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Extended group analysis of the wave equation Ma, Alex Yim-Cheong 1990

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E X T E N D E D G R O U P ANALYSIS OF T H E W A V E E Q U A T I O N By Alex Yim-Cheong Ma B. A. Sc. (Electrical Engineering) University of British Columbia, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS and INSTITUTE OF APPLIED MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1990 © Alex Yim-Cheong Ma, 1990 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 M a t - W a t - i r g The University of British Columbia Vancouver, Canada Date A P r i l 5> 1 9 9 0 DE-6 (2/88) Abstract A comprehensive study of potential symmetries admitted by partial differential equa-tions is given using the wave equation utt = c2(x)uxx as a given prototype equation, R. Methods are given for the construction of various conserved forms for R. Potential sym-metries for R are nonlocal symmetries realized as local symmetries of auxiliary systems obtained from conserved forms of R. The existence of potential symmetries for R can be determined algorithmically and automatically by the use of a symbolic manipulation program. Most importantly, the potential symmetries obtained through one auxiliary system may or may not include and/or extend those obtained through another auxiliary system. The work in this thesis significantly extends the previously known classes of potential symmetries admitted by R and results in a better understanding of the limits in the construction of potential symmetries for R. 11 Table of Contents Abstract ii Acknowledgement vi 1 Introduction 1 1.1 Symmetry Method for Differential Equations 1 1.1.1 L ie Groups of Transformations; Symmetries 1 1.1.2 Infinitesimal Transformations 3 1.1.3 Multiparameter Lie Groups 4 1.1.4 Extended Transformations 4 1.1.5 Invariance of Differential Equations 6 1.1.6 Invariant Solutions 7 1.2 Potential Symmetries for P D E ' s . 9 1.2.1 Overview 9 1.2.2 F ind ing Potential Symmetries 11 1.3 F ind ing Symmetries of Differential Equations 13 1.3.1 Algor i thm 13 1.3.2 Classification Problems 14 1.4 Invariance Properties of the Wave Equation 16 1.4.1 Group Analysis of the Wave Equation c2(x)uxx = utt 16 1.4.2 Group Analysis of the System vx = ut/c2(x), vt — ux 17 1.4.3 Forms of the Wave Speeds 19 i i i 1.5 Noether's Theorem and Conservation Laws 21 1.5.1 Euler-Lagrange Equations 21 1.5.2 Variational Symmetries 21 1.6 New Potential Symmetries for the Wave Equation 24 1.7 Chapter Summary 26 2 Cascading Potential Symmetries 27 2.1 Introduction to Cascaded Systems 27 2.2 Three Cascaded Systems for the Wave Equation 30 2.3 Induced Symmetries 32 2.3.1 Cascaded System T\{x, t, u, t>, <j>} 32 2.3.2 Cascaded System T2{x, t, u, u, w} 33 2.3.3 Cascaded System T3{x, i , u, v, w, <f>} 33 2.4 Group Classification of the System: <j>x = v, <f>t = u, vx — ut/c2(x). . . . 35 2.5 Group Classification of the System: wx = u/c2(x), wt = v, vt = ux. . . . 40 2.6 Group Classification of the System: <f>x = v, 4>t = u, wx = u/c2(x), wt = v 46 2.7 Group Analysis of (c2(x)vx)x = vu 47 2.8 Relationships Between the Two Wave Equations: c2(x)uxx = uu and (c2(x)vx)x = vtt 51 2.8.1 Associated System S{x, t, u, v} 51 2.8.2 Cascaded System Ti{x, t, u, v, </>} 52 2.8.3 Cascaded System T2{x, t, u, v, w) 53 2.8.4 Cascaded System T3{x, t, u, v, w, <f>} 54 2.9 Potential Symmetries of the System S{x, t, u, v} 55 2.10 Chapter Summary 56 iv 3 Nonlinear Conserved Forms of the Wave Equation 57 3.1 Introduction to Conservation Laws and Conserved Forms 57 3.2 Variational Symmetries for a Lagrangian of the Wave Equation 59 3.3 Construction of Conservation Laws 63 3.4 Group Classification of vx = (ut)2/c2(x) + (ux)2, vt — 2uxut 65 3.5 Higher Order Conservation Laws 68 3.6 Chapter Summary 70 4 Linear Conserved Forms of the Wave Equation 71 4.1 Introduction to Linear Conserved Forms 71 4.2 Group Classification of vx = xut/c2(x), vt — xux — u 74 4.3 Group Classification of vx = (tut — u)/c 2(x), vt = tux 80 4.4 Chapter Summary S2 5 Discussion 84 5.1 Conclusions 84 5.2 Future Research 88 Bibliography 89 v Acknowledgement I thank Professor George Bluman, my thesis supervisor, for his introduction of the sub-ject and his guidance and valuable suggestions during the preparation of the manuscript. I am indebted to Greg Reid for his computer assistance and his software packages that saved me tens of days of calculations. Finally, I would like to thank Professor Brian Seymour for his reading of the final draft of the thesis. vi Chapter 1 Introduction 1.1 Symmetry Method for Differential Equations 1.1.1 Lie Groups of Transformations; Symmetries Symmetry methods for differential equations were originally developed by Sophus Lie approximately a century ago. Lie introduced the notion of continuous groups, known as Lie groups or symmetry groups, for applications to differential equations. Based on invariance of a system of differential equations under a L ie group of transformations, a symmetry group is a group of transformations which maps any solution of the system to another solution of the same system. The symmetry group admitted by a system of differential equations can be found systematically using Lie's algorithm. Applications of symmetry groups to differential equations include reduction of order of ordinary differ-ential equations, reduction of the number of independent variables of partial differential equations, construction of invariant solutions to ordinary and partial differential equa-tions, construction of invariant solutions to boundary value problems, construction of conservation laws using Noether's theorem, mapping a given class of differential equa-tions (e.g. nonlinear equations) to a target class of differential equations (e.g. linear equations), and more. A n algorithmic approach to symmetries and differential equations can be found in a recent book by Bluman and Kumei [1]. In the studies of a system of partial differential equations ( P D E ' s ) one lets x = ( x 1 ? . . . , xn) 6 IRN denote the independent variables, and u = ( u 1 , . . . , u m ) G IR M denote 1 Chapter 1. Introduction 2 the dependent variables. For each (x,u) lying in region D C l R " + m one can define a set of transformations x* = Xi(x,u;e), i = l , 2 , . . . , n , (1.1a) (u")* = C/M(a;,ii;e), ^ = 1 ,2 , . . . , m , (1.1b) where e is a continuous parameter lying in some set S C IR. Then (l.la,b) is a one-parameter (t) local Lie group of point transformations on D if the following axioms hold: 1. The transformations (l.la,b) are one-to-one onto D for all e £ S, and (x*,u*) lies in D. 2. S forms a group with a law of composition (j) between elements in S satisfying the following: (a) For any e,6 € S, <f>(e,6) G S. (b) For any elements e, 6,7 (E S, <j>(e,<f>(8,f)) = 4>(4>(t,8),^). (c) There exists a unique identity element e of S such that for any e £ S, <f>(e, e) = <f>(e,e) = e. (d) For any e £ S there exists a unique inverse element e _ 1 £ 5 such that 3. If e = e is the identity element, then (x*,u*) = (x,u). 4. If e = e - 1 is the inverse element, then x = X(x*, u*; e _ 1), u = U(x*, u*; e _ 1). 5. If (x*,u*) are given by (l.la,b) and (x**,u**) = (X(a:*, «*; <!>), U(x*, u*; 6)), then u**) = (X(x, u; 6), C7(x, u; <5))). 6. e is continuous in 5, i.e. 5" is an interval in IR. Chapter 1. Introduction 3 7. (X, U) is infinitely differentiable with respect to (x, u) in D and an analytic function of e in S. 8. <j>(t, (5) is an analytic function of e and (5 in 5. Without loss of generality, one may take e = 0 to be the identity element of the group, [i.e. ( x*,ii*) = (x ,u) when e = 0], and e _ 1 = —t to be the inverse element of the group [i.e. x = X(x*, u*;—e),u = U(x*,u*; —e)]. Furthermore, one may assume that the law of composition of the group parameters is addition, that is, if x* = X(x\ e) and x** = X(x*;8) then x** = X(x;e + 6). The symmetry ( l . l a , b ) is a local symmetry since at any point x the transformations near the identity are determined if u(x) is sufficiently smooth in some neighbourhood of x. The transformations ( l . l a , b ) are called point transformations since points (x ,u) are mapped to new points (x*,u*). 1.1.2 Infinitesimal Transformations Since (X, U) is assumed to be analytic in e, one can expand ( l . l a , b ) about the identity e = 0 using Taylor's theorem. Then ( l . l a , b ) becomes x* = x,- + e£,-(x,u) + 0(e 2 ) , (1.2a) (u")* = u " + er/"(x,u) + 0(e 2 ) . (1.2b) Equations (1.2a,b) are called the infinitesimal transformations of the group ( l . l a , b ) , and c ( \ d X i , r / M (x,u) c=0 are called the infinitesimals. Using Lie 's fundamental theorem, one can find the global transformations ( l . l a , b ) if the infinitesimals of the group are known. Hence, the local Lie group of transformations ( l . l a , b ) is completely characterized by its infinitesimals. Chapter 1. Introduction 4 The studies of infinitesimal transformations are enhanced by the corresponding in-finitesimal generator which is a first-order differential operator denoted by x=^u)^t+^u)£-^ (i-3) The infinitesimal generator (1.3) is also called a vector field. 1.1.3 Multiparameter Lie Groups In general one can have an r-parameter Lie group of transformations in which case the infinitesimals form a n r x n infinitesimal matrix with entries dx* de Q = l,2 , . . . , r , i = l,2 , . . . , n , e € IRr. t=0 The corresponding infinitesimal generators are X a = tai-z— 5 a = 1,2,... , r . OXi The commutator of two generators, X Q and X^, is another first-order operator, [XcX^g] = XaX() — X ^ X Q = C ^ X - p where C2p are called the structure constants. A Lie algebra £ is a vector space spanned by the infinitesimal generators. There exists a law of composition in C, namely the commutator, and most importantly C is closed with respect to the commutation, i.e. if Xa,Xp £ C, then [X0X/3] € C. 1.1.4 Extended Transformations Given the Lie group of transformations (l.la,b) one can determine how the derivatives of u(x) are transformed. Explicit formulas for the extended transformations, or prolon-gations, of the group (l.la,b) are given in [1]. Chapter 1. Introduction 5 Let u denote the set of coordinates corresponding to all fcth-order partial derivatives of u with respect to x. A typical coordinate in u wi l l be of the form u -,,,a-** dxhdxi2---dxtk with p = 1 ,2 , . . . , m , ij = 1 ,2 , . . . , n , and j = 1 ,2 , . . . , k. The kth extension of a one-parameter Lie group of transformations and the corresponding kth extended infinitesimal transformations are given by x* = Xi(x, u; e) = x{ -f e£t(x, u) + 0(e2), (u»)* = £/"(x,u;e) = u'i + e7?'J(x,U) + 0(e2), « ) * = c/r(x,u,u;e) = ur + e7 ? ! 