX 2. PROOFS OF PROPOSITIONS 9-12. 1. Proof of Proposition 9. From Proposition 2, we have LBV=BVL and LBV=BVL. Then, y y y y [L,L] = (LB Z - LB Z )8 7 V = (B VLZ M-B VLZ M)8 7 V ' y y ^ Z v v y y ^ Z v (B VB KZ y-B VB KZ y)8 7 V = (CB,B]VZy)3vV. v K y K y ^ Z v y J ZV y we obtain [B,B]=-B. • By hypothesis, we have [L,L]=L=(S^ Z M)8 z V. Comparing these, 2. Proof of Proposition 10. It is sufficient to prove for the explicit form 4vzy=0 of f =0. By Proposition 2 and the invariance condition (1.36), we have 4 VZ y=0 - L^ VZ y = AVB KZy = 4 VZ' y = 0, y y k y y i4VZ1 y = 0 - LAVZ'y = LL4VZy = 0, y y y i.e., . 4 vZ y=0 + LL4vZy=0. • y y 3. Proof of Proposition 11. As in the proof of Proposition 10, we have L4 vZ y=4 vS yZ K=0. We also have B v4 yZ K=0 since y y K y K 4yZK=0. From these two we obtain (4vSy - Bv^y)Zl< = 0 K y K y K provided AuZK=0, i.e. U,B]Z = 0 provided AZ=0. • K 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, 4vByUK(X)=0, i.e. 4BU(X)=0. Repeating the y K same argument, we find A(S)mU(X)=0. • APPENDIX 3. A DETERMINATION OF AN INVARIANCE GROUP OF THE EQUATION (z ) 2- 4z z = 0. ^ ^ xyJ x y To illustrate the process of determining an invariance + group of a differential equation, we consider the equation f = (zv ) 2 " 4z z = 0. (1) ^ xy' x y To simplify notations in the following computation, we let ZX=P, zy=q, z x x = r ' z x y = s a n d zyy = t" T h e n > Eq-C1) 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 x y f This equation is' taken from Forsyth [373 page 198 , 2 {§xy s + Sxz^5 + Sxp s 2 + §xq s t +g ZyP s + SzzP^5 +-SzpP s 2 + SzqPst + §z s 2 2 +g rs + g qrs + g rs + g r s t + g s s Py &pz H 6pp &pq 5p x 2 2 3 2 +g s + g qs + g s +g s t + g ss qy &qz H qp qq q y - 2gyP - 2gzpq - 2gpps - 2gqpt - 2gxq - 2gzpq - 2gpqr - 2gaqs}. (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 - 4rq - 4ps = 0, .X. -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,sx,sy 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 p 2q 2) + t(g +g p + 2'g p 2q 2) PR PY pzH ppF M sqx 5qz F &qq F M + 2 g x y p 2 q 2 + 2gxzq(pq)2 + 4gxppq + ZgzyPtpq)*2 + 2g z z (pq)^ 2" + 4g z pp 2q 2 3 + 4g q yP cl + 4g q zP (l + 8Sqp(^Pcl^ " 2§yP " 2gx q' ^ We note that r and t appear only in the first three terms of (5). Thus their coefficients must vanish: 1, g = 0, g + g a + 2g (pa)2 = 0, % g + g p + 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 V  V-0 = eXyCpq)2 + e x zq(pq) 2 + 2axpq 3 ^ J + e y zp(pq) 2 + ezz(pq) + 2bypq (byq + ey)p - (axp + ex)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)9Z, (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  one-parameter 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) O ) du dt (1) J_ dx  I~  dx with a variable conductivity C(u), have appeared in prob-lems related to plasma and solid state physics.1,2 Interest in such processes has long occurred in other fields  such as met-allurgy and polymer science.3"5 Some exact solutions are well known for  such equa-tions.6 These can be shown to be included in the class of  all similarity solutions to such equations obtained from  invari-ance 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 a finite  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 conserva-tion laws'3). Well known nonlinear partial differential  equa-tions admitting LB transformations  include the Korteweg-deVries,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  so-litons,14 and existence17 of  Backlund transformations 18) are related to their invariance under LB transformations. Any linear differential  equation which admits a nontri-vial 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 associat-ed 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 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) a five-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 transforma-tion 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, we find  an operator connect-ing two infinitesimal  LB transformations  leaving Eq. (3) in-variant. We prove that this operator is a recursion operator which generates an infinite  sequence of  one-parameter in-finitesimal  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 opera-tor 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) corre-sponding to invariance under LB transformations. 1019 J. Math. Phys. 21(5), May 1980 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 LB transformations  include Lie groups of  point trans-formations  and finite  dimensional contact transforma-tions. 1 1 The algorithm for  calculating infinitesimal  LB trans-formations  leaving differential  equations invariant is essentially the same as Lie's method8 for  calculating infini-tesimal 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*), lead to determining equations Cl/„.„= 0, t* = t, where u, = ffu/dx',  i = 1,2,-. Let du/dt  = u,, du,/dt = u„, au/du  = U 0, dU/du,  = {/„  d 2U/du,du t = U iJt C • = dC  /du,  and C " = d 2C /du 2. In the above notation Eq. (1) becomes u, =C'(u,y  + Cu 2. (6) Under Eqs. (5) the derivatives of  u appearing in Eq. (6) trans-form  as follows: («,)* = «, +eU , + 0(e2), (u,)'  = u, + eU'  + 0{  e2), (u 2)* = u2 + eU"  + 0(e 2), where au (7) (9) (10) "C'U„u,  = 2C X • ;•= o Solving Eqs. (9) and (10) we find  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) + (1 - n)C'£„_, u, - —n(n  +3)C' (C) ,1/1 )"u2 4 at i U* = D,U=^+ ±UiUi +,, dx i E o rtt2 [To dx + X + .">+! + S U'"<*2-i.j=0 ( = 0 /), 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 deriva-tives 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 equa-tions 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 vanish-es and (he coefficients  of  (i/„ 4 , )' and u„ t l , respectively, + a [ j n 2 ( C ' ) 2 ( C ) n / 2 1 " - jn(n+2)C' (C)" / 2 >"](u , ) 2 = 0. (13) Solving Eqs. (12) and (13) we find  that U=a  [ ( C ) ( , / 2 ) X + ln(n  +3)C'  (C)" /2, n-' u,u„_, ] + F(u)u„_,  +G(u,u  u „ _ 2 ) , (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- (15) Hence, it is necessary that C (u) — a-(u + 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) at  dx\  dx) (17) This particular equation has been considered as a model equation of  diffusion  in high-polymeric systems.4 5 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|)\ (18) For n — 4 we obtain two linearly independent LB trans-formations  U' 1' and V' 2' = u-*u< -14u-5u,u, -10u"5(u,)2 +95u-6(«,)2u2-90u-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: - 107 -(3)kB=U {k\ k = 1,2,—. (20) The character of \B,U"\U' 2'\ leads one to consider for 3 the form & pD, + q + KD x)\ (21) where D„ is a total derivative operator, (D X)-(D X)~'  is the identity operator, and J p,q,r]  are functions  of\u,u„u2\. Then one can show that 2B — U' 1' if  and only if p = u\ •*•" (22) and q[u 2u2 —2u"'(u ,)2] + ru"2u, = -3u-4uiU 2+6U- 5(U i)3. (23) • Furthermore, (3 )2B=U' 2' if  and only if q=  —2u"2u„ (24) and r = -u-2u2+2u-3(«,)2. (25) A more concise expression for  the operator is & =(Dxf.(u {).{DxyK  (26) We now show that the constructed operator 3 is in-deed a recursion operator. Let the operator A = £ B,(DJ i = 0 = u\Dx)2 -4u-3u, D, +6!/""(u,)2 -2u}u2 = (D,)W 2, (27) where B, = (d/du,)B.  Olver's work16 shows that 3> is a re-cursion operator for  Eq. (17) if  and only if  the commutator [A-D,,3] = 0, (28) for  any solution u = 0(x,t) of  Eq. (17). Moreover, if 3 is a recursion operator, then the sequence | U  '" ,£/ '2 ',••• j given by Eq. (20) is an infinite  sequence of  LB transformations  leav-ing Eq. (17) invariant. It is straightforward  to show that A and 3 satisfy  Eq. (28). The nature of  U' 11 and the form  of  a general V  given by Eq. (11) show that for  n = / +2, there are at most &</ lin-early independent LB transformations  leaving Eq. (17) in-variant since U must depend uniquely on u, + ! . The proof  that 3 is a recursion operator demonstrates that k = 1 and hence we have found  all possible LB transfor-mations leaving Eq. (17) invariant, namely, | V tl'), k= 1,2,-. 4. A MAPPING TO THE LINEAR DIFFUSION EQUATION As far  as we know all p.d.e.'s invariant under LB trans-formations  have a recursion operator and, moreover, can be related to linear, p.d.e.'s. This suggests the possibility of  seek-ing a transformation  relating Eq. (17) to a linear equation. This leads us to consider the linearization of  Eq. (17), namely, (A  -<9/<?r)/=0, (29) where/4 is given by Eq.(27)foranysolutioni/  = 8 )ofEq. (17). Introducing a new variable u by /= f-(uS), dx we obtain from  Eq. (29) the equation l\  dx)  dx  dt] and if  we set d_ dx dx (31) (32) J_ dx' (33) 3 d .3 — = u u. dt  dt Eq. (31) becomes Hi =0 dx 2 dT Since/= 0 is always a solution of  Eq. (29), the relation (30) suggests that we set uu — constant. This and Eqs. (32) lead us to the transformation dx  — u dx  + u"2u, dt, dt  = dt,  (34) relating solutions u = 8 (x,t)  of  Eq. (17) to solutions u = 6 (x,7)ofEq.  (33). Choosing a fixed  point (x 0,l0), we have the following  integrated form  of  Eqs. (34): (35) It is easy to check that Eqs. (35) indeed transform  Eq. (17) to Eq. (33), and define  a map relating the solutions of Eqs. (17) and (33). Moreover, if  u > 0 (u > 0), Eqs. (35) define a one-to-one map since dx  /dx  > 0 for  each fixed  t.22 We now show that under the transformation  (34) the recursion operator 3 of  Eq. (17) is transformed  into the recursion operator 3 — Ds, (36) leading to an infinite  sequence of  LB transformations  of  the heat equation (33). The proof  is as follows: An LB transformation  of  the form  (5) induces an LB transformation  on the variables \x,t,u\  through Eqs. (34), namely, x* =x + ei+0(  e2), 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), V  — ~(M?V. (38) 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 correspond-ing infinite  sequence of  invariant infinitesimal  LB transfor-mations j t/"1) for  Eq. (33), namely, Uin = f - u-,i', where V ' = - ( i 7 ) 2 U ° \ (40) and u7 = (d/dx  )'u. From Eqs. (40) it is simple to show that =DiU"\  (41) leading to Eq. (36). Moreover, U< n = D;(ug')  = (D iyu-2, ' = 1,2,-. (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 Hopf-Cole transformation  relating Burgers' equation to the heat equation. Moreover, we can obtain the Hopf-Cole  transfor-mation by examining the linearization equation (29) corre-sponding to Burgers' equation. 5. PROPERTIES OF SOLUTIONS OF EQ. (17) FROM THE MAPPING 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,  (43) « = («)-•, with an integrated form := J_  udx + (uT h., dt\ ' = ' - h , (44) « = («)-', for  some fixed  point (x 0,t0). In the following,  we assume u > 0 (u> 0). Without loss of  generality, we set = ?„ = 0. A. Explicit formula connecting solutions; examples First we consider the problem of  giving a more explicit 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> 0, xe(x,^2). By Eqs. (43), x = X(x,7)  = pi*-,f) d?  + £ = o '.(45) This uniquely determines the function  A""1, x — X~'(x,t), where t = t. Now Eqs. (44) lead to the following  solution of Eq. (17): u = 0(x,r)= ^ —-! , 0(X'(x,t),t) on the domain Jt€(*,(/), x2(t)),  / > 0, where (46) xl(t)  = X(x„tU 1(.t)  = X(x 2,t). In a similar manner, Eqs. (35) map a solution u = 6 (x,t)  of Eq. (17) to u = 8(xj) = — _ , ' - -on the domain JPe(Jc,(/ )Jc2(t)), t > 0 where x,(n = X(x„7), x2(7) = X(x,,7), x = X(x,t)  = f  0(x\t  )dx' Jo d 1 (47) dx (0(x,!'))-'  df, (48) 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)' / 2e'\ on the domain — Ja <x < Ja, t > 0, where u(jc) is defined by (49) x = •r V7-Note that l im^_ ± i a 0(x , t )  = + oo. Example  2. The dipole solution of  Eq. (33), i.e., u = 0(x,7)  = - —la(4v7y V 2e-ii:">T ,), dx on the domain 0 <x < oo, F> 0, is mapped by Eqs. (45) and (46) into the following  self-similar  solution of  Eq. (17): on the shrinking domain 0 <x <a(4-t  ) " 2 , t > 0. (50) 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,;) be-tween initial conditions for  Eq. (17) and those for  Eq. (33). As for  the connection between boundary conditions, from the same formula  it is easy to see that x = s(f)  is an insulating boundary ofEq.  (17), i.e., [dd(x,t)/dx] T ,.«{ , = 0, if  and only if  the corresponding boundary if = s(t ) is an insulating boundary of  Eq. (33), i.e., the corresponding solution u = 6(x,7)  satisfies [dS(x,7)/dx] i _ s,-> ='0. Moreover, s(t) = const if  and only ifi(()  = const, i.e., there is a one-to-one correspondence between fixed  insulating boundaries of Eqs. (17) and (33). In general, a noninsulating boundary condition for  Eq. (17), on a fixed  boundary x = const = c, is mapped into a noninsulating boundary condition of  Eq. (33) with a corre-sponding moving boundary x = ) # const with speed where, as previously mentioned, u = 6 (x,t)  > 0. 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 U<"'+ c*C/ (" = 0, (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 , (53) where (o(r), b (t), c(r) | are arbitrary. Substituting Eq. (53) intoEq. (17) we find  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  simi-larity solutions of  Eq. (17) obtained from  invariance under a four-parameter  point Lie group.7 8 (b) The infinitesimal  transformations  (5) of  the four-parameter point group of  Eq. (17) are given by V"  = u + xux, U b =xu, + 2tu,, U'  = u,, V  = B. (54) Under the mapping (34), these are transformed,  respective-ly, to corresponding infinitesimals  of  invariant point group transformations  of  Eq. (33): U" = a, U h = xu-t + itUj, V'  = 0, U d = W=u }. (55) Conversely, the mapping (34) transforms  the six-parameter point Lie group of  Eq. (33) as follows:  The three-parameter subgroup of  infinitesimals  given by Eq. (55) transforms  to \U",V'\U d \ given by Eqs. (54) and U  = transforms  to U = 0; the remaining infinitesimal  point group transforma-tions U'' = xu +2 Tu;  and U r=  + \7)u + xiu-t + Pu,-are mapped, respectively, into infinitesimals  which depend on (;t,r.u.u|! and integrals of  u. (c) Generally speaking, the action of  a recursion opera-tor 9 ' on any infinitesimal  invariance transformation  Uof the form  (5) (whether of  point group or LB type) yields a new infinitesimal  transformation  U'  = .9' Vif  'J  U  #0 . For Eq. (17). we can show that = 3 V = 3 V = 0. (d)The heat equation is a special limiting case of  Eq. (3) obtained by setting a — b: and then observing limh . r b ~(u  + b)'- = 1. As one might expect if  a = b 2, for the corresponding recursion operator 3 , limA . a £/' = d/dx.  and the mapping formulas  reduce to identity mappings, : ' (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 transfor-mation of  the form x = K(x,t,u), 7=  L (x,t,u), u = M(x,t,u), relating solutions of  Eq. (1) and those of  Eq. (4). 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. '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. Thermophy-sical Properties  of  Matter  (Plenum, New York, 1970). Vol. 1. 4 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. New York. 1968); R. McGregor, R.H. Peters, and J.H. Petro-polous, 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  Equa-tions (Springer,  New York, 1974). °R.L. Anderson, S. Kumei, and C.E. Wulfman, Rev. Mex. Fis. 21, 1, 35 (1972); Phys. Rev. Lett 2«, 988 (1972). I 0H. Steudel, Ann. Physik 32, 205 (1975). "N. Ibragimov and R.L. Anderson, J. Math, Anal. Appl. J9. 145 (1977). ,!E. Noether, Nachr. Akad. Wiss. Goettingen Math. Phys. K1.2 234 (1918). "H. Steudel, Ann. Physik 32, 445 (1975) "S. Kumei, J. Math. Phys. 18, 256 (1977). "S. Kumei. 3. Math. Phys. 16, 2461 (1975). '"P. Olver, J. Math. Phys. 18, 1212 (1977). I7A.A. Fokas, Ph.D. Thesis, California Institute of Technology. 1979. Chap. V. Backiurtd  Transformations,  edited by R,M. Miura (Springer. New York. 1976). '"L.v. Ovsjannikov. Dokl. Akad. Nauk SSSR 125, 492 (1959). :"S. Lie, Arch. Math. 6, 328 (1881). *'An infinitesimal LB transformation x'  — x + €i  + 0(r  1, t ' = /  tT  •+ 0(r'l. u'  = u + f/i + O (r). acts on a surface F(x.t.u)  =-- 0 in the same manner as x" = * <• = t. «• = « + <l' + 0(r). where I'  — ft  — £u. — tu,. ! :G. Rosen, Phys. Rev. B 19, 2398 (1979). After submitting this paper we discovered the above reference through Nonlinear Science Abstracts. Here Rosen discovered transformation (35) and worked out some exam-ples. W e are also grateful to the referee for bringing the above paper to our attention. Invariance transformations, invariance group transformations, and invariance groups of the sine-Gordon equations* Sukeyuki Kumei Department  of  Physics, University  of  the Pacific,  Stockton,  California  95204 (Received 5 May 1975) 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 ux 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 well-known 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 waves . 1 One of  the basic properties of  the system which admits a soliton appears to be the possession of  an in-finite  number of  conserved quantities. As has been shown by Lax, 2 the existence of  such conserved quanti-ties 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 analysis5 and its generalization5" to the differential equations governing the waves. In this paper, we ap-proach 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 trans-formations  (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 invari-ance 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, 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 transforma-tion, 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 , (1) where z'  = (z l,z2 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 se-quence of  partial differential  equations which will suc-cessively determine a possible form  of  the u*. In parti-cular , the first  term u° must be a solution of  equation (1). If  the equation is linear, all the u"s must also sat-isfy  the same equation. In the case of  nonlinear differ-ential equations, however, all the equations are differ-ent. 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 u1; the nonhomogeneous term depends upon the u°, « ' , . . . , and their derivatives. By a deductive argu-ment, we expect that if  only the nonhomogeneous solu-tion is taken for  u2, 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 u2 i s found  to be 2461 Journal of Mathematical Physics. Vol. 16, No. 12, December 1975 Copyright © 1975 American Institute of Physics 2461 uj, - stn«°=0, (4, - u'cosu"=0, ul, -u !cosu° = - (u'l'slnu0. It is surprising that we can find  many solutions for  u1 and u2 which can be expressed as simple functions  of u" and its derivatives; a few  examples are u' = u° and u2 = u°,, = + and "2 = »L,,«+3K)2"°„I 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 care-ful  analysis of  invariance transformations. II. RELATION BETWEEN AN INFINITESIMAL AND A FINITE INVARIANCE TRANSFORMATION 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. We consider a set of  transformations  of  the coordi-nates of  the n-dimensional vector space R"(xl, ...,*") which analytically depends on the parameter a , and becomes the identity transformation  for  a = 0: x'-x'^X'ix'.a),  x' = X'(x',0). (4) We also consier an equation f(*')  = Fix 1, • • • ,x") = 0 which is defined  in the subspace R"(xl,... ,*") of R". The equation F(x')  = 0 defines  a manifold  S, or hyper-surface  in Rm. We define  the invariance transformation in the following  way; Transformation  (4) i s a continuous invariance trans-formation  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 trans-formation  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. Under the condition we have imposed on the trans-formation,  we can expand X'ix 1, a) in a Taylor series in a by : x'=X'(x>,a)  = x' + t Ci = {0„m*', «)}„.„• Defining  the differential  operator U t by U^pt-'d,  , we can write (5) as (5) (6) (3a—d) We analyze the effect  of  this transformation  on an analy-tic function fix')  defined  in R". Expanding fix')  in a Taylor series In a , we obtain Ax')= t ^  <*>,(*'), *, = {(3.)*/(5')}»o = A,/(x'), where i •A„= 1 and where ps and q : are the integers satisfying  the conditions p p j q ^ 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 (q lt... ,q,)  satisfying  the conditions is not unique, 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 /(*')= (l+t £A^/ix') = Tia)fix l). (10) We note that Tia)x l = x' + a£j + + • • • = recovers the definition  we started with. Now we suppose that the continuous transformation T(a) leaves the equation Fix')  = 0 defined  in Rm invari-ant in the sense defined  above. Then, the following statement will be obvious: The transformation  T(a) is a continuous invari-ance 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 transforma-tion to be an invariance transformation,  it is very diffi-cult to get any clear view of  the structure of  the trans-formation  unless a considerable simplification  of  ex-pression (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 all the V k's are first-order  differential  operators. We write down the first  four  generators in this form: A,= U lt A2 = (£/1)2 +1^, A3=(C/ 1)3 + 3C/ 1i72 + r'3, At= (UJ*  +6(V,) 2'U 1 + 3(V 2)2 + + U t, ' (12a)—(12d) where 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.» - 12?!