1 ) ^ ,u , i i ; e ) + (9( e 2), ("ilia"-.-*)* = U?ii2...ik(x,u,u,... ,u; c) = <t-2...,-k + ^ !*£!..t-t(x,u,u,... ,u; e) + 0(e2), where i = 1 ,2 , . . . , n, it = 1 ,2 , . . . , n for / = 1 ,2 , . . . , k, wi th A; = 1 ,2 , . . . , and {Vi^...ik} are the kth extended infinitesimals. The corresponding kth extended infinitesimal gener-ator is denoted by + ^».*(af»u>V'---'if)^~J: ' fc = 1 > 2 > - - - - (1-4) One can derive explicit formulas for the extended infinitesimals given as follows: 7 /p = D , / - ( D ^ X , (1.5a) nit.* = D ^ K l , - P . ^ K . , . , ( f c _ i P , (i.5b) where i = 1 ,2 , . . . ,ra, ij = 1,2, . . . , n for / = 1 ,2 , . . . , k wi th k = 2, 3 , . . . , and _ D 5 u d u d u d is the total derivative operator. C h a p t e r 1. I n t r o d u c t i o n 6 1.1.5 Invariance of Differential Equations Any Mh-order system of partial differential equations with m dependent variables u = (u1,... ,um) and n independent variables x = (xj , . . . ,x n ) can be considered to be hy-persurfaces in (x, u, u,..., u)-space given by 1 k F"(x, u, u,..., u) = 0, fi = 1,2,... ,m. (1.6) 1 k The Lie group of transformations (l.la,b) is admitted by (1.6), or (1.6) is invariant under (l.la,b), if and only if for each v = 1,2,..., m, F*(x>W,...,tO = 0 1 k when F**(x, u, u,..., u) = 0, /z = 1,2,..., m. I That is, the transformed system of equations and the original system of equations must look the same. A criterion for invariance of a system of partial differential equations is given by the following theorem which can be found in [1], Theorem 1.1 The system of PDE's (1.6) is invariant under the Lie group of transfor-mations (l.la,b) if and only if X w F I / ( x , u , « , . . . , « ) = 0, v = 1 ,2 , . . . ,m, (1.7a) 1 A: when F"(x,u,u,...,u) = 0, u = 1 ,2 , . . . ,m, (1.7b) 1 k where X^V is the kth extended infinitesimal generator given by (1-4)-Assume that each P D E in the system (1.6) can be written in solved form uti2-k„ = F(x,u,u,..., u), (1.8) Chapter 1. Introduction 7 where / M does not depend on the /Mth order partial derivative tt£°,-2...,-, , <r = 1,2,..., m, for each /z = 1,2,..., m. Then, from Theorem 1.1 and (1.7a,b) one sees that the system of PDE's (1.8) admits infinitesimal generator (1.3) with kth extension given by (1.4) if and only if +$t*aZT—i M = l,2 , . . . , m , (1.9a.) w hen <V-.,„ = / " ( x , « , « , - . . , « ) , a = l , 2 , . . . , m . (1.9b) It happens in most applications that /M(x, u, u , . . . , u), /i = 1,2,..., m, is a polynomial 1 k in the components of w, . . . ,u . By substituting (1.9b) into (1.9a) one can show that the resulting equations are polynomials in the components of u, p < k, whose coefficients are linear homogeneous in & and 77^  and their partial derivatives up to pth order. Setting these coefficients to zero one obtains a linear homogeneous system of PDE's (overdetermined in most cases) for £ and rj called the determining equations. The infinitesimals (£i,f7M) for the symmetries (l.la,b) admitted by (1.8) can then be found by solving the system of determining equations. The determining equations can be set up automatically using a symbolic manipulation program. 1.1.6 Invariant Solutions Suppose a system of PDE's (1.6) admits a one-parameter Lie group of transformations with infinitesimal generator given by (1.3). Then u — 0(x) is an invariant solution of (1.6) if and only if u = 0(x) solves (1.6) and X(u" - 0"(x)) = 0 when u = 0(x), v = 1,2,..., m, Chapter 1. Introduction 8 i.e. d<dv £ , ( x , 0 ( x ) ) — = ,/"(*,0(x)), i/ = l , 2 , . . . , m . (1.10) OXi Equations ( 1.10) are the invariant surface conditions for invariant solutions correspond-ing to (1.3). The invariant solutions u = Q(x) can be found by solving the invariant surface con-ditions ( 1.10). The corresponding characteristic equations are given by dx} dx2 dxn du1 du2 dum If Xi(x, u),X2(x, u),...,Xn_y(x, u), u1(x, u), v2(x, u),..., vm(x, u), are independent con-stants of integration of ( 1.11) with the Jacobian d(v\v2,...,v™) d(u\u2,...,um) * ' then the invariant solution u = 0(x) is given implicitly by the invariant form v"(x,u) = <f>v(X1(x,u),X2(x,u),... ,Xn-x(x,u)), v = 1,2,..., m, (1-12) where <f>", v — 1,2, . . . , m , are to be determined. The independent constants Xi, i = 1,2,..., n — 1, and v", v = 1,2,..., m, are invariants of X of (1.3) such that XA r , = 0, i = 1,2, . . . , n — 1, and Xu 1' = 0, i/ = 1,2,..., m. Substituting ( 1.12) into the given system of PDE's (1.6) one gets a reduced system of PDE's with n — 1 independent variables (XL,X2,... ,Xn_i) and m dependent variables ( i A , u 2 , . . . ,vm). Solving this reduced system of PDE's yields the invariant solution u = 0(x) in the form ( 1.12). The independent variables (Xi, X2,..., X „ _ i ) are called similarity variables for the symmetry (1.3). Chapter 1. Introduction 9 1.2 Potential Symmetries for PDE's 1.2.1 Overview One can enlarge the classes of symmetries to nonlocal symmetries whose infinitesimals symmetries act on a larger space than the space of independent and dependent variables and their derivatives. In particular a symmetry is a nonlocal symmetry if its infinitesimals depend on integrals of the dependent variables. Bluman, Kumei, and Reid [2] systematically find nonlocal symmetries admitted by a given system of differential equations by realizing such symmetries as local symme-tries admitted by an auxiliary system of differential equations associated with the given system. The idea is illustrated by the following example found in [2]. Consider the linear wave equation depend on the global behaviour of u(x) at any point x. The infinitesimals of nonlocal (1.13) where c(x) is the wave speed. By inspection, one can express ( 1.13) as a conserved form (1.14) By introducing a potential v, one obtains an auxiliary system vt - u XI (1.15a) (1.15b) associated with ( 1.13). Equations ( 1.13) and ( 1.15a,b) are equivalent in the following senses: If {u(x,t),v(x,t)} solves ( 1.15a,b), then u(x,t) solves ( 1.13). Chapter 1. Introduction 10 If u(x, t) solves ( 1.13), then there exists a v(x, t) (not unique) such that {u(x, t), v(x, t)} solves ( 1.15a,b). Hence, one says that ( 1.13) is embedded in the auxiliary system ( 1.15a,b). Since if (u,v) solves ( 1.15a,b) then so does (u,v + C) for any constant C, the trans-formation from ( 1.13) to ( 1.15a,b) is non-invertible. A local symmetry admitted by ( 1.15a,b) maps any solution of ( 1.15a,b) to another solution of ( 1.15a,b). Since ( 1.13) is embedded in ( 1.15a,b), a local symmetry of ( 1.15a,b) also maps any solution of ( 1.13) to another solution of ( 1.13). As a result, a symmetry of ( 1.15a,b) induces a symmetry admitted by ( 1.13). This induced symmetry of ( 1.13) is a nonlocal symmetry if the infinitesimal corresponding to any of the variables (x,t,u) depends explicitly on the potential v. Any nonlocal symmetry obtained in this way is called a potential symmetry. The idea given in the above example can be generalized as follows: Suppose at least one P D E of a given system of PDE's R{x,u}, with independent variables x and dependent variables u, can be expressed in a conserved form with respect to some choice of its variables. Then one can introduce new variables v which are potentials for the conserved form. With these potentials and the original variables one can form an auxiliary system of PDE's S{x,u,v}. By construction, any solution {u(x),v(x)} of S{x,u,v} defines a solution u(x) of R{x,u}. The given system R{x,u} is then said to be embedded in the auxiliary system S{x,u,v}. Since the potentials v appear only in derivative form, the transformation from S{x,u} to R{x, u, v} -is non-invertible. Let Gs be a local Lie group of point transformations admitted by S{x,u,v}. Then any solution of S{x,u,v] is mapped to another solution of S{x,u,v] under the action of Gs- Since R{x,u} is embedded in S{x,u,v}, any symmetry in Gs also maps any solution of R{x, u} to another solution of R{x, u}. Consequently, Gs induces symmetries admitted by R{x,u}. A local symmetry in Gs is a nonlocal symmetry admitted by Chapter 1. Introduction 11 R{x,u} if the infinitesimal corresponding to any of the variables (x,u) depends explicitly on the potentials v. The nonlocal symmetries obtained in this way are called potential symmetries of R{x, u}. Since potential symmetries are nonlocal symmetries which are realized as local sym-metries in the (x,u,v)-space, they can be found by Lie's algorithm given in §1.1.5. 1.2.2 Finding Potential Symmetries A scalar P D E R{x, t, u} with two independent variables (x, t) and one dependent variable u is considered here. More general cases can be found in Bluman and K u m e i [1]. Suppose R{x,t,u} can be written in a conserved form Tx-~Ft=^ ( U 6 ) where / = f(x,t,u,u,..., u ) and g = g(x,t,u,u,..., u ). Then one can introduce a 1 k~1 1 k—1 potential v and form an auxiliary system S{x,t,u,v}: %-f, (.-».) | = , (1.17b) Assume S{x, t, u, v} given by ( 1.17a,b) admits a local Lie group of point transformations Gs: x* = Xs{x,t, u,v; e) = x + e£s(x,t, u,v) + 0 ( e 2 ) , (1.18a) r = Ts(x,t,u,v-e) = t + eTS(x,t,u,v) + 0(e2), (1.18b) u* = Us(x,t,u,v;e) = u + ens(x,t,u,v) + 0(e2), (1.18c) v* = Vs(x,t,u,v;e) = v-re(s(x,t,u,v) + 0(e2), (1.18d) where ^ 5 , Ts, ns, and (s a r e infinitesimals corresponding to x, t, u, and v, respectively. Chapter 1. Introduction 12 The corresponding infinitesimal generator for Gs is denoted by d d $ d X 5 = £s(x,t,u,v)— + TS(x,t,u,v)— + r]S(x,t,u,v)— + (s(x,t,u,v) — . (1.19) ox ot OU OV The infinitesimals {£s,T~s,r]s,(s} can be found using Lie's algorithm described in §1.1.5. A local symmetry in Gs maps any solution of S{x,t,u,v] to another solution of S{x,t,u,v}. Since R{x,t,u} is embedded in S{x,t,u,v}, Gs also maps any solution of R{x,t,u} to another solution of R{x,t,u}. Hence, a local symmetry in Gs induces a symmetry admitted by R{x, t, u}. This induced symmetry is a potential symmetry.which is a nonlocal symmetry if any of the infinitesimals {<fs> rs, ^s} depends explicitly on v. Let R{x,t,u) admit a Lie group of transformations GR with infinitesimal generator of the form d d d X;? = £R(x,t,u) — + TR(x,t,u)— + r)R(x,t,u)^-. (1.20) Ox Ot Ou Then one notes that the infinitesimals £R(x,t,u) and rR(x,t,u) of ( 1.20) may not be equivalent to £s(^ , t, u, v) and TS(X, t, u, v) of ( 1.19) in the sense that there exists no infinitesimal in GR such that (£R, rR) = (£S,T~S)- For the case of a linear P D E R{x,t,u) and a system of PDE's S{x, t, u, v} the