«!?!. ,t!. „ - St k {«»,«»'. , • (13a)—(13d) 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 opera-tors 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 condi- (C) tion yf(x')  = 0 on F(x')  = 0. 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 sat-isfy  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 £( [ / , )* = (15) fc>0 * ' Result (C) is significant  in studying the structure of  in-variance transformations  because it clarifies  the con-straints on and arbitrariness of  an invariance trans-formation.  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 transforma-tion 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 transforma-tions in R" which leave the equation F(x')  = 0 defined  in Rm invariant. We now introduce some functional  rela-tions among the coordinates, which are compatible with the equation F(x')  = 0; such relations will restrict fur-ther the domain of  manifold  S. We consider a function  u(x')  defined  in the {k  - 1 ) -dimensional space ^'"'(x1 , . . .,«*"') with k<m, and assume the following: The coordinate x* 16 determined by the relation x* = u(x'), and the coordinates At**1 x" are determined as the derivatives of u(x') with re- . . spect to the coordinates x1, For in-stance, x*" = Bsu, ... ,xu'l = dt.1u, *J'=3 151u, x^'^a .a ju , We suppose that R" is chosen in such a way that if  it con-tains 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 de-rivatives defined  by (16). Each solution will define  a submanifold  3 of  the manifold  S, called the solution surface. Now, we consider a continuous coordinate trans-formation  (4) under which a manifold  satisfying  condi-tion (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 coordi-nates and a complete set of  the first  through the jth order coordinates. In particular, we call the 0th extended space (x1 xk), the basic space. Basic transformation We call the transformation  of  a set of  basic coordi-nates, 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 is clear that under condition (16), the transforma-tion of  the basic coordinates will determine the trans-formation  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 trans-formation  leaves the equation f (x^=0  invariant. The geometrical meaning of  the invariance transformation is 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 con-sidered an invariance group transformation  of  the form x'-x^e^x'-x'  + a('(x)  + •••, i = l,...,k, x = ( x * 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  transforma-tion of  the coordinate x* does not involve any coordinate whose order is higher than the order of  xp. This guar-antees 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 trans-formation,  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 co-ordinates 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 re-stricted, 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 trans-formation. 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 ' , 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"=A tu" 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 written as u"(z')=A l lu0(z i), where h° is a solution of  the same equation and the operator Ak is con-structed by (11) from  the generators Q of  invari- , . ance group transformations  of  F=0  by which the independent variables z1 are unchanged. In par-ticular, uJ = Qu°. Furthermore, if  only the in-homogeneous solution is taken for  every h* , k > 1, then «* = (Q)*«°, and a resulting solution is expressed as h(z') = e'^u'U')• 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 clear-ly indicate the existence of  invariance groups, or sym-metries, of  the sine—Gordon equation. We will system-atically 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 x1 , x2 , and x3 , and F(x')  is chosen as F(x\  x2 , x3 , x4 , x5 , x6 , x7 , x») = x 7 - sinx3. As stated in (16), we establish the following  constraints on the coordinates: x 3 = it(x',x2), X4 = M1, X s —u2, XS = H11, x7 = M 1 2 , X 8 = K 2 2 X 9 = H U 1 , X ' ° = I / U 2 , X " = « I 2 2 , X = W 2 2 2 , X = M I I I I , X — U1112, X 1 5 ~ "1122, X L 6 = « 1 2 2 2 , X 1 —U 2222, X  ~ I'm  11 , -19 — ,, V20 — „ V21 — u r22 — II X  ~ <'11112,  X  — "iu22, X — «11222, —"12222, X23 = « 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 x2 are un-changed and the infinitesimal  transformation  of  .v3 de-pends on x 3 and the first  through the third-order coordinates: x' = x', x2 = x2, x 3 = x 3 + o ? 3 ( x 3 , x 4 , x 5 , x ' , x 8 , x 9 , x ' 2 ) . (20) We note that the inclusion of  the coordinates x",  xu\ and x" is redundant because we can replace them by the coordinates in £3 after  we have introduced the condi-tion F = 0. The infinitesimal  transformation,  induced by (20), of  the coordinate corresponding to the derivative (S1)m(d2)"u is calculated as 9 x W + aO.Ha^^x' + a?'. (21) Here, the partial derivative should be interpreted as (a1r02)"53 = 01)"02)'' X «»(« (* ' , j r* ) ,^ 1 , * 2 ) « 2 2 2 (x' ,x 2 ) ) (22) For instance 54 = 5 3 X 4 + ? 3 x « + ?|X7 + 53X9 + | 3 X " + ^ , 3 + ^ 2 X , s , (23) where is the derivative of  £3 with respect to the coor-dinate 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 ex-tended space as xi = (l + a4)x', with {'=^  = 0. (24) i-i 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 differen-tial equation because some of  the coordinates which appear in the equation are independent from  the coor-dinates in £3 (Appendix). By solving these equations, we find  four  independent solutions: t*=jr<, 5> = *» + *(*4)3, (26a) C = d = * , 2 + i<*5>3- (26b) Obviously, the test two solutions can be obtained from the first  two. by interchanging the roles of  x' and x2. The second extended operators associated with £3 and {{j are calculated from  (21)  and (24)  as Qa = .v<c3 + A-6c4 + x , c 5 + * 9 t 6 + j - ' 0 c 7 +j - n c e , (27a) Qt=ix°  + i(x")3}o3 + {x>3 + f  (x4)2x6}5< + {*» + j ( . V ) y } s 5 + {x'e + 3jr-cV)z+4 (x 4 )V}a 6 + {a-" + 3,vVV + f  Cv4)V0}57 + {x20 + 3x4(x7)2 + 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. This result explains the origin of  the solutions (3a)—(3d); they are obtained from  result (D): „ 2 = §a.v3 = .v< (28a) and h| = W„)V = A (28b) «> = §„ x' = x° + i(x<)2 (28c) and «l = (Q„)V = x24 + 3 (x4)2*'3 + |(x4)"x6 + 9.vVxB + 3(xe)3 . (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 trans-formation  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 form  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  gener-ators, 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 equa-tions 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-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 invariance transformation  connected to the identity transformation.  By reversing this, we argue that if  we have an invariance transformation  connected to the „ identity transformation,  then we can find  at least one. such operator. More precisely, we proceed in the fol-lowing way. Suppose we have an invariance transformation  of  the equation F(x') = 0, J W + E^A/tW + Eit?!, ' = 1 <29> in which all the £ j are known. Using the result (11), we can write, (> =Asx< = , s<=As>  = {&>)' + U,}x<, l i=Aje> = {(t/,)3 + 3 U & + t'3}*<, • • •, where all the Vk are first-order  differential  operators. From the first  equation, we obtain • (30) Feeding this into the second, we get U 2x'= t,'2-(U 1)2x', which provides l / 2 = £ U M < & i > 2 * ' } ] ? ( • (31) tel Next we substitute these for  the Z\ and U2 in the third equation to determine U3. Continuing this process we can obtain a series of  operators Ut, all of  which satisfy the invariance condition "QF=0  on F—0."  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 diffi-cult. Obviously, this method can be used to find  gener-ators 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. We start with the well-known Backlund transforma-tion of  the sine-Gordon equation,1 J4 - X 4 = 2a sin£(*3 + .r3), a (if 5 + x s) = 2 sinj(x3 - .v3), (32) with the convention established in (19). This transforma-tion 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 con-struct 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 A3, 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 trans-formation  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', t» = 12*B + 2(x<)', 53 = 48x1 3 + 48(x4)2x8 , 4 3 = 240x18 + 360(x4)2x® + 600x4(x6)2 +18 (x4)5, = 1440X24 + 2880(x4)2x13 +12 960x 4 xV + 3840(xV + 1440(x4)V { 3 = 10080x31 + 176 400(x 8 )V +95 760x ,(x9)2 + 141 120x4x8x1 3 + 25 200(x4)2x18 (34a-g) + 63000(x4)3(x5)2 +18900(x4)4x9 +450(x4)7 , • • where we have adopted the convention (19) and x 3 1 = M iuuu- 'his specific  Backlund transformation, the coordinates x1 and x 2 are unchanged, i . e . , x ' = x l , if 2 = x 2 or = ^ = 0 for  i » l . (35) The transformations  (33) and (35) form  the basic trans-formations,  and they provide all the necessary informa-tion to follow  the above prescription to find  Vt. We list the results up to £/,: & 2 = & , = t/, = 0, £/1 = 2x43„ Z/s = {4x» + 2(x4)3}3s, &, = {48x l s + 120(x4 )V + 120x4(x8)2 + 18(x4)'}3a, U, = {1440X31 + 25 200(x")2xs +15 120x4(x")2 + 20 160x4x8x1 3 + 5040(x4)2x18 +12 600(x4)3(x')2 + 6300(x 4 )V+450(x 4 ) 7}3 3 . (36a-e) Here, we have given the operators in the basic form; the operators in the extended form  can be obtained from (21) and (24). By continuing this process, we will be able to find  an infinite  number of  operators which satisfy the Invariance condition (25). We can associate one in-variance group transformation  of  the sine-Gordon equa-tion with each of  these operators. VI. A SERIES OF CONSERVATION LAWS AND INVARIANCE GROUPS In this section, we use notation (16), hence x* repre-sents a solution of  the differential  equation Fix 1) = 0. We consider an equation f(x')  = 0 which can be put Into a conservation form: S 3 , / ' = 0, / ' = / ' ( x ' x*" 1 ,x* , . . . ,x ' ) , (37) 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 Is Invariant under an r-parameter group with the property: x' = r , (a )x '=x ' for  x = l , . . . , f c - l  with T , ( a ) = e x ( 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 corre-sponding flux,  f=  ( / , / 2 , . . . ,/""*) is written as / ' = / ' ( x 1 ) . . . , x * - I , x * x') = r t (a) / ' (x ' x*"',x* x'). (39) The Implication of  the new flux  Is the same as the old one, except that it is now for  the new solution, However, its power series expansion in a tells us something new about the starting solution x*; because we have assumed that the transformation  T t(a)  exists at least for  some range of  la I, It acts as a generating function  of  fluxes; each term of  the expansion of  (39) in a ' a ' also forms  a divergent free  flux.  