infinitesimals depend only on (x, t). The potential symmetry with (£s, 7"s) ^ 0 is defined to be a nontrivial or type I potential symmetry if there is no infinitesimal in GR such that (£R,TR) = (£S,T~S)- Otherwise, the potential symmetry is called a trivial or type II potential symmetry if there is some infinitesimal in GR such that (£R,rR) = US,TS)-It should be noted that the similarity variables corresponding to a potential symmetry of type II of R{x,t,u} are identical to those for some point symmetry in GR. Hence, a type II potential symmetry is conjectured to be not useful in constructing new invariant solutions of R{x,t,u}. Chapter 1. Introduction 13 1.3 Finding Symmetries of Differential Equations 1.3.1 Algorithm We have seen in §1.1.5 that the problem of finding symmetries admitted by a given system of differential equations is equivalent to solving a system of linear homogeneous PDE's called the determining equations of the group. Symbolic manipulation programs have been developed in attempt to solve determining equations. Kersten [3] and Schwartz [4] are earlier workers in this area of research. Their programs, based on heuristics, work on only a limited number of examples that have been solved before. Recently, an algorithm has been developed by Reid [5] who systematically reduces a system of determining equations to a system of first-order PDE's called the standard form of the determining equations. By examining the standard form one can determine the dimension of the solution space of the determining equations and hence the number of parameters in the group. In this section we will give an overview of the algorithm developed by Reid [5] to solve the determining equations. Suppose one has set up a system of determining equations using Theorem 1.1. The following notations are used: xn+fi = v.", ^ = 1,2, . . . , m , Vi = t » »' = l , 2 , . . . , n , Vn+I1 - 77", u = 1,2,... ,m. By introducing new variables defined as derivatives of the infinitesimals Vi, one can Chapter 1. Introduction 14 express the determining equations as a first-order system of the form TT 1 = E ^ W ^ ' * e x, j e J „ fce«, (1.21) for some index sets X, Ti, and Ji C {1,2,..., n -f m) for each i E X. The integrability conditions d2Vi d7Vi i €X, j,k £ Ji, j < k, (1.22) dxjdxk dxkdx^ of (1.21) are appended to ( 1.21) to form a new system of linear PDE's. Then, if necessary, one relabels the derivatives of the V"s and forms a new first-order system of the same form given by ( 1.21) with the index sets X, Ti, and Ji being modified accordingly. The algorithm repeats the same step of appending to ( 1.21) the integrability condi-tions ( 1.22) of ( 1.21). This process will either stop when all the integrability conditions are identically satisfied or it will never terminate. If the algorithm terminates after a finite number of steps, it will produce the standard form ( 1.21) with Ti C X, and the size of the group is equal to the number of Ws appear in ( 1.21). The algorithm will not terminate if the group is infinite dimensional. The algorithm has been developed and implemented by Reid using the symbolic language MACSYMA. A new version in the symbolic language MAPLE is currently under development. 1.3.2 Classification Problems An important application of Reid's algorithm is to solve group classification problems where a given system of PDE's involves one or more variable coefficients. A typical example of such problem is the classification of the wave equation ( 1.13) which contains a variable coefficient, namely, the wave speed c(x). The size of the group admitted by the wave equation ( 1.13) depends on the functional form of the wave speed c(x) which Chapter 1. Introduction 15 is usually characterized by some ordinary differential equations called the classification-equations. If a given system of PDE's involves variable coefficients then the determining equa-tions of the group admitted by the system also involve these coefficients and their deriva-tives. Hence, a set of determining equations can be expressed in matrix form M A = 0, (1.23) where M is a matrix whose entries may involve variable coefficients and their derivatives, and A is a column vector consisting of the infinitesimals and their derivatives J ^ . During the process of transforming the determining equations into the first-order system ( 1.21) the linear system ( 1.23) is triangularized into an upper triangular form. The leading coefficients (the pivots) of this triangular system involve the variable coefficients and their derivatives. In order to row-reduce the system one divides the system by the pivots which are assumed to be nonzero. Pivots generated in the process are stored. After the algorithm terminates with the standard form ( 1.21) it has stored a list of pivots which have been assumed to be nonzero. The program runs into completion and one gets a standard form ( 1.21) which holds for arbitrary variable coefficients in the given system. The pivots are in general nonlinear ordinary differential equations for the variable coefficients. One can set one or more of these pivots to zero and rerun the program again. A new standard form is obtained and is valid subject to the specified ODE's and a new list of nonzero pivots. The nonlinear ODE's are called the classification equations since they determine the standard form and hence the size of the group. Various subcases arise by setting different pivots to zero. Chapter 1. Introduction 16 1.4 Invariance Properties of the Wave Equation 1.4.1 Group Analysis of the Wave Equation c2(x)uxx = uit A complete group classification of the wave equation c2(x)uxx = uu, (1-24) has been given in a paper by Bluman and Kumei [6]. Here we summarize some important results. It has been proved in Ovsiannikov [7] that a single second-order hyperbolic P D E with two independent variables admits a Lie group of point transformations with at most four parameters; otherwise, it admits an infinite parameter group. It has also been proved in [7] that if a scalar second-order linear P D E admits a Lie group of transformations then the corresponding infinitesimal generator must be of the form d d d X = t(x,t)— + r(x,t)- + f(x,t)u—. (1.25) The following theorem has been proved in [6]: Theorem 1.2 The wave equation ( 1.24) admits a four-parameter Lie group of point transformations if and only if the wave speed c(x) satisfies the following fifth-order ODE dx or, equivalently, LI* H'" 3{2(H')3 - 2HH'H" - (H")2) 2H' + H2 (2H' + H2)2 = 0, (1.26) C 2 ( Q ' - Ha)' a (1.27) where a(x) = [2i/' + H2] 2, H = c'/c, and a is an arbitrary real or imaginary constant. In this case the infinitesimals are given by £ = a(x)\pe°* + qe-*], (1.28a) Chapter 1. Introduction 17 T = o--l[a'- HctWpe*1 - qe-at) + r, (1.28b) / = ^aHlpe^ + qe-^ + s, (1.28c) wnere {p, q, r, 5} are the group parameters. The group becomes infinite if and only if c(x) = (Ax + B)2, where A and B are arbitrary constants, fn this case the infinitesimal £ satisfies U - ( l / c 2 ) £ « - = 0, (1.29) and {T,/} are gz'wen 6?/ r = / « . - * « * , / = ^ - d-30) J4// other arbitrary wave speeds lead to a trivial two-parameter group of translations in t and scalings of u (i.e. X i = and X 2 = ' m § ^ ) -The solutions of the classification equation ( 1.26) and the corresponding invariant solutions have been constructed in [6]. 1.4.2 Group Analysis of the System vx = ut/c2(x), vt = ux In the same paper [6] Bluman and Kumei applied the idea presented in §1.2 to find potential symmetries of the wave equation ( 1.24). They examined an auxiliary system vx = ut/c2(x), (1.31a) vt = ««, (1.31b) associated wi th ( 1.24). One can show that ( 1.31a,b) admits an infinitesimal generator of the form d d d d X = t(x,t)— + r(x,t) — + [/0M)u + 9(x,t)v]— + [k(x,t)v + l(x,t)u}—. (1.32) The following theorems have been proved in [2] and [6]: Chapter 1. Introduction 18 Theorem 1.3 The system ( 1.31a,b) admits a four-parameter Lie group of point trans-formations if and only if the wave speed c(x) satisfies the fourth-order ODE [cc'(c/c')"\' = 0. (1.33) ff the wave speed c(x) does not satisfy ( 1.33), then the system ( 1.31a,b) admits only a two-parameter group of translations in t and uniform scalings in both u and v (i.e. x i = ft a n d x i = uiL + v-§-J-Theorem 1.4 For any wave speed c(x) satisfying ( 1.33), there exists a potential sym-metry of the wave equation ( 1.24). Theorem 1.5 ff the wave speed c(x) satisfies the third-order ODE (c/c')" = 0, (1.34) whose solution is c(x) = (ax + py (1.35) for arbitrary constants {a,(3,j}, then the wave equation ( 1.24) admits a trivial potential symmetry of type ff and admits no nontrivial potential symmetries of type f. ff the wave speed c(x) satisfies cc'(c/c')" = const ^  0, (1.36) then the wave equation ( 1.24) admits two nontrivial potential symmetries of type f and admits no trivial potential symmetries of type II. The following theorem has been proved in [8]. Theorem 1.6 A wave speed c(x) simultaneously satisfies ODE's ( 1.26) and ( 1.33) if and only if c(x) satisfies either ( 1.34), whose solution is given by ( 1.35), or c2c'c"' + c(c'fc" - c2(c")2 - j(c')4 = 0. (1.37) ft Chapter 1. Introduction 19 The solution of ( 1.37) consists of two families of solutions y/c — arctanC\/c= Ax + -B, (1.38a) 2 ^ + l o g | ( v ^ - C ) / ( V ^ - r C ) | = Ax + B, (1.38b) where A, B, and C are arbitrary constants. The common solutions for ODE's ( 1.34) and ( 1.37) are c(x) = (Ax + B)\ c(x) = (Ax + B)2/z. 1.4.3 Forms of the Wave Speeds Integrating the classification equation ( 1.33) once yields cc'(c/c')" = const = p. (1.39) Solutions of ( 1.39) have been classified in [6] as follows: If u = 0, to within arbitrary scalings and translations in x, the solution of ( 1.39) reduces to either c(x) — ex or c(x) = xc, where C is an arbitrary constant. The corresponding invariant solutions have been constructed in [6]. If u 7 ^ 0, one can set u = ±1 by scaling c and x appropriately. Then the solution of ( 1.39) reduces to the following canonical forms: c' = v~x sin(i/logc); (1.40a) c' = ^-1sinh(j/logc); (1.40b) c' = logc; (1.40c) c = v'1 cosh(i^ log c); (1.40d) where v ^ 0 is an arbitrary constant. If c(x) = <f>(x,v) is a solution of any one of the equations ( 1.40a-d) then the corresponding general solution of ( 1.39) is c(x) — K(j>(Lx + M, u), where K2L2 = \u\ for arbitrary constants L, M, and v. Chapter 1. Introduction 20 In a paper by Bluman and Kumei [9] various invariant solutions of the system ( 1.31a,b) have been constructed for wave speeds satisfying ( 1.40a). These wave speeds are bounded away from zero and can be used to model two-layered media with a smooth transition. The invariant solutions are superposed to solve general initial value problems. Chapter 1. Introduction 21 1.5 Noether's Theorem and Conservation Laws 1.5.1 Euler-Lagrange Equations The problem of the calculus of variations involves the determination of an extremum of the functional J[u] = J L(x,u,u,... ,u)dx, (1-41) where L is called a Lagrangian and the functional J[u] of ( 1.41) is an action integral. The value of u(x) is usually subject to a set of boundary conditions prescribed on the boundary of the domain Q. Assume L is (k + 1) times differentiable. Then one can show that for a smooth function u(x) to be an extremum of the action integral J[u] of ( 1.41) it is necessary that u(x) satisfies the Euler-Lagrange equations = H - + D . D , | | + ' • • + ( - D ^ D , , • • • D ^ g j g - = 0, (1.42) for fi = 1,2,..., m, where « l « 2 - « l t is defined as the Euler operator. 1.5.2 Variational Symmetries One of the most important application of infinitesimal transformation methods is the construction of conservation laws by the use of Noether's theorem. When a system of differential equations arises from a variational formulation, Noether [10] proved that for every infinitesimal transformation which leaves the action integral of a Lagrangian invariant one can construct a conservation law for the system. There are two formulations Chapter 1. Introduction 22 of Noether's theorem: Boyer's formulation and Noether's formulation. B o t h formulations can be found in [1]. Here only Boyer's formulation of Noether's theorem is discussed. Noether's formula-tion can be found in [1]. Noether first considered infinitesimal transformations of the form x* = x + e£(x,u,y,... ,u) + 0 (e 2 ) , (1.44a) u* = u + et](x,u,u,... ,u) + 0(e2), (1.44b) which are now called Lie-Bdcklund transformations. One can show that any transforma-tion of the form ( 1.44a,b) is equivalent to a transformation of the form x* = x, (1.45a) u* = u + e[r](x,u,u,... ,u) — Ui£i(x,u,u,... ,u)] -f 0 ( e 2 ) , (1.45b) which leaves the independent variables x invariant. The advantage of using ( 1.45a,b) is that the extended transformation for ( 1.45a,b) is easy to obtain. One can show that the kth extended infinitesimal generator corresponding to the H h extension for ( 1.45a,b) is u<" - +* +<s4+"""+^tz' (L46) where T#.... = D . D , • • • D . i / " . Boyer used transformations of the form ( 1.45a,b) to derive Noether's theorem. A transformation of the form ( 1.45a,b) is a variational symmetry of the action integral ( 1.41) if for any u(x) there exists some vector function A(x, u, u,..., u) = (A\A2,...,An) such that U W Z = D.-A''. (1.47) For arbitrary u and r) one can establish the identity V{k)L = E„(L)T1>i-rDiWi[u,T1}, ^ = l , 2 , . . . , m , (1.48) Chapter 1. Introduction 23 where is the Euler operator given by ( 1.43) and W = (W1, W2,..., Wn) is some vector function. Hence, if U is a variational symmetry for L and u(x) is a solution of the Euler-Lagrange equations, i.e., E^L) = 0, u = 1,2,... ,m, then by equating ( 1.47) and ( 1.48) one obtains a conservation law D,-(W* — A') = 0. This leads to Boyer's formulation [1] of Noether's Theorem as follows: Theorem 1.7 If\] = n^-^ is the infinitesimal generator of a variational symmetry of an action integral ( 1-41) such that U^L = D,vl' holds for anyu(x), then for any solution u(x) of the Euler-Lagrange equations given by ( 1.42) one can construct a conservation law TJi(W'[u,rj] - A*) = 0. The following theorem shows the relationships between a variational symmetry for the Lagrangian L and a Lie-Backlund symmetry admitted by the Euler-Lagrange equations. Theorem 1.8 If the Lie-Backlund transformation ( 1.45a,b) is a variational symmetry for the Lagrangian L such that ( 1.47) is satisfied, then ( 1.45a,b) is a Lie-Backlund symmetry admitted by the corresponding Euler-Lagrange equations E^L) = 0, p = 1,2,..., m. The proof of Theorem 1.8 is given in [1]. It should be noted that the converse of Theorem 1.8 is not true. That is, a Lie-Backlund symmetry admitted by the Euler-Lagrange equations is not necessary a variational symmetry for the Lagrangian L. However, one can first find all Lie-Backlund symmetries admitted by the Euler-Lagrange equations and then determine whether a Lie-Backlund symmetry is a variational symmetry by checking if the symmetry satisfies condition ( 1.47). In chapter 3 we will apply Boyer's formulation of Noether's theorem to construct conservation laws of the scalar wave equation ( 1.24). Chapter 1. Introduction 24 1.6 New Potential Symmetries for the Wave Equation In the next few chapters we wi l l further investigate the invariance properties of the wave equation ( 1.24). We wi l l consider various auxiliary systems associated with ( 1.24). Our objective is to find a larger class of symmetries admitted by ( 1.24). In particular, we wi l l find new wave speeds which lead to new potential symmetries of the wave equation ( 1.24). Moreover, we wi l l establish relationships among symmetries admitted by various auxiliary systems under consideration. In chapter 2 the system ( 1.31a,b) associated with the wave equation ( 1.24) is ex-pressed as other conserved forms by introducing appropriate potentials. We call these new conserved forms cascaded systems. A point symmetry admitted by a cascaded system may be a potential symmetry of the system ( 1.31a,b) and hence a potential symmetry of the wave equation ( 1.24). We wi l l show that the classification equation for one of these cascaded systems is a fifth-order O D E different from the O D E ( 1.26) associated wi th the scalar wave equation ( 1.24). We wi l l show that this new fifth-order O D E is the classification equation associated with another scalar wave equation related to ( 1.24). This related scalar wave equation wi l l be analyzed. We wi l l discuss how symmetries are induced from one system to another, and how to enlarge the symmetries admitted by the wave equation ( 1.24) by using one of the cascaded systems. In chapter 3 we construct conservation laws for the wave equation ( 1.24) using Noether's theorem described in §1.5. We apply the variational principle by consider-ing the wave equation ( 1.24) as the Euler-Lagrange equation for a Lagrangian. B y checking whether the point symmetries admitted by the Euler-Lagrange equation are variational symmetries for this Lagrangian, we construct three conservation laws for the wave equation ( 1.24). These conservation laws lead directly to three nonlinear conserved forms for the wave equation ( 1.24). We wi l l also see how to obtain a sequence of higher Chapter 1. Introduction 25 order conservation laws by the use of a recursion operator. In chapter 4 we consider linear conserved forms for the wave equation ( 1.24). We wi l l first examine a general linear system wi th two dependent variables and arbitrary coefficients. In order to be a conserved form for the wave equation ( 1.24) the coefficients of the linear system must satisfy a set of P D E ' s . We wil l show that there exists infinitely many linear conserved forms for the wave equation ( 1.24). Two such linear conserved forms wi l l be analysed in details. One of these two conserved forms leads to the discovery of a new classification equation for the wave speed c(x). In chapter 5 we summarize all the results presented in this thesis and propose some open problems. Chapter 1. Introduction 26 1.7 Chapter Summary In this chapter we have given a review of Lie group of transformations and its application to differential equations. We have seen how to find local and nonlocal symmetries which leave invariant a given system of PDE's. We have discussed Reid's algorithm for solving determining equations and how it can be applied to classification problems. Earlier works by Bluman et al. on the invariance properties of the wave equation ( 1.24) have been summarized in §1.4. A brief introduction of Noether's theorem and its application to construction of con-servation laws has been given in §1.5. Extensions to the invariance properties of the wave equation ( 1.24) developed in this thesis have been outlined in §1.6. Chapter 2 Cascading Potential Symmetries 2.1 Introduction to Cascaded Systems We have seen in §1.2 that if a given system of P D E ' s can be written in a conserved form then one can form an auxiliary system associated wi th it by introducing appropriate potentials as auxiliary variables. The auxiliary system often gives rise to new symmetries called potential symmetries admitted by the given system of P D E ' s . In this chapter we extend the idea given in §1.2 and construct a sequence of auxiliary systems associated wi th the given system of P D E ' s . Some of these auxiliary systems give rise to new potential symmetries admitted by the given system of P D E ' s . Consider a kth order P D E R{x,t,u} wi th independent variables x and t and a de-pendent variable u. Suppose R{x,t,u] can be written in a conserved form ^ - ^ = 0 (2 1) dx dt ' { l ) where / = f(x, t, u, u,..., u ) and g = g(x, t, u, u , . . . , u ). Then one can introduce a 1 k—1 1 k—1 potential v and form an auxiliary system S{x, t, u, v}: I-/. (2.2.) If ( 2.2a) can be written in a conserved form 27 Chapter 2. Cascading Potential Symmetries 28 where / = f(x,t, u, v, u,..., u ) and g = g(x, t,u,v,u,..., fcu2)> then one can introduce a potential <^> and form an auxiliary system Ti{x, i ,u, v,<f>}: 4>t = / , (2.4a) '• • ' 4 = 5, (2.4b) fx = 5- (2.4c) If ( 2.2b) can be written in a conserved form where / = f(x,t, u, v, u,..., u^^ ) and g = g(x, t,u,v,u,..., fcu2)) then one can introduce a potential w and form an auxiliary system T2{x,t,u, v, w): wt = f, (2.6a) wx = g, (2.6b) vt = f- (2.6c) If both ( 2.2a) and ( 2.2b) can be expressed in conserved forms given by ( 2.3) and ( 2.5), respectively, then one can combine Ti{x,t,u,v,(f>) and T2{x, t, u, v, w] to form another auxiliary system T3{x, t, u, v, w, <f\: 4>t = / , (2.7a) <t>* = 9, (2.7b) wt = f, (2.7c) wx = g. (2.7d) The auxiliary systems T^{x, t, u, v, <j)}, T2{x, t, u, v, w] and T3{x, t, u, v, w, <f>} obtained above are called cascaded systems associated with the given P D E R{x,t,u}. Chapter 2. Cascading Potential Symmetries 29 It is obvious that the cascading process can be continued if any of the equations ( 2.7a-d) can be written in a conserved form. The number of cascaded systems one can get is limited by the order k of the given P D E R{x, t, u}, since at each level of cascading the order of the cascaded systems is reduced by one. In the