We state this as follows: If  a differential  equation F(x')  = 0 admits an in-variance group with property (38), and if  a flux f  of  the form  (37) exists, then, for  any polynomial (E) function  G(Q l Qrj  of  the generators of  the group, the vector Gf  forms  a divergent free  flux. Here, we see two basic patterns for  a series of  diver-gent free  fluxes  to arise: one associated with a series Qjf,  i =  l r  and one associated with a series (Q,)"t, n = 1,2, •• • . It will be reasonable to say In general, that the former  is more fundamental  than the latter because the series of  the second type can be mechanically con-structed If  Qt Is known, although the reverse is not possible. One, however, should not think that the fluxes of  the second type are tr ivial . 1 0 We now apply this analysis to the sine-Gordon equa-tion, J = x 7 - sinx3. The equation can be put into the conservation form  by multiplying by x ! ; 3 l / ' + 3 2 / 2 = 0 wi'h f=(/',/ 2) = (i(x')2 , cosx3), and the generators (36b)—(36e) can be used to derive new fluxes.  We list a few  of  them, (using the notation t, = b,t)• f , : / {  = 2 x V , / f  = - 2x4slnx3 , f,  :/J = {4x14 + 6(x4)2xV. fl = ~ {4x8 + 2(x4)3} slnx3 , U -fi  = 24{2x2' + 10x 4 xV + 5(x4)2x1 4 + 5x7(x")2 + 1 0 * 4 x V ° + ^ ( x 4 ) V } x ! , / ; = -24{2x 1 0 + 5(x4)2x» + 5x4(x8)2 + | (x 4 ) ! }s lnx 3 , • / i , = ( ^ ) V ' = ISlx5.*32 + 3(x4) W s + (x1 4)2 + {3(x4)Jx7 + 9 x 4 x V } x 1 4 + 6 x V x V 3 + {9x5(x6)2 + 9 x 4 x V + | ( x 4 ) V } x 1 0 ( 4 0 a _ d ) + 9 x V x V + 9 (x 4 ) 3 xVx 7 + |(x4)4(x7)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 results are clearly different  from  the fluxes  given in the paper by Scott el al." Their results, however, can be obtained by taking a linear combination of  fluxes with the form  (£/,)"(!/,,)' • • • (U rYt.  In fact,  by using (11) and (36a)-(36e), we find  that A,t and AJ  recover their results. For instance, AJ 1 = {(T/,)3 + 37/,% + U,}/ 1 = 6 { 2 x V 4 + 4 x V ° + (x*)2xV} = 6{2u2«111j + iuL2ullz + (u1)3u2u12}, where we interpret TJl as a generator extended to a 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 con-servation law associated with a solution. The remark-able feature  of  the sine-Gordon equation is that it possesses a series of  basic fluxes. SUMMARY 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 transforma-tion, and we stated the explicit relation between a con-tinuous invariance transformation  and a continuous in-variance 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 sine-Gordon Eq. [(36b)—(36e)]. In Sec. VI, we gave a group theoretic criteria for  the existence of  a series of  con-servation laws associated with solutions of  a differen-tial 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 frame-work 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 type . 1 3 Note  added  in proof:  The transformation  (10) with A„ defined  by (11), (12a—d) has been found  to be the power series expansion, in or, of  the expression T(a) = e"5' e<2!e"u' e""-1*4®* ... ACKNOWLEDGMENTS I wish to express my grateful  acknowledgment to Professor  Carl E. Wulfman  for  helpful  discussions, his continuing encouragement and many valuable com-ments on the manuscript. I also thank the Research Cooperation and the Physics Department of  U.O.P. ' for  the support of  this research. 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 Cole . 5 c Us ing/ for  £3 , the deteter-mining equations for  our problem are the following: /». V^  +A +/., +/.. 1**° +/„. i 1^ +/». i2*16=o, /s.o*5 +US +/,.»*a +/„,»*" = 0, A*7 +A*" + V +A*19 + fi, (A.+/„,,*7 +/5.+/.,,*10 +/,.,*12 6, 8,12 +/,.,*»)=o, with supplementary conditions: x 7 = sinx3 , x 1 0 = x 4 c o s x 3 , x" =** cos*3 , x 1 4 = x " coax3 - (x4)2 sinx3 , x " = x*cosx' - (x3)2 sinx3 , x " = x ° cosx3 - 3x4x f  sinx3 - (x4)3 cosx3 , x 2 2 = x " cosx3 - SxV  sinx3 - (x5)3 cosx3 , where f, = 3,f  and , = d fij. •This work was supported by a Research Cooperation Grant . 'A .C . Scott, F . Y . F . Chu, a n d D . W . McLaughlin, P roc . IEEE 61, 1444 (1973) (review ar t ic le) . l P . D. Lax, Comm. Pure Appl. Math. 21, 467 (1968). s I t has been suspected that some transformation  property of the differential  equation governing the wave motion is r e spon-sible for  the existence of  a s e r i e s of  conservation laws. In fact,  the res t r i c t ed Backlund transformations  (R. B. T.) have provided a sys temat ic way of  deriving a s e r i e s of  conserva-tion laws. However, the derivation involves a p rocess of power s e r i e s expansion of  a solution with r e spec t to some pa rame te r . Such a method only exemplifies  the existence of  a s e r i e s , but does not explain the origin  of  individual conse rva -tion law. On the discussion of  R. B. T. in the theory of  so l i -tons , we refer  to (a) G. L. Lamb, 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 Estabrook, Phys . Rev. Lett . 31, 1386 (1973). We  add in proof  the follow-ing papers on the Backlund transformations'.  G. L. Lamb, J r . , J . Math. Phys . 15, 2157 (1974); M. Wadati, H. Sanuki, and K. Konno, P r o g r . Theor . Phys . (Kyoto) 53, 419 (1975). ' R . L . Anderson, S. Kumei, and C . E . Wulfman,  Rev. Mex. F is . 21, 1 ,35 (1972); J . Math. Phys . 14, 1527 (1973). *For L i e ' s work and its l a te r development, we refer  the r e a d e r to (a) S. Lie , Transformationgruppen  (Chelsea, New York, 1970), 3 Vols. , Reprints of  1888, 1890, and 1893 eds. , S. Lie, Differentialgreichungev  (Chelsea, New York, 1967), repr in t of  1891 e d . , S. Lie , Continuierliche Gruppen (Chelsea, New York, 1967), repr in t of  1893 ed. (b) L . V . Ovsjannikov, Group theory of  differential  equa-tions (Siberian Sec. Acad, of  Sc i . , Novosibirsk, USSR, 1962). This book has been t ranslated into English by G.W. Bluman, Department of  Mathematics , University of  Brit ish Columbia (unpublished). L .V . Ovsjannikov. Some problems arising in group analysis of  differential  equations (Proceeding Conference  on Symmetry, Similari ty and Group Theoret ic 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  Differen-tial  Equation  (Springer , New York, 1974). (d) R . L . Anderson, S. Kumei and C . E . Wulfman,  Phys . Rev. Lett. 28, 988 0972), ®We note that the well-known contact t ransformations  of  o rd i -nary differential  equations, which were extensively studied by Lie , a r e a real izat ion of  the derivative-dependent t r a n s -formations  in which only the f i r s t -order  derivative appears , 'if  the equation is an ordinary differential  equation, it is always possible to find  a closed space . 8Several yea r s ago, Professor  G.M. Lamb kindly ra i sed the question of  the relat ion between this generalization and the Backlund t ransformation,  which depends on f i r s t -order  de -r iva t ives . The basic difference  is the fact  that the Backlund transformation  is not a group transformation  in genera 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 re complemen-tary in the sense that neither of  them subsumes the other . L ie ' s infinitesimal  approach, however, will be super ior in the s t ruc tura l analysis of  continuous invariance t ransformations. *The general formula  of  the expression of  the extended opera-tor will be found  in the paper by R. L. Anderson and S. Davison, J . Math. Anal. Appl. 48, 301 (1974). 1 0We define  " t r iv ia l" flux  in the following  way. We consider .a set S{ f , , f 2 , . . • ,fj}  which cons is t s of  divergence free  fluxes, ft,..., f| and their der ivat ives of  any o rde r . We note that the der ivat ives a re a lso divergence free.  Now, a flux  f  is said to be t r ivial with respect to the set S, if  f  can be expressed as a linear combination of  the members of  the set S.  In this s ense , the flux  <?*f  is , in general , nontrivial with respect to the set S{f,Q(,Q :tt... .Q^'f}.  For instance, the flux  f 3 | , of (40) is nontrivial with respec t to the set S{f,Ujf}. " E q . VI. B. 7 in Ref.  1. l 2 T h i s is due to the special charac te r of  the operator the operation of  Uj  on the variable x{,  i >2, is equivalent to the differentiation  of  the function  u with respec t to xA. For in-s tance , l\xz-xit (C/|)2x3=ar6 and I/Jx5 = x" a re the t ransforma-tions w — u , , u~u|i and t/2 — « 1 2 . Because of  this proper ty , the fluxes  obtained from  (i / jJ ' f  a re all t r iv ia l . 1 3 S. Kumei, "Group theoret ic studies of  conservation laws of  nonlinear d ispers ive waves" (II, III, IV) (submitted for publication). 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  type equation + a(/)/,+/, = 0 and of the coupled nonlinear Schrodinger equation /„ + Q(J,g)+  if,  = 0, Su + a(&<f) — 'Si ~ 0 o n e 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 KdV, the cubic Schrodinger, and the Hirota equations. INTRODUCTION In this paper, we study group theoretic aspects of time evolution equations of  nonlinear waves, particular-ly 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 generalization1 of  the Lie-Ovsjannikov2"4 theory of  invariance groups of  differential  equations, and applied it to a number of  quantum mechanical sys-tems to systematically study dynamical groups.5 Re-cently it has been shown by Ibragimov and Anderson6 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 in/ = 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 waves8 from  the standpoint of  the generalized theory, and to gain a clear-er 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. 