rest of this chapter we examine several cascaded systems associated with the wave equation ( 1.24). In particular we will study the invariance properties of these cascaded systems and their relationships with the wave equation ( 1.24). Chapter 2. Cascading Potential Symmetries 30 2.2 Three Cascaded Systems for the Wave Equation Let R{x,t,u} be the wave equation: c2(x)uxx = utu (2.8) and S{x, t, u, v} be the associated system: vt = ux, (2.9a) vx = (u/c2(x))t. (2.9b) Note that the potential v satisfies an associated wave equation R{x,t,v}: {c2(x)vx)x = vtt. (2.10) Equations ( 2.9a,b) are already in conserved forms. Hence, we can form three cascaded systems as follows: 4>x = v, (2.11a) <f>t = u, (2.11b) vx = (u/c2(x))t- (2.11c) T2{x,t,u,v,w}: wx = u/c2(x), (2.12a) wt = v, (2.12b) vt = ux; (2.12c) Chapter 2. Cascading Potential Symmetries 31 T3{x,t,u,v,w,<t>}: <f>x = v, (2.13a) fa = u, (2.13b) wx = u/c2(x), (2.13c) wt = v; (2.13d) We see that the potential <j> satisfies the wave equation R{x,t,<j>} c2{x)cj>xx = 4>u, (2.14) and the potential w satisfies the associated wave equation R{x,t,w} (c2(x)wx)x = wu. (2.15) Examining ( 2.13a-d) we see that (q>,w) satisfy the associated system S{x,t,d>,w} wt = <f>x, (2.16a) wx = (<f>/c2(x))t. (2.16b) In the next section we will examine each of the cascaded systems and see how the symmetries of the cascaded systems relate to the symmetries of R{x,t,v}, and S{x, t, u, v}. Chapter 2. Cascading Potential Symmetries 32 2.3 Induced Symmetries 2.3.1 Cascaded System Ti{x, t, u, v, <j>] Let Ti{x, t, u, v, <f>} given by ( 2.11a-c) admit a local Lie group of point transformations x* = X(x,t,u. , v, <f>\ t) = x + e((x, t, u, v, <j>) + 0(e 2), (2.17a) t* = T(x,t,u, v, <f>; t) = t + er(x, t, u, v, <f>) + 0(e 2), (2.17b) u* = U(x,t,u, v, </>; e) = u + erju(x,t,u,v,(f)) + 0(e 2), (2.17c) = V(x,t,u, v,(f>;e) = v + er)v(x,t,u, v, <t>) + 0(e2), (2.17d) = <&(x,t,u, v, <f>; e) = <j> + 01*(x, t, u, v, <f>) + 0(t2). (2.17e) d v by <j>t and <px, respectively, we have * x = X(x,t, <f>t, <f>x, 4>; e), (2.18a) t* = T(x,t,<j)U<f>x,<f>; e), (2.18b) P = $(x,t,<f>t,(j>x,(t>; e), (2.18c) * u = (4>tY = U(x,t,<f>t,4>x,<f>;e), (2.18d) V* = (<t>x)* = V(x, t, 4>t, 4>x, <f>; e). (2.18e) A group of transformations of the form ( 2.18a-e) defines a Lie group of contact transformations if the contact condition du = udx is preserved. If we know (f>* then we can find (<f>t)* a n d (<f>xY u s m g the extension formula given by (1.5a). Once ( (4)* , (<f>tY) are known we can find (u*,v*). Hence, if we can find a contact symmetry admitted by R{x, t, cf>}, then we can obtain a point symmetry admitted by Ti{x, t, u, v, <f>}. We say a contact symmetry of R{x,t, <f>} induces a point symmetry of the cascaded system Ti{x, t, u, v, 4>}. Conversely, a point symmetry of Xi{x, t, u, u, (j)} must correspond to a contact symmetry of R{x, t, <f>}. Chapter 2. Cascading Potential Symmetries 33 2-3.2 Cascaded System T2{x, t, u, i>, w} Let T2{x, t, u, v, w] given by ( 2.12a-c) admit a local Lie group of point transformations * X = X(x,t,u. ,v,w\e) = x + t£(x,t,u,v,w) + 0(e2), (2.19a) t* = T(x,t,u, v, w; e) = t + CT(x,t,u,v,w) + 0(e2), (2.19b) u* U(x,t,u, v,w;e) = u + enu(x, t, u, u, w) + 0(e2), (2.19c) V* — V(x,t,u, v, w; e) — v + eqv(x,t,u,v,w) + 0(t2), (2.19d) w* = W(x,t,u ,v,w;e) = •- w + erjw(x, t, u, v, w) + 0(e2). (2.19e) Replacing u and v by c2(x)wx and wt, respectively, we have x* = X(x, t, wx, wt, w; e), (2.20a) t* = T(x, t, wx, wt, w; e), (2.20b) w* = W(x,t,wx,wt,w; e), (2.20c) u* = (c2(x)wx)* = U(x,t,wx,wt,w;e), (2.20d) v* = (wt)* = V(x,t,wx,wt,w;e). (2.20e) Using the same argument given in §2.3.1 we see that a contact symmetry admitted by R{x,t,w} induces a point symmetry of the cascaded system T2{x, t, it, v, w} and, conversely, a point symmetry of T2{x, t, u, v, w} must correspond to a contact symmetry of R{x, t, w}. 2.3.3 Cascaded System T3{x, t, u, v, w, <f>} Let T3{x, t, u, t), w, <f)} given by ( 2.13a-d) admit a local Lie group of point transformations x* = X(x,t,u,v,w,<j);e) = x + e£(x,t,u,v,w, (f>) + 0(e 2), (2.21a) Chapter 2. Cascading Potential Symmetries 34 t* — T(x,t,u,v,w,<f>;t) = t + CT(x,t,u,v,w,(f>) + 0(e2), (2.21b) u* = U(x, t, u, v, w, <f>; e) — u + er]u(x, t, u, v, w, (j>) + 0(e2), (2.21c) v* = V(x,t,u,v,w,<f>;e) = v + erjv(x,t,u,v,w,<}>) + 0(e2), (2.21d) w* = W(x,t,u,v,w,(f);e) = w 4 - erjw(x,t,u,v,w, (f>) -f 0(e2), (2.21e) = $(x,<,u,u,u;,^ ;e) = + C T / ^ X , u, u, to, </>) + 0(e2). (2.21f) Replacing u by c ^ i ) ^ or <f>t and u by wt or <^>x, we have x* = X(x,t,wx,wt,<j>x,<f>t,w,<f>;e), (2.22a) f = T(x,t,wx,wt,4>x,<f>t,w,<j>;c), (2.22b) * — W(x,t,wx,wt,<f>x,<f>t,w,<i))e), (2.22c) = $(x,t,wx,wu<j>x,(t>t,w,<j);e), (2.22d) = (c2wx)* = (<j)t)* = U(x, t, wx, wt, <j>x, <j>t, w, <j>; e), (2.22e) V* = (wty = (4>xy = V(x,t, wx, wt, <f>x, (f>t, w, <f>; e). (2.22f) Since one can show that a one-to-one contact transformation for two or more depen-dent variables must be a one-to-one point transformation [11], ( 2.22a-f) defines a Lie group of point transformations acting on (x, t, w, (^ )-space. The transformations of (u, v) can be determined by knowing how the first partial derivatives of w and <f> transform. Hence, a point symmetry admitted by S{x, t, <f>, w} induces a point symmetry of the cascaded system T3{x,t, u,v, w, <f>}. Conversely, a point symmetry of T3{x,t,u,v,w,(f} must correspond to a point symmetry of S{x,t,u, v}. In the next few sections we will give a complete group classification of each of the cascaded systems. Chapter 2. Cascading Potential Symmetries 35 2.4 Group Classification of the System: cj>x = v, d>t = u, vx = u t/c2(x). Let the cascaded system T\{x, t,u, v,<j>} admit an infinitesimal generator d d d X = £(x,t,u,v,<f>)— + r(x,t,u, v,<f>)— + nu(x,t,u,v, <j>) — ox ot ou () d -rnv(x,t,u,v,<t>)-- + T)+fx,t)Uiv,d>)— (2.23) Ov 0q> corresponding to the Lie group of point transformations ( 2.17a-e). Examining the de-termining equations from invariance of Ti{x, t, u, v, <j)} under ( 2.23) one can show that £ = t{x,t), T = r(x,t), Vu = h(x,t)u + 9\(x,t)v + hi(x,t)4>, T]V = f2(x,t)u +g2(x,t)v + h2(x,t)<f>, •q4' •= f3(x,t)u +g3(x,t)v + h3(x,t)<j>. Hence, the infinitesimal generator ( 2.23 ) becomes X = ((x,t)— + r(x,t) — + (/j(z,<)u + gi(x,t)v + h!(x,t)<f>) — OX Ot OU + (f2(x,t)u + g2(x, t)v + h2(x, *)<£)^ + (/3(x, t)u + g3(x, t)v + h3(x, i)4>)j^- (2.24) Using Reid's algorithm described in §1.3 we classify the cascaded system Ti{x,t,u,v, <f>} as follows: Case I: 2cc" - (c')2 = 0. (2.25) The general solution of ( 2.25) is c(x) = (Ax + B)2, (2.26) Chapter 2. Cascading Potential Symmetries 36 for some arbitrary constants A and B. For the wave speed given by ( 2.26) the cascaded system T\{x,t, u, v, (j)} admits an infinite-parameter Lie group of point transformations. Case II: H' 2H' + H2 + 3(2(H')3-2HH'H"-(H")2) (2H' + H2)2 0, dx y where H = c'/c. In this case we obtain the following standard form of the determining equations cV" - 2cc'c" + (c')3 (2.27) dx di dt dr dx dr dt dh dx dt dgi dx dgi dt h 92 h2 fs 93 2c2c" - c(c')2 : -gi, —o9i, cc 2cc" - (c')2 (-4c3c"" - {c'Y)c" + 2c2(c')2c"" + 6c3(cw)2 8c3(c")2 - 8c2(c')2c" + 2c(c')4  (3c(cQ3 - 6cVc")c'" - 4c2(c")3 + 4c(c/)2(c")2 8c3(c")2 - 8c2(c')2c" + 2c(c')4 2c2c"' + 2cc'c" - (c')3 (2.28a) (2.28b) (2.28c) (2.28d) -9u Ac2c" - 2c(c')2  c2c"' - 2cc'c" + (c'f 2c2c" - c(c')2 ( - 2 c V -f c3(c')2)c"" + 3c4(c"')2 + (2c2(c')3 - 4c3c'c")c"' •9u 4c2(c")2 - 4c(c')2c" + {c'Y -2cg^ -cc'c'" + 2c(c')2 - (c')2c" 4c2c" - 2c(c')2 •6 0, 0, (2.28e) (2.28f) (2.28g) t, (2.28h) (2.28i) (2.28J) (2.28k) (2.281) (2.28m) (2.28n) Chapter 2. Cascading Potential Symmetries 37 cd" * 8 = " 2 ^ T ( ^ ^ + / l - ( 2 - 2 8 0 ) Equations ( 2.28a-h) form a Frobenius system which has a unique solution. Moreover, the number of parameters in the solution space of ( 2.28a-h) is equal to the number of dependent variables in the system ( 2.28a-h). In this case there are four parameters. The classification O D E ( 2.27) is identical to the classification O D E (1.26) for the scalar wave equation R{x,t, u}. The infinitesimals { £ , r , /13} are given by (1.22a-c), re-spectively. Using these three known infinitesimals one can easily determine the rest of the unknowns in ( 2.28a-o). Case III: c(x) arbitrary. In this case we have t = 9i = hi = fi = h2 = h = g3 = 0, and T = r, / 1 = g2 = h3 = s, where r and s are arbitrary constants. Here we have a t r ivial two-parameter group of translations in t and uniform scalings in u , D, and <f>. Hence, we have proved the following theorems: Theorem 2.1 The cascaded system Ti{x,t,u,v, cf>} admits an infinite-parameter Lie group of point transformations if and only if the wave speed c(x) satisfies c(x) = (Ax+B)2 for arbitrary constants A and B. Theorem 2.2 The cascaded system Ti{x, t, u , v, <j>} is admits a four-parameter Lie group of point transformations if and only if the wave speed c(x) satisfies the fifth-order ODE (2.27). Theorem 2.3 For any other wave speeds the cascaded system Ti{x,t,u,v,<f>} admits a trivial two-parameter Lie group of translations in t and uniform scalings in u, v, and (j). Chapter 2. Cascading Potential Symmetries 38 We have shown in §2.3.1 that a contact symmetry of R{x,t, </>} induces a point sym-metry of the cascaded system Ti{x, t, u, u, </>}. A point symmetry of Ti{x, t, u, v, <j>] must correspond to either a contact or point symmetry of R{x,t, <f>}. The fact that £ and r do not depend on (u, v, (ff) and 77* is linear in <f) (since f3 = g3 = 0) implies that there exists no contact symmetry for R{x,t,<f>). Hence, a point symmetry of Ti{x, t, u, v, <j)} corresponds to a point symmetry of R{x, t, </>}. Moreover, for any wave speed c(x) satisfying ( 2.27), a point symmetry of R{x,t, (f)} induces a point symmetry of Ti{x, t, u, v, <j>} which in turns induces potential symmetries of R{x,t,u} and R{x,t,v), since the infinitesimals for u and v depend explicitly on the auxiliary variable <f>. The potential symmetries of R{x,t,u} obtained through the use of Tt{x, t, u, v, <f>} are all trivial potential symmetries of type II, since the infinitesimals £ and r admitted by T\{x,t,u,v,(f)} are the same as those admitted by R{x,t,<f>}. A type II potential symmetry is not useful in constructing new invariant solutions of R{x, t, u} since the similarity variables for a type II potential symmetry of R{x,t,u} are identical to those for some point symmetry of R{x,t,u}. On the other hand, the potential symmetries of R{x,t,v] obtained through the use of Ti{x, t, u, v, <j)} can be shown to be all nontrivial potential symmetries of type f, since the infinitesimals £ and T admitted by T\{x, t, u, v, <f>) are essentially different from those admitted by R{x,t, v}. These new potential symmetries of R{x,t,v} are beyond those obtained through the use of the associated system S{x,t,u,v}. A group analysis of R{x,t,v} will be given in §2.7. In summary we have the following theorems: Theorem 2.4 For any wave speed c(x) satisfying the fifth-order ODE ( 2.27) a point symmetry of the wave equation R{x,t,u} induces a point symmetry of the cascaded sys-tem Ti{x, t, u, v, <f>}. Two of the four local symmetries of Ti{x, t, u, v, <f>] correspond to Chapter 2. Cascading Potential Symmetries 39 potential symmetries of the wave equation R{x,t,u} and the associated wave equation R{x,t,v). Theorem 2.5 The potential symmetries of the wave equation R{x,t,u} obtained through the use of the cascaded system Ti{x,t,u,v,<j)} are all trivial potential symmetries of type ff. Theorem 2.6 The potential symmetries of the associated wave equation R{x,t,v] ob-tained through the use of the cascaded system Ti{x,t,u,v,<f>} are all nontrivial potential symmetries of type f beyond those obtained through the associated system S{x, t, u, v}. Chapter 2. Cascading Potential Symmetries 40 2.5 Group Classification of the System: wx = u/c2(x), wt = v, vt = ux. It can be shown that the infinitesimal generator corresponding to ( 2.19a-e) admitted by the cascaded system T2{x, t, u, v, w} is of the form d d d d + (f2(x,t)u + g2(x,t)v + h2(x,t)w) — ov Q + (fsix, t)u + g3(x, t)v + h3(x, t)w)—. Ow (2.29) Using Reid's algorithm we classify the cascaded system T2{x, t, u, v, w} as follows: Case I: 2cc" + (c')2 = 0. (2.30) The general solution of ( 2.30) is c{x) = {Ax + B)2/3, (2.31) for arbitrary constants A and B. For the wave speed given by ( 2.31) the cascaded system T2{x, t, u, v, w} admits an infinite-parameter Lie group of point transformations. Case II: 'b(H')2 + 2HH" + H'" 3(3HH' + H")2' dx \ [ 3H2 + 2H' (3H2 + 2H')2 where H = c'/c. Equation ( 2.32) can be integrated once to give '5(H')2 + 2HH" + H'" 3(3HH' + H")2 0, (2.32) (3H2 + 2H')2 3H2 + 2H' where a is a real or imaginary constant. In this case we obtain the following standard form of the determining equations: (2.33) dx c2c'" - (c')3 2 c V + c(c')2 (2.34a) Chapter 2. Cascading Potential Symmetries 41 dt dr 1 9n (2.34b) dx c 2 — 7 < 7 i , ( 2 . 3 4 c ) dr cc'" 4- 2c'c" a/i c " ( -4c 3 c , " / - 3(c') 4) - 2c 2 (c ' ) 2 c"" + 6c 3 (c '") 2 dx 8c3(c")2 + 8c2(c')V + 2c(c')4 (lOcVc" - lc(c'f)c"' - 12c2(c")3 + 12c(c')2(c")2 8c3(c")2 + 8c2(c')2c" -1- 2c(c')4 dh 2cV" + 6cc'c" + {c'f e, (2.34e) dt 4c2c" + 2c(c')2 dgi c2c"' - (c ' ) 3 9u (2-34f) 9u (2-34g) (2.34h) dx 2c2c" + c ( c ' ) 2 5£i _ c"(-2cV"' - 2 c ( c ' ) 4 ) - c3(c')2c"" + 3 c 4 ( c " ' ) 2 5 * ~ 4 c 2 ( c " ) 2 + 4 c ( c ' ) 2 c " + ( c ' ) 4 ( 4 c 3 c ' c " - 4 c 2 ( c ' ) 3 ) c ' " - 4 c 3 ( c " ) 3 + 6 c 2 ( c ' ) 2 ( c " ) 2 4 4c 2 (c") 2 + 4c(c')2c" + (c') - c W + 2c2(c")2 - c(c')2c\ fcl = 4cc» + 2 ( ^ ( 2 - 3 4 l ) / 2 = 4/i, (2.34J) c c 92 = h - - Z , (2.34k) c h2 = (2.341) / 3 = 0, (2.34m) 53 = 0, (2.34n) 2 c V " + 8cc'c" + 2(c ' ) 3 , , h> = ^ T w ( 2 - 3 4 o ) From ( 2.34a-h) we see that T2{x,t,u,v,w} admits a four-parameter group if and only if the wave speed c(x) satisfies ( 2.32). We note that only £ and gx appear in the right hand side of ( 2.34a-h). Thus, we first need to solve the coupled equations ( 2.34a,b,g,h) for £ and g-y. Chapter 2. Cascading Potential Symmetries 42 Integrating ( 2.34a) yields t = F{x)G(t), where F(x) = (3H2 + 2H')-V\ H = c'/c, for some G(t) to be determined. It immediately follows from ( 2.34b) that 9 l = -F(x)G'(t). Note that ( 2.37) is consistent with ( 2.34g). Substituting ( 2.37) into ( 2.34h) we have " 5 ( F ) 2 + 2 M " + F " 3(3# H' + H")2 (2.35) (2.36) (2.37) G"(t) = c2 ZH2 + 2H' ( 3 # 2 + 2H'Y G(t), (2.38) where H = c ' /c. Equat ion ( 2.38) is separable such that G"{t) = c 5(H')2 + 2HH" + H'" _ Z{ZHH' + H")2 3H2 + 2H' (3H2 + 2H')2 (2.39) G(t) Two subcases arise: Case Ila: a = 0. In this subcase G(t) satisfies G"(t) = 0, i.e., G(t) = p + qt, where p and q are arbitrary constants. Case l ib: a ^  0. In this subcase G(t) satisfies G"(t) = <r2G(t), i.e., G[t) = ptat + qe~°\ where p and q are arbitrary constants. It then becomes straightforward to solve the rest of the equations. For a / 0, the solution of ( 2.34a-o) is i = F{x)\pe^ + qe-"% T = o-~l[F' - HF)\peat - qe'"1] + r, fi = [-F, + lHF]\pe*t + qe-t) + si (2.40a) (2.40b) (2.40c) Chapter 2. Cascading Potential Symmetries 43 9\ = -uF\peot -qe-"\ (2.40d) hi = -l-c\HF)'\pe"i + qe-''% (2.40e) h = -^-F\pe^ - qe-"% cz (2.40f) 92 = [-F' + ^HF}\pe<>t + qe-°t} + s, (2.40g) h2 = -^HF\pe^-qe-% (2.40h) h = o, (2.40i) 93 = o, (2.40J) h3 = -iHFlpe^ + qe-^ + s, (2.40k) where {p, q, r, s} are arbitrary constants corresponding to the four parameters of the group. Case III: c(x) arbitrary. In this case we have £ = 9i = ^ = f2 = h2 = f3 = 93 = 0, and T = r, fi = g2 = h3 = s, where r and s are arbitrary constants. Here we have a trivial two-parameter group of translations in t and uniform scalings in u, v, and w. Hence, we have proved the following theorems: Theorem 2.7 The cascaded system T2{x,t,u,v,w} admits an infinite-parameter Lie group of point transformations if and only if the wave speed c(x) satisfies c(x) = (Ax + E>)2/3 for arbitrary constants A and B. Theorem 2.8 The cascaded system T2{x,t,u,v,w] admits a four-parameter Lie group of point transformations if and only if the wave speed c(x) satisfies the fifth-order ODE ( 2.32). Chapter 2. Cascading Potential Symmetries 44 Theorem 2.9 For any other wave speeds the cascaded system T2{x,t,u,v,w] admits a trivial two-parameter Lie group of translations in t and uniform scalings in u, v, and w. Since £ and r do not depend on (u,v,w) and f3 = g3 = 0 , a point symmetry of T2{x,t,u,v,w} corresponds to a point symmetry of the associated wave equation R{x,t,w). Hence, there exists no contact symmetry of R{x,t,w}. Moreover, for any wave speed c(x) satisfying ( 2.32), a point symmetry of R{x,t,w} induces a point sym-metry of T2{x, t, u, v, w} which in turns induces potential symmetries of R{x,t,u} and R{x,t,v), since the infinitesimals for u and v depend explicitly on the auxiliary variable w. The potential symmetries of R{x,t,u} obtained through the use ofT2{x,t,u,v,w] are all nontrivial potential symmetries of type f, since the infinitesimals £ and r admitted by T2{x,t,u,v,w}, or equivalently R{x,t,v}, are essentially different from those admitted by R{x,t,u}. Most importantly, these type f potential symmetries of R{x,t,u} are new potential symmetries which are not obtainable through the use of the associated system S{x,t, u,v}. On the other hand, the potential symmetries of R{x,t,v} obtained through the use of T2{x, t, u, v, w} are all trivial potential symmetries of type II, since the infinitesimals £ and r admitted by T2{x, t, u, v, w] are the same as those admitted by R{x,t,v]. The classification O D E ( 2.32) for the cascaded system T2{x, t, u, v, w} is different from the classification O D E ( 2.27) for the wave equation R{x, t, u}. The general solution of ( 2.32) is different from that of ( 2.27). Hence, we can find new wave speeds such that the wave equation R{x,t,u} admits potential symmetries. In summary we have the following theorems: Theorem 2.10 For any wave speed c(x) satisfying the fifth-order ODE ( 2.32) a point symmetry of the associated wave equation R{x,t,v} induces a point symmetry of the Chapter 2. Cascading Potential Symmetries 45 cascaded system T2{x, t, u, v, w}. Two of the four local symmetries of T2{x,t,u,v,w} correspond to potential symmetries of the wave equation R{x,t,u} and the associated wave equation R{x,t,v). Theorem 2.11 The potential symmetries of the wave equation R{x,t,u} obtained through the use of the cascaded system T2{x, t, u , u, w} are all nontrivial potential symme-tries of type f beyond those obtained through the use of the associated system S{x,t, u, v). Theorem 2.12 The potential symmetries of the associated wave equation R{x,t,v} ob-tained through the use of the cascaded system T2{x,t,u,v,w} are all trivial potential symmetries of type ft. Chapter 2. Cascading Potential Symmetries 46 2.6 Group Classification of the System: <f>x = v, <pt = u, wx = u/c2(x), wt — v. It can be shown that the infinitesimal generator corresponding to ( 2.22a-f) admitted by the cascaded system T3{x, t, u, v, w, <f>) given by ( 2.13a-d) is of the form x = t(z,t)— + T(x,t)— + (fl(x,t)u + g1(x,t)v + h1(x,t)w + k1(x,t)<l>) — Ox ot ou Q + {f2{x, t)u + g2(x, t)v + h2(x, t)w + k2(x, t)4>)-Q-+ (f3(x, t)u + g3(x, t)v + h3(x, t)w + k3(x, t) (fl-ow + (/4(x, t)u + g4(x, t)v + h4(x, t)w + k4(x, (2-41) We have shown in §2.3.3 that any point symmetry of S{x, t, <f>, w} induces a point symmetry of T3{x, t, u, v, w, </>} and vice versa. For any wave speed c(x) satisfying the fourth-order ODE (1.33), a point symmetry of T3{x,t, u, v, w, <f>} induces potential sym-metries of R{x,t,u} and R{x,t,v}, since it can be shown that the infinitesimal for u and v depend explicitly on the auxiliary variables {w,(fl. The type of the potential symmetries admitted by R{x,t,u} can be classified according to Theorem 1.5. It is important to note that the potential symmetries of R{x,t,u} obtained through the use of T3{x, t, u, u, w, (fl are different from those obtained through the use of Xi{x, t, u, v, (fl or T2{x,t,u,v,w], even though T3{x,t, u, v, w, (fl includes both Ty{x,t,u,v,4>} and T2{x,t,u,v,w}. Chapter 2. Cascading Potential Symmetries 47 2.7 Group Analysis of (c2(x)vx)x = vtt We have shown in §2.5 that any point symmetry of the cascaded system T2{x, t, u, v, iv] corresponds to a point symmetry of the associated wave equation R{x,t, w}. In this section we give a group classification of R{x,t,w] using the information obtained from the studies of the cascaded system T2{x, t, u, v, w] in §2.5. Consider the infinitesimal generator ( 2.29) admitted by the cascaded system T2{x, t, u, v, w}. Since any point symmetry of T2{x,t,u,v,w} corresponds to a point symmetry of R{x,t,w}, it follows that R{x,t,w) admits an infinitesimal generator of the form X = t(x,t)— + r(x,t)— + h3(x,t)w—. (2.42) ox ot dw We classify R{x,t,w} as follows: Case I: 3H2 + 2H' = 0, where H = c'/c, i.e., 2 A c = {Ax + B)2'\ H = ~ 3 Ax+ 5' where A and B are arbitrary constants. In this case R{x,t,w} admits an infinite-parameter group. Case II: dx | 5(H')2 + 2HH" + H'" 3(3HH' + H")2' ( 3H2 + 2H' (3H2 + 2H')2 where H — c'/c. Equation ( 2.43) is the classification ODE ( 2.32) for the cascaded system T2{x, t, u, v, w}. Integrating ( 2.43) yields 'b(H')2 + 2HH" + H'" 3{3HH' + H")2 c2 3H2 + 2H' (SH2 + 2H')2 which is the same as ( 2.33). (2.44) Chapter 2. Cascading Potential Symmetries 48 For <7^0, the infinitesimals of ( 2.42) are given by ( 2.40a), ( 2.40b) and ( 2.40k), namely, e = F{x)]peot + qe-1), T = a-x[F' - HF}\peoi -qe-at] + r, h3 = -HFlpe't + qe-^ + s, (2.45a) (2.45b) (2.45c) where F = (3H2 + 2H')~1/2, H = c'/c, and {p, q, r, s) are arbitrary constants correspond-ing to the four parameters of the group. We note that ( 2.45a-c) is essentially different from the four-parameter group (1.28a-c) admitted by the wave equation R{x,t,u}. The local symmetries given by ( 2.