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 / „ , + a(f)f x + / , = 0, one invariance group i s associated, hence the KdV equation admits an infinite 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 solu-tion) 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 in-variance 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 equa-tions H(t,x',f,  ft, f ilt••')+/, = 0. It will be shown that (1) A generator U of  an invariance group of  H + / , = 0 always satisfies  the relation [Hi  f ]  + iV/dt  = 0, where ff  is a Lie operator associated with H\  (2) For a class of  nonlinear time evolution equations, the equation (d/dt)(V)  = ([//,  I/] + ?U/ct>  can be generalized; in par-ticular, if U  is a generator, then (d/dt)(V}  = 0. 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. (2) 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'o, v„), y = («oo, u0/r,...,uin,u„i «„), V  = (WoO>  Vol  v0Jf  vJtO>  vNi>  " '  I vNK)I 2 (3) - 119 -Now we consider an infinitesimal  transformation  in V z=z+(Z, Z = (0,7], C,n, [,...,!),{, •••), (4) 1 1 > » where j) = i](z;c), £ = E(z;c), (5) and the components of TJ and £ are to be determined by the formula Vi-'.^ 'iV"'."' ''.v-'r^ v','' (6) where Dtl...m =D,D, •••Dm with D< = + M.  + v,8v) + (un dUj + v„d V j) + • • • + («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 expres-sion of TJ or £. We write (4) compactly in the usual way as Z=(l + eC/)z, (8) with foa. + J3„) + + f,3„() + • • •+ (17,...,3  + + (9) 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  func-tions /(*) and g(x), F'(z;c) = 0, i = l,2, (10a) «=/(*), v=g(x),  '<={(*), y=g(x),  ft  = 1,2, (10b) where fix)  and g(x)  are functions  of  the (N+  l)*-tuple k k f{x) = (/„, /, f„), g(x)=-(g 0,gl g„), 1 1 fix) =  (/oo fsn), g(x) = (gm, . . . ,gff.y), 2 2 (11) with/ , . . . / = 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 solu-tion manifold. It is well known2"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 , (12) vhere (• •) I j indicates to evaluate the quantity under he conditions F 1 — Of D,r.,tF<  = 0, i = 1,2, fe  = l,2 ». (13) The operator U is then a generator of  an invariance group. We define  ((-conjugation of  a quantity A(z;c) =A(x,u,v,...,u,v;c)  by A(z;c)*=A(x, v,u,... ,v,  u;c*), where the asterisk represents a complex conjugation. Ah important subclass of  Eq. (10) is F'(z;c) =  0 with F 2 = ( F l ) # , ' ri * « v. «=/(*), v =f(x)  , . For this equation, the generator U  takes the form (16) (18) In this paper, we consider the infinitesimal  transfor-mations 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  trans-formation  of  a more general t y p e 2 - 6 i=z+tz, z=(i,i,t,5,£,•••), e = I1,...,i-v), 1 1 (17) where It can be proved11 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  (10), then for  an arbitrary choice of  f,  ij, and £ subjected to the conditions TJ — £ F = rj. i -V 1', = £> the trans ormation  (17) is also an invariance transfor-mation of  Eq. (10). Conversely, if  (17) is an invariance transformation  of  (10), then so is (4) for TJ  = FJ  - L'U T, C = S-Vr,. In the following  sections, we write the operators (9) and (16) as C/ = I)3. + £3„, U  = TJ3„ + TJ*3V (19) They, however, must be always interpreted as their infinite  prolongation. Also, we use the following abbreviation: [A(z)]„^ „, = [A(n,t>)]/f and / [A(u, v)\,dx  = JA(u,  v)dx. II. A GROUP THEORETIC ANALYSIS OF THE KdV EQUATION • The equation of  our interest is f ut +ff t + / 0 = 0 . 1 2 - 1 6 The equation is 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 definition  i), = D(t), r] l t =DIDtrj,  etc. Also, because the equation involves a single real function,  all the f's  in the first  section are to be ignored. A. A Lie algebra of  an Invariance group of  the KdV equation We write the equation as F = u„x + uux + u,= 0, u=f(x,t),  u^=_f x(x,l),  «,=/,(*,/),• (20) (21) We look lor an operator £/ = j)B„ which satisfies  condi-tion (12) for  this equation. We assume the transfor-mation to be a generalized Lie type1 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 algorithm3'4 for  finding  gen-erators leads to the following  results: lA = (tux- i)a„, V 2 - j {arw, - 3/(h„, + mi,) + 2ii] a„, £/3 = K,a„, I/4 = (*„, + ««,)a„, u5 = (i + «»*** + 2iv„ + W« x ) a„. The generators form  a nonsemisimple algebra (see Appendix B for  the definition  of  a commutator) [t/ 1, U2]  = 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) ear equation H.if)  f  + H,(f)  0 , + • • •+#.<„,(/) 0("> + 0 , = 0, (24) where//„(/)  = (a,HL/> etc. and 0"" = (d,r<pix,  I). We 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). ls For the constant of  motion of  integral type, i. e . , /( /) = fp(f)dx,  the gradient has a simple expression: As-suming p(it)  =p{x,t,u  k"'), r(u) =P,-D,p, x + Dip^x +••• + (- (26) In this case, we have (T(f),<p)=  f  T(f)<t,dx. Lax observed (r( / ) , 0) is a constant of  the motion. (27) (28) + a{f\f,  + f,  = 0 and B. Constants of motion of f KMX its groups Now we prove a theorem which establishes a relation-ship between a constant of  motion of  the KdV type equa-tion and its invariance group. We consider an equation h "if)fx  +/<  = (29) This set of  generators is well known. 1 4 , 1 7 The genera-tor ifi,  however, is new and its properties will be analyzed later. Let us consider operators dU/BI  = (5,tj) 3u 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 known13 that the KdV equation admits an in-finite  number of  conservation laws. To study a possible connection between the present groups and the conser-vation laws, we need to know effects  of  infinitesimal invariance transformations  on constants of  motion. In his analysis of  constants of  motion of  the time evolution equation H(x,  t, u,ti,, u „ , . . . , « ' " ' ) + « , = 0, «'"'=«Srt7T  , u=f(x,t),  Lax15 considered an infinitesimal transformation  of  a solution fix,  t) into a solution u =f(x,  t) + «t>ix,  t).  The function  0 must satisfy  the lin-where aif)  is a function  of / .  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\(ii)  = Dxr(u)  is a generator of  an invariance group of the equation. Proof:  It is sufficient  if  we prove {£/(«„, +a(n)", + »,)}/ = 0. We consider a transformation  of  a solution/ to a solution/ + £0. Then, by (24), 0 „ , + o( / )0 , + a„(/)/ ,0 + 0 , = o. Thus, O = / r ( / ) ( 0 , „ + (j0, + oB f ,0  + 0,)djr. In-tegrating this by parts and assuming null contribution from  the boundary terms, we obtain 0 = / { - D 3 r - Dx(aT) + Tajt x-D,T} l^dx  + (d/dl)  IT  <j>dx.  The second term vanishes because of  (28). Because we can prescribe an arbitrary admissible function  for  0 at initial time t0 , this equation implies {D3r + Dx(aT)  - Tajt x + D,!"}, = 0. Differentiating  this with respect to x, and defining V=DJ~, we find  {£>3rj + r)aji x + (£> xtj) a + D,n}, = {f  (n„, + aux + «/,)}, = 0. This theorem establishes a relationship between con-stants of  motion and invariance groups of  Eq. (29), l(f)  = f  p(f)dx  — r(u)—~{U=»)3.,  j)=i»,r}. (30) The process from  U to I involves an integration process and not all the generators are integrable to I. In Sec. IV, we provide another scheme to connect a group to a constant of  motion which can supplement such a non-integrable case. The application of  the theorem to the generators (21) leads to (within constant factors), l'  = f(itu 1-xu)dx,  I^fi^dx, P = j&J-u\)dx,  (31) 15 = j au'-3uul + l„l)dx. The generator l/1 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 is 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  number of  con-stants of  motion for  the KdV equation means that the equation is invariant under an infinite  number of  groups; the situation is similar to the case of  the sine-Gordon equation / „ - s in/ = 0.' 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 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 follow-ing unique properties: (1) If r'(r<) is a gradient of  / ' = Jp'(f)dx,  2, then r'(u)D xr'(u)=J"  with J"  = polynomial in «> »*> "xx, • " • (2) Every solitary wave solution h = 3c sech^VtT (x-ct)  = 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 at-tention on three constants I3 , i 4 , and T5. For these constants, the gradients are r3 = u, r* = u1 + 2uxx, r 5 = a3 + 3i4 + 6K«„ + f u„„, (34) and correspondingly, r3(s) = s, r4(s) = 2cs, ri(s) = f c ! s . (35) Another remarkable property of  r (u)  of  the KdV equa-tion is due to Gardner, (3) If  we define  an operator W' associated with r'(u) off',  i> 2, by W = (15,1") 3„ + (Djr1) 3., + (Djr1) 3^, + • • •, (36) then [ W , W"] = 0. 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. Neverthe-less Gardners result implies that two generators U' and U' associated with l ' and I ' commute, [ l / ' , t / ' ] = 0, i,j> 2. (37) This is obviously the reflection  of  the fact  that the KdV equation is a completely integrable Hamiltonian sys tem. 1 8 > 1 9 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 /  vV </*  = /£  (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  con-stants 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. E. Properties of e M ' 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,  (40) 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 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'/ t!)(  U') k-1h"},dx  = !{p'} idx. 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 sup-pose r to be a linear combination of  r ' associated with the constants of  motion / ' of  integral type. Differentiat-ing 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 solu-tion (32), we hav^ the operator identity V  = yDx, Conse-quently, the group' operation e'v has the effect  of  trans-lation 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,) a s / - " . For such a wave profile,  interactions between solitons are small, hence at least for  small a we may assume {^""l/b,. {e° t '"}, l<«,.