^5a-c) of R{x,t,w] induce local symmetries of T2{x, t, u, v, w}. Two of the local symmetries of R{x,t,w} induce nontrivial potential symmetries of the wave equation R{x,t,u}. The infinitesimal generators corresponding to ( 2.45a-c) are given by X p = e at X, = e -at F%- + a-\F' - HF)?- + l-FHw~ ox dt 2 dw F | _ _ a-i{F> _ HF)f> + \FHW£-dx dt 2 dw w d_ dw' The nonzero commutators of the corresponding Lie algebra are [Xp, X,] = 2«7-M(F' - HF)2 - (*F/c)2]Xr, [X r,X p] = crXp, [X r,X 9] = — aXg It immediately follows that (F' - HF)2 - (aF/c)2 = const = K. (2.46) Chapter 2. Cascading Potential Symmetries 49 The third-order O D E ( 2.46) for c(x) is invariant under a two-parameter Lie group of transformations x* = e£2(x + ea), (2.47a) c* = e'2c, (2.47b) which is the same group admitted by ( 2.44). Hence, ( 2.46) can be reduced to a first-order O D E plus two quadratures. For the subcase a = 0, ( 2.46) is invariant under a solvable three-parameter Lie group of transformations x* = ee2(x + C l ) , (2.48a) c* = e£3c, (2.48b) and hence it can be reduced to three quadratures using the reduction algorithm given in [1]. Case III: c(x)arbitrary. In this case we have r = const = r, / = const = s, (2.49) and hence R{x,t,w} is invariant only under translations in t and scalings in w. In particular for any wave speed c(x) which does not solve ( 2.44) for any <r, R{x,t, w} admits only a two-parameter Lie group of transformations with the infinitesimals given by (2.49). We summarize the invariance properties of the associated wave equation R{x,t, v] in the following theorems: Theorem 2.13 The associated wave equation R{x,t,v} admits an infinite-parameter Lie group of point transformations if and only if the wave speed c(x) satisfies c(x) = (Ax + B)2/3 for arbitrary constants A and B. Chapter 2. Cascading Potential Symmetries 50 Theorem 2.14 The associated wave equation R{x,t,v) admits a four-parameter Lie group of point transformations if and only if the wave speed c(x) satisfies the fifth-order ODE ( 2.43). Two of the local symmetries of R{x,t,v] induce nontrivial potential sym-metries of the wave equation R{x,t,u}. Theorem 2.15 For any other wave speeds the associated wave equation R{x,t, v} admits a trivial two-parameter Lie group of translations in t and scalings in v. Chapter 2. Cascading Potential Symmetries 51 2.8 Relationships Between the Two Wave Equations: c2(x)uxx = utt and (c2(x)vx)x = vtt. We discuss the relationships between the symmetries of the wave equation R{x, t, u} and the associated wave equation R{x,t,v}. There exist potential symmetries admitted by R{x,t,u} or R{x,t,v} which can be realized as point symmetries admitted by the associated system S{x,t,u,v} or one of the cascaded systems T \ { x , v , <T3}, T2{x,t,u, v, w} and T3{x, t, u, v, w, <f>}. We consider each of the auxiliary systems separately. 2.8.1 Associated System S{x,t,u,v} If (u,v) satisfies the associated system S{x,t, u,v} then u solves the wave equation R{x,t,u} and v solves the associated wave equation R{x,t,v}. Since both R{x,t,u} and R{x, t,v} can be written in the same conserved form given by S{x, t, u, v} (either u or v can play the role of the potential variable), a point symmetry admitted by S{x, t, u, v} induces potential symmetries admitted by R{x,t,u} and R{x,t,v}, provided that the wave speed c(x) satisfies the fourth-order ODE (1.33). In this case the two wave equations are embedded in the associated system. Any solution (u,v) of S{x,t,u,v} defines a solution u of R{x,t,u) and a solution v of R{x,t,v}, and to any solution u of R{x,t, u), or any solution v of R{x,t,v}, there corresponds a function v, or u, such that (u,v) defines a solution of S{x,t,u,v}. Moreover, a boundary value problem (BVP) posed for the wave equation R{x,t, u} or the associated wave equation R{x,t,v} can be embedded in a BVP posed for the associated system S{x, t, u, v}. If (u,v) solves the associated BVP for S{x, t, u, v}, then u solves the BVP for R{x,t,u}, or v solves the BVP for R{x,t,v}. Invariance of the associated BVP under a point symmetry leads to the construction of the solution of the Chapter 2. Cascading Potential Symmetries 52 BVP for the wave equation R{x,t,u], or the associated wave equation R{x,t,v). Solu-tions of the wave equation R{x,t,u] constructed from invariant solutions of S{x, t, u, v] can be superposed to solve general initial value problems (Bluman and Kumei [7]). 2.8.2 Cascaded System T\{x, t, u, v, <f>) If (u,v,<j>) solves the cascaded system T\{x, t, u, v, <j>} then each of u and <f> solves the wave equation R{x,t,u} and v solves the associated wave equation R{x,t,v}. We have shown that in general a contact symmetry admitted by the wave equation R{x,t,<fi} induces a point symmetry of the cascaded system Ti{x,t,u, v,<f>}. By examining the form of the infinitesimals admitted by the cascaded system Ti{x,t,u,v,<f>} we have found that the most general contact symmetry admitted by the wave equation R{x,t,u] must correspond to a point symmetry of R{x,t,u}. Thus a point symmetry is admitted by R{x,t,u] if and only if a point symmetry is admitted by Ti{x,t,u,v,<f>}. From the group classification of the cascaded system Ti{x, t, u, v, </>} we see that the infinitesimal for v depends explicitly on the auxiliary variables (u, <f>) for any wave speed c(x) satisfies the fifth-order ODE ( 2.27). Hence, for any wave speed satisfying ( 2.27) a point symmetry of t,u, v,4>} is a potential symmetry of R{x,t,v}. Consequently, a point symmetry of the wave equation R{x, t, u] induces a potential symmetry of the associated wave equation R{x,t,v} through the cascaded system Tx{x,t, u, v, <f>). Moreover, we see that through the cascaded system T\{x,t, u, v, <f>} the associated wave equation R{x,t,v} is embedded in the wave equation R{x,t,u] in the following sense: Any solution ti of R{x,t,u} defines a solution v of R{x,t,v}, and to any solution v of R{x, t, v} there corresponds a function u(x) such that u defines a solution of R{x, t, u}. We note that the classification equations for T\{x,t,u,v, <f\ and R{x,t,u} are the same. That is, the wave speeds which lead to invariance of T\{x, t, u, v, cf>] and i?{x, t, u] are the same. Therefore, Ti{x, t, u, v, <j>] and R{x,t,u} are equivalent in the sense that Chapter 2. Cascading Potential Symmetries 53 studying one system does not lead to new information of another. 2.8.3 Cascaded System T2{x, t, u, v, w) If (u,v,w) solves the cascaded system T2{x, t, u, v, w] then u solves the wave equation R{x,t,u] and each of v and w solves the associated wave equation R{x,t,v}. We have shown that in general a contact symmetry admitted by the associated wave equation R{x,t,w) induces a point symmetry of the cascaded system T2{x, t, u, v, w}. By exam-ining the form of the infinitesimals admitted by the cascaded system T2{x, t, u, v, w] we have found that the most general contact symmetry admitted by the associated wave equation R{x,t,v} must correspond to a point symmetry of R{x,t,v}. Thus a point symmetry is admitted by R{x,t,v} if and only if a point symmetry is admitted by T2{x,t,u,v,w}. From the group classification of the cascaded system T2{x,t,u,v,w} we see that the infinitesimal for u depends explicitly on the auxiliary variables (v, w) for any wave speed c(x) satisfies the fifth-order ODE ( 2.32). Hence, for any wave speed satisfying ( 2.32) a point symmetry of T2{x,t,u,v,w} is a potential symmetry of R{x,t,u). Consequently, a point symmetry of the associated wave equation R{x, t,v] induces a potential symmetry of the wave equation R{x,t,u} through the cascaded system T2{x, t, u, v, w}. Most im-portantly, these are new potential symmetries for R{x,t,u} beyond those obtained with the associated system S{x, t, u, v}. Moreover, we see that through the cascaded system T2{x, t, u, u, w} the wave equation R{x,t,u} is embedded in the associated wave equation R{x,t,v} in the following sense: Any solution v of R{x,t,v} defines a solution u of R{x,t, u}, and to any solution u of R{x,t,u} there corresponds a function v such that v defines a solution of R{x,t,v}. We note that the classification equations for T2{x, t, u, v, w} and R{x,t,v} are the same. That is, the wave speeds which lead to invariance of T2{x, t, u, v, w) and R{x, t, v} Chapter 2. Cascading Potential Symmetries 54 are the same. Therefore, T2{x, t, u, v, w} and R{x,t,v} are equivalent in the sense that studying one system does not lead to new information of another. 