«,) as (45) In view of  (43), we can write this as .v {eo t ,t(}/iT,i)~E s,(t - c , / + 6,-t-ay(c,)) a s / - « > . (46) 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,). F. Invariant solutions of the KdV equation One curious question would be whether there exists a solution which is mapped onto itself  under the trans-formation  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 equa-tion is called an invariant solution (or generalized sim-ilarity solution).1 The necessary and sufficient  condi-tion for  f  to be the invariant solution of  eaU  is obviously {Vu} f  = 0. (47) One of  the best known invariant solutions will be the Green's function  of  the heat equation / „ - / , = 0, / = (417/)-1'2 exp( -* 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 m +//,)  + qf x = 0. An integration of  this equation with respect to x, assum-ing/(± «>, 1) = 0, leads to the fourth  order equation ob-tained by Kruskal and Zabusky,12 and Lax. 1 5 The nature of  the solution was carefully  studied by Lax, and the solution was shown to be the doublet solution. From a group theoretic viewpoint, therefore,  the doublet  solu-tion of  the KdV  equation is the invariant  solution  of  the group 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 equa-tion. 8 , 1 8 , 1 9 In this section we study group theoretic aspects of  conservation laws associated with a class of nonlinear Schrodinger equations. (44) 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),  u,=f,(x,t),  « , = / , ( * , / ) , • • • , (48 v=g(x,  t), vx =gx(x,  t), v,=g,(x,  <),•••, where a function  a is subject to the condition "„(/,«•;£) = 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„ f f  = * „ (so) where bH/bg  and 6 $ / 6 / are Frechet derivatives of ft  - ! E(f,  g) dx,  £ = energy density. Equation (48) re-duces 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 operator V  = iifil/bv)  ?„ - iifil/ 5h) 3„ is a generator of  an invariance group of  the equation. 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  0 + ~ 4') dx el(f,g)  + e61. Thus, d/dtbl  = 0. Next obviously, On integrating by parts this yields 0 = I(P<t>  + <?#) dx  . +. (d/dt)  5/ where P = - Mi>„  + a(u,  v,c)* - iw,]} /> f + 'laj/,g;c)-a u(g,/;c*)]j f, Q = - {t/[« = + "(«,  v;c) + iti,h,c Because (d/dt)  61=0, we obtain f  (P<j>  + Qii)d:r  = 0. (*) One can prescribe arbitrary admissible functions  for <t> and i at an initial time. Thus, the Eq. (») implies that P  and Q are identically zero. Furthermore, the second terms of  P and Q are zero because of  condition (49), hence, P = 0 and <? = 0 yield the equations to be proved. This theorem enables us to find  constants of  motion if  we know the invariance groups of  Eq. (48); the pro-c e s s involves a straightforward  integration process (61/Sf,  il/6g)—l.  However, we note that there may be a generator which i s not integrable to a constant of motion. This theorem can be extended to a general Hamiltonian s y s t e m . ' 0 B. Invariance groups of the cubic Schrodinger equation and its conservation laws We look for  the operator of  the form  (16) which satis-fies  the invariance condition (12) for  F l =f xx+f 2f + if,  = 0 and F 2 = (F 1)* = 0. Assuming the transforma-tion 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 gen-erators [writing only the first  term of  (16)]: (/,  = (-  jixu + tux) a„, l/j = (iti<„  + itu 2v + ixux + 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 + imxvxx + 2vuxuxx + 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„  V 2 = - xd x — 2/3, + u3„, 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, / = exp[- i(ax  + a>t/2)/2]f(x  + at, t), f  = af(ax,  <ft),  f  = exp(ia)f(x,  t), (52) /=/(*  + a,t), f  =f(x,  t + a). The remaining three generators are of  the generalized type, and there exists, at present, no analytic method of  finding  corresponding global transformations. 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-tmx), p2 = vv, p' = iti,v, Ps = i(" xi' x - s w V ) , pe = i(u xxxv + h'U Iv1), p7 = u„v„ + j u V - 2(«,i> + uvx)2 - 3tt}v xuv, p" = "xxxx,"  + 5 (ui'mV  + wixvxx + 2r,x„xxv + ,m„vx + u2vx) The operator U 2 i s not integrable. These constants of motion, except the first  one, agree with the ones ob-tained by Zakharov and Shabat.9 The phase shift  opera-tor U 3, the x translation operator t'4, and the t transla-tion operator ifi  have given rise to the probability den-sity p3 , the momentum density p', and the energy den-sity p5 . The first  constant / ' also has a simple meaning if  we consider the cubic Schrodinger equation as the Schrodinger equation for  a particle with negative mass - The / ' represents the initial position of  the particle, <*o) = 0r-/V> = jf = V  = velocity. Let us define  the Lie Hamiltonian by ^'if) 3»" (!'S) e" = C/5' W = energy = /5. (54) Then, we find  that the operator V' of  (51) satisfies  the relation [H, V'] + 3t/ ' / i t = 0 with SU St We note that the second generator U2 which is not re-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 com-mutation relations among U l are [u', U y ] = 0 for 3 IV. GENERAL PROPERTIES OF GENERATORS OF INVARIANCE GROUPS OF TIME EVOLUTION EQUATIONS Let us assume that Eq. (15) is a time evolution type: x° = time coordinate, F 1(z;c)  = H(z;c)+u a = 0, F 2(z;c)  ~ [H(z;c)]^  + v0 = 0. (55) To carry out a consistent analysis, we must take into account the relation (13), +«o> = °, k = \,2 «. (56) We define  two operators associated with H121 and U by H=H9,  + H*d„  (57) dV dP (58) - 124 -As was mentioned in the first  section, they must be interpreted as their infinite  prolongation. By the definition  of  a time evolution equation, H  is not a function  of  the coordinates corresponding to ^-der iva-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. A key in the present analysis is to write Eq. (55) as W  + D„)v =  0. (59) We first  prove: Lemma 3: If U  is a generator of  an invariance group of  the equation H + u0 = O, then under condition (56) we have [U,H]  + BV/Bx"  = 0. Proof:  We have [C/,//] + dU/ix <> = aiv + a*Bv  with a = UH  -HJ)  + 3 oT). It i s sufficient  if  we prove that a van-ishes under (56). Indeed, 0 = UiH  +1<0) = UH  + £>0 n = VH + + "on- + "el. + • •- = VH  + 3 0tj - Hi u - H*3,, - • • • = un-rS^-Hn. Now, we define  the following  quantity: {V)  = Re f ivUu)^ ftx) dx' dx2 • • • 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 H  + tt0 = 0 satisfies  the equation if* + vH.  + uiHJ*  - C,[t>WB| + h(H„()#] + • • • + (-[ttf . +•'(«„ )#] = 0 (61) and if  all the boundary integrals an V./*tl) and .,]..,<,, v'dn s " * !./*(*> vanish for  v = ( v t , — , v") = normal vector on the boundary surface,  then d d? 00 =([£/,//]-3 U Y7, (62) Proof:  For brevity, we write (60) as (l/) = ReJ vUudx. Then, we have d/dx°(U)  = Re !(v0Uu + vD t Uu)dx = Re/[-W*ri + «(3l / /3x 0- / / ty)«)dx. Here, we have used 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 id/dx"){V)  = <[£/,//]  + BU/Bx"). The combination of  Lemma 3 and 4 leads to a method to associate a conserved quantity with an invariance group of  the equation: Theorem  3: If  the operator V  defined  by (16) is a gen-erator 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, i. e., d/dx"(V)  = 0. We note that in proving this we did not assume the r quantity / fix) fix) dx' •••dx1" to be independent of  time. Lemma 3 can be generalized to a set of  nonlinear time evolution equations of  the form H ' + ul = 0, ff'=  #<(*•,u,u,u, ...,u), ( = 1 , 2 , . . . , M t (63) where u = (u 1,u',.. ,,ti"),  u = ( u ' j u 2 , . . . ,u") n fc k * nd u' = / ' (*) , etc. In this case we have Lemma 3': If V  = is a generator of  an invariance group of  Eq. (63), then we have [u,H] + BU/Bx°  = 0 where H  = H i2„t and dV/dx°  = i?x<,v')  3„i. For Hamilton's equations of  a field  [t/1 = P=pix), i/3 = Q = ?(*)] | + P 0 = O, -g +« o = 0. we obtain the familiar  expression with j c a . The theorem above can be specialized to a real dif-ferential  equation: If Hix.  u,u.u...., u). in the equation 1 2 r H + wo = 0, satisfies  an equation H + uH„-D,(uH Ui) + --- +(- ) = 0 (64) and if  all the surface  integrals /s["t7,...»W|,j ...Jiu/ii) dn vanish for  S = boundary, then the quantity ( t j = ittiUi(]„,, dx1 • ••dx*  is a constant of  motion. 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 - t [ E ( ? , , ) V + / ' / * ] + » . / = 0. Hirota equation8 a(3,)'/+ib(Bx)\f + cff'BJ  + idf*f'  + 3 , / = 0. However, the heat e q u a t i o n / „ - / , = 0 and Burgers equation/„ +ff x-f,  = 0, both of  which represent a dis-sipative system, do not satisfy  Eq. (64). The application of  TTieorem 3 to the KdV equation and to the cubic Schrodinger equation has turned out to produce only a few  constants of  motion, KdV equation: m = -f«dx,  <U 2) = fi« 2dx, {Ui) = 0, f o r i > 2 . cubic Schrodinger equation: (V 2) = fi mi* dx,  <t/<) = 0 for  f  > 2. 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 pres-ent approach leave the differential  equation invariant whereas the groups in Noether's theorem leave an action integral invariant. In the following  communication, a generalization of Theorems 1 and 2 will be discussed. ACKNOWLEDGMENTS I am sincerely grateful  to Professor  Carl E. Wulfman for  many helpful  discussions and for  a number of  valu-able comments on the manuscripts. I also thank the Research Corporation for  supporting this research. APPENDIX A: PROOF OF LEMMA 1 It is sufficient  if  we prove D0UA  = UD 0A. To avoid complex indices, we represent a set of  indices i •• • k appearing in the expressions (7) and (9) of D0 and U  by a circle " or by a dot • , and write D0 and V  as 0^ = 3,0+? ("o^  + fo'V-where the sign indicates a summation over all the parenthesized quantities in (7) and (9). Then, by the definitions  of D0 and U, =£ bDaA^+ZD^) +E[U>01o) A„o + (DoS.Mj. Using D0rja = j]0o = Uu 0o, D0£0 = [Oo = Uv 0o, D0UA=E  (tJoA^+E.-DoAJ +E[(t/«0„)A.o+(I/t>0„)AvJ. (») The first  term is +?M„ S + v^A^)} + EoU0„o +E + =Zb, A + £oA«.J+?t'o•EhA-.'e+ZA-.J  ) = UA 0 +Lu„ UA^  UA V m. Hence, (») gives Dt UA  = t/[>l0 ("0oi4„o + u0oA„o)] = 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!...,)] We write this as V  = [t/1, U 2] = t)3. + £3V + • • • + T,,..., 3„(ii>t We prove that this satisfies  the condition imposed on (9), i . e . , the condition (6). In fact,  by applying Lemma 1, T),..., = U 2V\... t = I/'B(...,r,! - U 2D r,' = D,...t(UW  -t/V)=.D (...»1-Similarly £ , .„ , = D, . . . ,£ . Therefore,  the operator ob-tained from  the commutator of  two operators of  the form  (9) also assumes the same form. •Th i s study has been supported by a Resea rch Corporation grant . ' R . L . Anderson, S. Kumei. and C . E . Wulfman,  Phys . Rev. Let t . 28, 988 (1972). I have learned that the idea of  the derivat ive dependent infinitesimal  transformation  is not new. For instance, H. Johnson, P roc . Am. Math, Soc. 15, 675 (1964). I am grateful  to Professor  A. Kumpera of  University of  Montreal for  bringing this paper to my attention. 2 S. L ie , Transformationgruppen  (Chelsea, New York, 1970), 3 Vols . (Reprints of  1888, 1890, 1893 eds . ) ; Differential-greichungen  (Chelsea, New York, 1967) (Reprint of  1897 ed.) ; CoTitinuierliche  Gruppen  (Chelsea, New York, 1967) (Reprint of  1893 ed . ) . 5 I . . V. Ovsjannikov, Group Theory  of  Differential  Equation (Siberian Section of  the Academy of  Sciences, Novosibirsk, USSR, 1962). [This book has been t rans la ted into English by G.W. Bluman (unpublished).] *G. W. Bluman and J . D . Cole, Similarity  Methods  for  Differ-ential  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. Phys . 14, 1527 0973) . e N . H . Ibragimov and R . L . Anderson, "Lie—Biicklund Tangent Trans format ions . "  J . Math. Anal. Appl. (accepted for  publi-cation). (See Sov. Math. Dokl. 17, 437 (1976) for  excerpt ion.) J S. Kumei, J . Math. Phys . 16, 2461 (1975). ' A . C . Scott, F . Y . F . Chu, and D. W. MacLaughlin, P roc . IEEE 61, 1443 (1973). ' V . E . Zakharov and A .B , Shabat, Zh. Eksp. Teor . F i z . 61, 118 (1971) [Sov. P h y s . - J E T P 34, 62 (1972)]. 1 0 The notation v is from  Ovsjannikov (Ref.  3). 1 < S . Kumei (unpublished). Also, L .V . Ovsjannikov (Ref.  3). 1 2 N . J . Zabusky, in Nonlinear  Partial  Differential  Equations, edited try W. Ames (Academic, New York, 1967). " R . M . Miura , C .S . Gardner , and M.D. Kruskal , J . Math. Phys . 9, 1204 (1968). " M . D . Kruskal , R . M . Miura , C .S . Gardner , and K . J . Zabusky, J . Math. Phys . 11, 952 (1970). 1 S P . D . Lax, Commun. Pure Appl. Math. 21, 467 (1968); 28, 141 C1975). " C . S . Gardner , J . Math. Phys . 12, 1548 (1971). " H . Shen and W. F . Ames, Phys . Lett . A 49, 313 (1974). I owe th is reference  to Professor  R . L . Anderson. " V . E . Zakharov and S .V. Manakov, Teor . Mat. F iz . 19, 332 0 974). 1 9 Y . Kodama, P rog . Theor . Phys . 54, 669 (1975). « S . Kumei (unpublished). nH  and the energy H  in Sec. m should not be confused. On the relationship between conservation laws and invariance groups of nonlinear field equations in Hamilton's canonical form a ) Sukeyuki Kumei b l 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 form flp/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 gives rise to 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 In p r e v i o u s p a p e r s , 1 , 2 we have s tud ied i n v a r i a n c e 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 evolut ion equa t ions by app ly ing the t h e o r y of  g r o u p s of  L ie—Backlund t a n -gent t r a n s f o r m a t i o n s 3 (not to be confused  with the 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 te ra ture" 1 ) and we have shown tha t e ach of  the we l l -known 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 the s i n e - G o r d o n equa t ion , the Kor teweg—de V r i e s equa t i on , and the n o n l i n e a r S c h r o -d i n g e r equa t ion 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 the c o r r e s p o n d i n g different ial  e q u a -t ion i n v a r i a n t . T h e - g r o u p g e n e r a t o r s ob ta ined in t h e s e p a p e r s depend 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 , so tha t they a r e not of  the type 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 con t inuous g r o u p s of  t r a n s f o r m a t i o n s .  The q u e s t i o n n a t u r a l l y a r i s e s : T o what ex ten t can the p r e v i o u s r e s u l t s be g e n e r a l i z e d ? In the p r e s e n t p a p e r , we s tudy 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 equa t ions gove rn ing the t i m e evolut ion of m u l t i c o m p o n e n t f ields  p„i\), qa(x), p a = - 6 H ' & , u , « 0 = 6//.'6/>0, a = l, 2, ... ,A\ (1) w h e r e x = (v", .v ' , .v2 , .v3) and />„ = ?,o/>„, q a = ? / / / „ • We a s s u m e that an e n e r g y dens i t y H a s s o c i a t e d with H can depend on c o o r d i n a t e s .v ( including .vn), 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 ma in i n t e r e s t of  the s tudy i s : to e x a m i n e the r e l a t i o n s h i p be tween 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 Eq . (1) and c o n s e r v a t i o n l aws obeyed by the f ie lds .  We wi 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 be s t a t e d h e r e a s a t h e o r e m . T h e no ta t ions in the 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 the f ields  and a r e funct ions  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 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. o r d e r ; £>, r e p r e s e n t s a d i f ferent ia t ion  with r e s p e c t to x ' , and the quant i ty 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 Hami l ton ian equa t i ons (1), t he funct ions  AU,p a,q0,"r) and J ' ( . v , p a , q a , " ' ) obey the c o n s e r v a t i o n law D„A + = 0 , then the p ro longa t i on 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 in -v a r i a n c e g r o u p of  the Hamil fonian  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  the form  A' = (6A'/6p a) ?flo - (M',< w h o s e p ro longa t i on b e c o m e 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  the Hami l ton i an e q u a t i o n s , t h e r e e x i s t s a flux  _?' which t o g e t h e r with a dens i ty 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  Hami l ton ian s y s t e m s with finite  d e g r e e s of  f r e e d o m  gove rned by the equa t i ons — dH/?p a, pa = - cH/cq a h a s been ob ta ined by P e t e r s o n . 6 We will p r o v e the 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). To e s t a b l i s h the L ie b r a c k e t f o r m u l a t i o n ,  one 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  the s y s t e m . Such a f o r m a l i s m  is known for Hami l ton ian 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 the following  we wil l deve lop a s i m i l a r f o r m a l i s m  for the field  e q u a t i o n s (1) by app ly ing the t h e o r y of  L i e -Backlund tangen t 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 be v e r y a p p r o p r i a t e in s tudy ing the connec t ion of i n v a r i a n c e g r o u p s of  Eq . (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 ac t ion i n t e g r a l ]L tlx: We dea l d i r ec 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 ferent ia l  e q u a t i o n s . All the r e s u l t s in the following  s e c t i o n s r e m a i n va l id 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 We consider groups of  Lie-Backlund tangent trans-formations  generated by the operators® U  = - c.a,. + U),F m)amaii - iD.c.te,.., _ { ( - D ( ) . . . ( - D y ) G a } + where represents a total derivative operator (2) = aTij (3) /><,,i..., i 9<,,i...i represent coordinates associated with derivatives ?xi • • • pa(x), •'" ®a(.v). Throughout the paper we adopt a summation convention for  repeated indices: a greek index runs from  1 to AT 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 palx), q a(x), and any of  their derivatives of arbitrary  order. In the study of  Eq. (1) which is a time evolution type we can assume without a loss of generality that the F a and Ca are not functions  of  time derivatives of pa(x) and q a(x). This will be assumed in the following  for  all the operators of  the form  (2). To avoid a complex expression we write the operator (2), which we call a Lie—Backlund operator, as U  = FJ  - G „ ? . (4) a 9ft  a Pa' We must always consider this to be the infinite  series given by (2). We denote a set of  operators of  the form (2) by A. It is known that the U  have the properties (a) If  U\  U'eA,  then U3 = [U\  t / 2 ]e A with F' a = U 1F\ - U*Fl  and Ca = U'G* a - U'G' a. 2-s (b) If  V1, V , [ ) ! € A , the Jacobi identity holds9: [[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 follow-ing 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 by r e p l a c i n g F a a n d G a in (2) by S r 0 F a a n d a ^ G , , . Now, let us consider a variational problem of  a functional J\p,<!,x°)  = jj(x,p,q)dx',  dx' = dx1dx2dx3. (6) The density $(x,p,q)  depends on x and pa, q a and their derivatives of  arbitrary order except ones involving time derivatives. For the variation paix) - />„(*)+ £!>.(*) we have - 6J  = e / c iha dx'  + surface  integral w i t h (7) (8) Similarly, for  a variation q a(x) — <7„(.v) + e0„(.v), we have ¥-=9  -D.Q ,+DMj + ( _ D ( ) . . . ( - D J ) < ( ? a a i . ..., + ••• . (9) We adopt (8) and (9) as the defining  equations of 6J/6p a and 6,7/bqa. We call ty a density of  J. With the functional J  we associate an operator J which is obtained from  (2) by substituting 6J/6p a and bj/bq a for  F a and G„: In simplified  notation M S (10) We designate the operators of  this particular form  by boldface  letters. Then, with the energy functional  H, the following  Lie—Backlund Hamiltonian operator will be associated: («/>„) C</ft)?'°' perator corresponding b nd to be equivalent to • ' - M S K - M S K an The o to a functional is fou(12) Let us denote the set of  all the operators of  the form (10) by SJ. We can prove that n closes under the com-mutation operation defined  in (a) above: Proposition-.  If  two operators A and B belong to f2, the commutator C = [B,A] also belongs to n , and its density C is given by any one of  the following: C2 = B/4, C3 = -aS. (13) 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 [A,B\. We note that the canonical commutation relations among pa and q a are not carried over to the operator formalism:  The operators cor-responding to pa and q a are P a = dq  and Q a — - Sp and they all commute. II. INVARIANCE GROUPS OF HAMILTON'S EQUATIONS AND CONSERVATION LAWS We now turn our attention to the theorem stated earlier. The well-known equation which describes the time evolution of  a functional  A = IA dx'  is ^A = {H,A}  + fB xo/ldx'. (14) We associate an operator K = [H,A] +A 0 with the quan-tity 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: ^fMHfm^ 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 fol-lowing properties of  a generator of  an invariance group of  Eq. (1): Lemma 1: The Lie-Backlund operator V  defined  by (2) satisf ies  the equations ([H, U]  + U :f i)pa | „ = 0 , ([H, U]  + U xo)qa | w = 0 , (16) if  and only if  U  is a generator of  an invariance group of  Hamilton's equation (1). Proof:  In view of  the definition  of  H, under the con-dition (• • • )l w we have an identity D0= 3,0 + H. Using this relation, we obtain ([H, £/] + Uxo)pa\ w = { -HG„ + £7(6/7/6q„)- 3,0G J l „ = {l / (6fl /6 9 < 1) - D 0 G „ } | r = V(6H/tq, + pa)\w. Similarly, ([H,C/]+ V^)q a]w=U(-6H/6p a + q a ) l „,. These relations obviously prove the statement. In the following  analysis, it is often  helpful  to con-sider an initial value problem of  Eq. (1). We say that functions  /„(*•') and ga(x'),  x'  = (*', x2, x 3 ) , are admis-sible if  the initial value problem pa I =/„,  qa I =ga has a solution. A set of  all such admissible functions will be denoted by I. The following  lemma states that Eq. (16) holds without the condition I v . 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 ga(x')  which belong to I. Proof  .'We  have, by definition,  [H, U]  + f  = -N aBt, where MJx,p,q)  = H F a - U(bH/bp a) + 3xoF a, N a(x;p,q)  = -RGa-V(t,H/bq 0L) + B^G.  By L e m m a 1, if Pa' 9<x a r e solutions of  Hamilton's equation, then M a = N a = Q. We let x°-1 = initial t ime. At t, both M 0 and N a are well defined  (note that F a and Ca 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 value, we may replace x by (x°, x2, x 3 ) to obtain the desired result. Remark:  If  / „ , ga 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^eO holds even pointwise: For any given values of  x, pa, qa, Pa it  la i>**"> the operator vanishes. This should be true as long as there exis ts a solution which takes the designated values at the given point x. It is also clear that we can allow f a and ga to be functions  of  x instead of  x'  because x", if  it appears in f a and ga, 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 assume, 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 + BjtA^DcA, 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  arbi-trary functions  f a(x')  and ga{x').  Thus, noticing that the left-hand  side of  the equation Of)  i s the /C2 of  (*»), we have K,  = -D{J'.  This implies that 6K/6p a and 6K/6q a vanish identically, and, as a result, [H,A] +A^>s0 by (»*). In view of  Lemmas 1 and 2, we see that this is the necessary and sufficient  condition for  A to be a generator of  an invariance group of  Eq. (1). Conversely, if  A i s a generator of  an invariance group, in view of Lemma 1 we obtain two equations 6K/6p a = 0 and 6K/6q a = 0. According to Lemma 2, these vanish identically. This implies that the density K  in (**) must have a divergent form;  for  instance, K 2 — (H + d ^ A = -Dj'  with J ' = 3'(x,p,q).  Now if  we let />„ and qa be solutions of  Eq. (1), the quantity in the middle of this equation becomes equal to D^A, and the equation leads to the desired result DqA + D , J ' = 0. III. INTEGRABILITY OF GENERATORS TO CONSERVED DENSITIES We have proved that with every conservation law 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, C a = 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 func-tional 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, (18) - 130 then, because of  the definition  {(r//rfe„)/»[/,  + t,01J},.„ = 6 ( y fixed),  this relation is written (or fixed  a and j as, This is the integrability condition of  the set S„ to a con-served 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 : . ( 6 ? , S.) = S<!> ? , 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 equa-tion is obtained by letting q = », H  = \p2 + j®2 - cosq: <7i=/>, /> , - -9„+s in< / . (21) In the previous paper,1 we have shown the equation ux 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 2 = M , -<-<7„ + si-V/)?,. v i = l 4 » n , " Sfl.cos?  + + h j > 2 ) \ ~ ( - 4/>„, + 1px cos® - f ® ^  - - 3qxqxJ>V e, v4 = (4/>„  " />  cos9 + \q2J>  + - (-  4qxxx + 5®xx cos® - §q\sin® - sin® cosq ip'sinq-ZqJ>J>-\p 2qxx)*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 den-sities 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 - ip2 cos q + \q2J> 2 + ip" ~11\cosq + i cos2® + iq*. by Lamb,'2 and their group theoretic aspects have been studied by the author1 and by Steudel.13 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 prop-erty of  the equation. The present results provide a uni-fied  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 con-servation 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  invari-ance 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  systems. 1 4 Clearly, the present approach to conservation laws via a Lie bracket formalism  is quite different  from  con-ventional 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 communica-tion 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 re-sults similar in part to those obtained here . 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 form/+/( + / 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) These conserved densities are related to those obtained This relationship is entirely independent from  Eq. (1). We prove the f i r s t  re la t ion: the s e c o n d fo l lows  s i m i l a r l y . To prove ( A l ) , w e a s s u m e that functions  p a and g a decay suff iciently  fast  a s [ (*') ' + ( x 2 ) 2 + ( x * ) 2 ] " 2 s o that all the sur face  i n t e g r a l s which appear in the p r o -c e s s vanish . Let u s c o n s i d e r an integra l Jv(x')l/(6M/6<i s) dx'  where v i s s o m e a r b i t r a r y function  except that it does not d i v e r g e at inf inity .  If  w e w r i t e 3,i • • ' 3 , / f  = f ( - J I then integrat ing o v e r the w h o l e s p a c e = JvUD-^/H^dx-  = JvD' imlV/n, ti j dx' =!v<. lvK<-t dx' - Gav,.,d tl l.P~r.i<Mp„f.,.)dx' (us ing [Z)„ < v A „ J = °> Now, w e have the right hand quantity of  Eq. ( A l ) in the last integrand. Both in the s t a r t i n g and in this f inal  form of  the integra l the function  v a p p e a r s a s a factor . B e c a u s e v i s a r b i t r a r y , th i s equation n e c e s s a r i l y i m p l i e s ( A l ) . Nex t , by def init ion,  [B , A ] = C « ° 3 , o - C ' « 3 , with In v iew of  the e q u a l i t i e s (A2), w e obtain +(4) ~ fer) fc)»., J = D ' -> [© (S)_ ©] S i m i l a r l y , w e obtain C ' s D J . ^ C , ] , . T h u s , w e have proved 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 note that they a r e r e l a t e d to C :t>yC,=C,  + Df and C,=Ci  + D^  w h e r e / ' and g*  a r e functions  of x, pu, q a , and of  the ir d e r i v a t i v e s and the index i runs from  1 to 3 . The fact  that functional  d e r i v a t i v e s of  the functional  / d i v h ( x , p , q ) d x '  a l w a y s van i sh l e a d s to the d e s i r e d r e s u l t s . ' S . Kumei, J . Math, Phys . 16, 2461 0975) . *S. Kumei, J . Math. Phys . 18, 256 0977) . ' N . H . Ibragimov and R . L . Anderson, Soviet Math. Dokl. 17, 437 0 976); N . H . Ibragimov and R . L . Anderson, "Lie— Backlund Tangent Trans fo rmat ions , "  to appear Id J . Math. Anal. Appl. As these authors have c lear ly shown, the t r a n s -formations  which have been considered to previous pape r s LR .L. Anderson, S. Kumei, and C . E . Wulfman,  Phys . Rev. Le t t . 28, 988 0972) , and Ref.  1 and 2 above) form  groups of infinite  o r d e r contact t ransformations;  for  instance. If  we take our p r e sen t problem as an example, the opera tor U  de -fined  by Eq. (2) Is a tangent vec tor and genera tes a t r ans for -mation which is a local one -pa rame te r group of  contact t ransformat ions  defined  in a vector space of  Infinite  d imen-sion with coordinates ( p a , q a , p „ , i , g a , i , p a , n , 1 ^ i i • ' " 1 1 1 th is space , no finite  dimensional subspace exis ts which c loses under the t ransformation  except for  special ca ses where the t ransformation  becomes an ord inary point transformation o r a f i rs t  o rde r contact t ransformation.  The bas ic Idea of  the method of  calculating group genera tors i s the same a s the one due to Lie; for  instance, see G.W. Bluman and J . D. Cole, Similarity  Methods  for  Differential  Equations  (Springer, New York, 1974). *For Instance, see R . M . Miura , Ed. t Backlund  Transforma-tions, The  Inverse  Scattering  Methods,  Solutions,  and  Their Applications  (Springer, New York, 1976). *A method of  cas t ing Euler— Lagrange equations into Hami l -t on ' s canonical form  Is well known for  the case where Lagranglan densi t ies involve no der ivat ives of  fields  whose o r d e r s a r e higher than one. The case where Lagrangian den -s i t ies depend on higher der iva t ives 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 he s i s . Univers i ty of  the Pacific,  1976 (unpublished). ' R . Abraham Bad J . E . Marsden , Foundations  of  Mechanics (Benjamin, New York, 1967). •These o p e r a t o r s , a s they appear , do not genera te t ransfor -mat ions In Independent var iab les x . However, such t r ans fo r -mations a re contained in them in dtsgutse as s tated in the previous paper (Lemma 2 in Ref.  2 above], and we a re not excluding any of  such t ransformat ions .  See the example in Sec . Ill for  instance. *H.H. Johnson, P roc . Am. Math. Soc. 15, 433, 675 0964) . " N . H . Ibragimov, Dokl. Akad. Nauk SSSR 230, 26 0 976). u F o r Instance, s e e L . V . Ovsjanlkov, "Group proper t i es of differential  equa t ions , " Ozdat. S ib i rsk . Otdel. Akad. Nauk SSSR, Novostvlrsk, 1962 (In Russian) . " G . W . Lamb, J r . , Phys . Let t . A S2 , 251 0970) . " H . Steudel, Ann. d e r Phys . 32, 205 0975) . uNote  added  in proof:  Recently, It has been shown that many of  the t ime evolution equations which a r e solvable by an In-v e r s e sca t te r ing method a r e wri t ten In Hamil ton 's canonical form;  Y. Kodama, P r o g . Theor . Phys . 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). LIST OF PUBLICATIONS 1. C. Wulfman and S. Kumei, "Highly Polarizable Singlet Excited States of Alkenes", Science 172_ 1 9 6 1 (1971) 2. 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) transforma-tions", Phys. Rev. A £ 2309 (1974) 8. S. Kumei, "Invariance transformations, invariance group transforma-tions 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. Phys. 18_ 256 (1977) 11. 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)"2u1 - u = 0", J. Math. Phys. 21 1019 (1980) ®X L- X 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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