2.8.4 Cascaded System T3{x, t, u, v, w, <f>} If (u,v,w, <f>) solves the cascaded system T3{x,t, u, v, w, </>} then each pair of (u,v) and ((f>, w) solves the associated system S{x, t, u, v}. We have shown that a point symmetry is admitted by S{x, t, u, v} if and only if a point symmetry is admitted by T3{x, t, u, v, w, <f>}. The group classification of the cascaded system T3{x,t, u, v, w, <f>} is equivalent to the group classification of the associated system S{x, t, u, v}. A point symmetry admit-ted by T3{x, t, u, v, w, </>} induces potential symmetries admitted by both R{x,t,u} and R{x,t,v}, provided the wave speed c(x) satisfies the fourth-order ODE (1.33). We note that the use of T3{x,t, u, u, w, <f>) to find potential symmetries of R{x,t,u} or R{x,t, v} is redundant since we can obtain the same results using S{x, t, u, v}. Chapter 2. Cascading Potential Symmetries 55 2.9 Potential Symmetries of the System S{x,t,u,v} So far we have considered only the potential symmetries admitted by the wave equation R{x,t,u) and the associated wave equation R{x,t,v} through the use of the associ-ated system S{x,t,u,v) and the cascaded systems T\{x,t, u, v, <^>}, T2{x,t,u,v,w} and 73(0;, t, u, v, w, <f>}. However, the system S{x, t, u, v] can also admit potential symme-tries through the use of some auxiliary systems, namely, the three cascaded systems, associated with it. Using the results obtained in §2.4, §2.5 and §2.6, we can prove the following theorems: Theorem 2.16 For any wave speed c(x) satisfying ( 2.27) there exist two nontrivial potential symmetries admitted by the system S{x, t, u, v). Theorem 2.17 For any wave speed c(x) satisfying ( 2.32) there exist two nontrivial potential symmetries admitted by the system S{x,t,u,v}. Theorem 2.18 For any wave speed c(x) satisfying (1.33) there exist two trivial potential symmetries admitted by the system S{x,t,u,v}. Chapter 2. Cascading Potential Symmetries 56 2.10 Chapter Summary In this chapter we have considered three cascaded systems associated with the wave equation R{x,t,u}. Each of these cascaded systems has been analysed in detail. Cascaded system T\{x, t, u, v, <f>} yields no new potential symmetries for the wave equation R{x,t,u}. But Ti{x, t, u, v, (f>) leads to new potential symmetries admitted by the associated wave equation R{x,t, v} through R{x,t,u}. For two local symmetries of T\{x, t, u, v, <f)} there exist two nontrivial potential symmetries of the associated system S{x,t,u,v}. Cascaded system T2{x,t,u,v,w} yields new potential symmetries for the wave equa-tion R{x, t, u} through R{x,t, v}. For two local symmetries of T2{x, t, u, v, w} there exist two nontrivial potential symmetries of the associated system S{x,t,u,v}. Cascaded system T3{x, t, u, v, w, (f>} provides no new potential symmetries for the wave equations R{x,t,u} and R{x,t,v}. It yields trivial potential symmetries for the associated system S{x, t, u, v}. The relationships between the two equations R{x,t,u} and R{x,t,v] have been dis-cussed. We have shown that nontrivial potential symmetries of the system S{x,t,u,v) can be obtained by embedding 5{x,i, u, v} in either T\{x, t, u, v, <f>} or T2{x, t, u, v, w}. Chapter 3 Nonlinear Conserved Forms of the Wave Equation 3.1 Introduction to Conservation Laws and Conserved Forms A conservation law for a kth order PDE R{x,t,u] with independent variables x and t and a dependent variable u is of the form D x / - D t f = 0, (3.1) where / = /(x, t, u, u,..., u ) and g = g(x, t,u,u,..., u ). 1 k—1 1 k—1 A conservation law ( 3.1) is a conserved form of the given PDE R{x,t,u} if and only if ( 3.1) is another kth order PDE equivalent to R{x, t, u]. In general, any (k — l)st order conservation law for a kth order PDE is a conserved form of the PDE. As we have seen in the previous sections on potential symmetries, a given PDE R{x,t,u) can be embedded in an auxiliary system S{x,t,u,v} given by vt = f, vx = g if and only if ( 3.1) is a conserved form of R{x,t, u}. In this chapter we will construct conservation laws for the wave equation c2(x)uxx - utt = 0, (3.2) by an application of Noether's theorem given in §1.5. In particular these conservation laws are nonlinear. The invariance properties of the corresponding nonlinear systems are analysed. 57 Chapter 3. Nonlinear Conserved Forms of the Wave Equation 58 We will also find higher order conservation laws for the wave equation (3.2) by the use of a recursion operator corresponding to a symmetry of (3.2). Chapter 3. Nonlinear Conserved Forms of the Wave Equation 59 3.2 Variational Symmetries for a Lagrangian of the Wave Equation We have seen in §1.4 that the wave equation (3.2) admits a four-parameter Lie group of point transformations with the corresponding infinitesimal generators given by 1 dt' X 2 = e X 3 = ct d , rT \ d 1 TT d a— + a (a - Ha) — + -affu— dx dt 2 du 0 - 1 / / TT S3 1 TT 0 a- a (a-Ha)— + -aHu— dx ot 2 du d X 4 = u—, du where a = (2H' + H2)'1^2, H = c'/c, if and only if c(x) satisfies (3.3a) (3.3b) (3.3c) (3.3d) c2(a' - Ha)' = a a (3.4) We have seen in §1.5 that an infinitesimal generator of the form is equivalent to d d d X = {(x,() 5- + r ( x , ( ) ^ + , ( x , ( , U ) 5 ; X = ( , - u l f - u , r ) f Ou Thus one can show that the infinitesimal generators ( 3.3a-d) admitted by (3.2) are correspondingly equivalent to Y 9 X X = - U f T - , du 1 d X 2 = e^l-aHu - aux - a~1(a' - Ha)u^\ —, 2, Ou 1 d X 3 = c-at[-aHu-aux + o-\a'- Ha)ut} —, i au X 4 = u—. du (3.5a) (3.5b) (3.5c) (3.5d) J Chapter 3. Nonlinear Conserved Forms of the Wave Equation 60 A Lagrangian for the wave equation (3.2) is 1 = K ) 2 " s t ) { U t ) 2 - ( 3 - 6 ) The Euler-Lagrange equation for L of ( 3.6) is the wave equation (3.2). We know from the studies of the calculus of variations that every extremum u(x) of the functional J[u] = f[{ux)2 - -±-{ut)2)dxdt, (3.7) must satisfy the Euler-Lagrange equation, or equivalently, the wave equation (3.2). We will find variational symmetries for the Lagrangian L given by ( 3.6) and then construct conservation laws of the wave equation (3.2) by applying Boyer's formulation of Noether's theorem given in §1.5. A variational symmetry for L leaves the action integral ( 3.7) invariant. We determine which of the symmetries ( 3.5a-d) is a variational symmetry by checking if for any u(x) there exists a vector function A = (A1, A2) such that the condition X™L = D ^ 1 + BtA2, (3.8) is satisfied. The first extensions of the corresponding infinitesimal generators ( 3.5a-d) are given by: d d d X[^ - - M ( 7 u x t - , (3.9a) ou oux out X(2l) = eat^ctHu-ctux-c-l{a'- Ha)ut)^-du 1 d + h(aHu)x ~ (ctux)x - o-'1^' - Ha)ut)x]~— 2 Oux + \-oaHu — o~aux — (a' — Ha)ut + —c\Hut — ctuxt - a~\a'- Ha)utt\-^V (3.9b) Chapter 3. Nonlinear Conserved Forms of the Wave Equation 61 X 3 1 } = e~at i[\aHu-aux + o - \ a ' - Ha)ut)-^ y i ou 1 d + [-(ctHu)x - (aux)x + a-\{a' - Ha)ut)x} — -f [-^aaHu + aaux - (a' - Ha)ut) + ^ctHut - auxt + a - V - Ha)utt]£-\, dut X (i) d d UTT + u x - ]- ut-—. du dux dut We consider each of the symmetries ( 3.5a-d) separately. (3.9c) (3.9d) • Xi: This corresponds to translations in t. Since the Lagrangian L does not depend explicitly on t, Xi obviously leaves the action integral ( 3.7) invariant. One can show that for any u(x) 1 XS,)X = D«- c2(x) (uty - (Ux)2 . (3.10) Hence, ( 3.8) is satisfied and X\ is a variational symmetry for L. X 2 : One can show that for any u(x) X^L = D x \ eoi \{*H)'u2 - a(ux)2 + ^ y " K > 2 at -<T~\a' -Ha)(ux)2- 2c2{x) c2(x) (3.11) Again, condition ( 3.8) is satisfied and hence X 2 is a variational symmetry for L. X 3 : The infinitesimal generators X 3 and X 2 differ in the sign of a. Replacing a in ( 3.11) with —a we have \{aE)'u2 - a(ux)2 + ^ y « ( u t ) 2 Chapter 3. Nonlinear Conserved Forms of the Wave Equation 62 -at o~ „ o a 1 (3.12) Hence, A3 is also a variational symmetry for L. • X 4 : This corresponds to scalings in u which does not leave the action integral ( 3.7) invariant, as we see that J[u*] = f3J[u] if u* = ftu, (3 ^ 1. Thus we have found three variational symmetries {Xi ,X2 ,X 3} for L of ( 3.6). In the next section we will construct conservation laws corresponding to these variational symmetries by an application of Noether's theorem. Chapter 3. Nonlinear Conserved Forms of the Wave Equation 63 3.3 Construction of Conservation Laws Consider an infinitesimal generator of the form and its first extension X = r}(x,t,u,uXlut)—, Ou ou oux out If L is given by ( 3.6) then X^L = 2uxDxrj utDtT] cz(x) = Dx{2uxr}) - Dt \ -§-TUtr) J - 2uxxn + uttrj. (3.13) yc^x) J c'(x) But we have the Euler-Lagrange equation E(L) = ^ - D x ^ - D t — du x dux dut 2 — ~2uxx -\—— -Uu c'(x) = 0, (3.14) which is equivalent to the wave equation (3.2). Substituting ( 3.14) into ( 3.13) we have X^L = T>x(2uxn) - Dt {^jutvj • (3-15) It should be emphasised that ( 3.15) holds only for those u(x) satisfying the Euler-Lagrange equation ( 3.14), or equivalently, the wave equation (3.2). We construct a conservation law for each of the variational symmetries { X ] , X 2 , X 3 } as follows: Conservation Law I: For infinitesimal generator X l 5 ( 3.15) becomes Xi1 } = Bx(-2uxut) + Dt (^y(u<)2) • (3-16) Chapter 3. Nonlinear Conserved Forms of the Wave Equation 64 Equating ( 3.16) and ( 3.10) we have a conservation law Dx{2uxut) - Dt (ux)2 + 2 , M2 ' c2(x), = 0. (3.17) Conservation Law II: For infinitesimal generator X 2 , ( 3.15) becomes X^L = Dx{e°t[aHuux-2a(ux)2 -2a~\a - Hcx)uxut)} ( <>ot c2(x) [-aHuut + 2ctuxut + 2a~1(a' - Ha)(ut)2] } . (3.18) Equating ( 3.18) and ( 3.11) we have a conservation law ^-(aH)'u2 + a(ux)2 + * .a(ut)2 - cxHuux + 2a 1(a' - Ha)uxut °~aHu2 - o-\cx' - Ha){ux)2 - ~ Ha)(ut)2 2c2(x) c2(x) 1 2 + -Tr^-aHuut ^—rctuxut cl(x) cz(x) 0. (3.19) Conservation Law III: For infinitesimal generator X 3 , we replace a in ( 3.19) with —a to yield another conservation law \-(aH)'u2 + a(ux)2 + —|—a(u t) 2 - aHuux - 2a 1(a' - Ha)uxut £ C I X I + D« e -at ^ a H u 2 + a~\«> - Ha){ux)2 + ^ ( a ' - Ha)(ut)2 1 rr 2 H—rr^ctHuut —— auxut = 0.(3.20) c2(x) c2(x) Hence, we have used Noether's theorem to construct three nonlinear conservation laws of the wave equation (3.2). It should be emphasised that these conservation laws hold only for those wave speeds c(x) which satisfy the classification ODE ( 3.4). Chapter 3. Nonlinear Conserved Forms of the Wave Equation 65 3.4 Group Classification of vx = (ut)2/c2(x) + (ux)2, vt = 2uxut. We consider the conserved form given by ( 3.17). By introducing a potential variable v, we can write ( 3.17) as the following nonlinear system: vx = -^—{ut)2 + (ux)2, (3.21a) C [X) vt = 2uxut. (3.21b) Let ( 3.21a,b) admit an infinitesimal generator d d d $ X = ((x,t,u,v) — + r(x,t,u,v) — + r)u(x,t,u,v)— + r)v(x,t,u,v)—. (3.22) ox Ot Ou ov We classify ( 3.21a,b) as follows: Case I: c(x) = (Ax + B)2 for arbitrary constants A and B. The nonlinear system ( 3.21a,b) is invariant under a seven-parameter Lie group of point transformations. (We recall that for this wave speed the wave equation (3.2) admits an infinite-parameter group.) Each of the point symmetries admitted by the system ( 3.21a,b) corresponds to a point symmetry admitted by the wave equation (3.2). Hence, the nonlinear system ( 3.21a,b) induces no potential symmetry of the wave equation (3.2). Case II: (a' - Ha)' = 0, (3.23) where a = (2H' + # 2)" 1 / 2 and H = c'/c. Equation ( 3.23) has been solved in [6]. The general solution of ( 3.23) is c(x) = (Bx2 + Cx + D)exp[(A - C) J(Bx2 + Cx + D)~xdx\, (3.24) where A, B, C and D are arbitrary constants. The nonlinear system ( 3.21 a,b) is invariant under a six-parameter Lie group of point transformations if and only if the wave speed c(x) is given by ( 3.24). We note that ODE Chapter 3. Nonlinear Conserved Forms of the Wave Equation 6 6 ( 3.23) is a special case of the classification O D E ( 3.4) for the scalar wave equation (3.2) with the integration constant a = 0. For the wave speed c(x) given by ( 3.24) the scalar wave equation (3.2) is invariant under a four-parameter group [5]. For A 7^ C the infinitesimals of ( 3.22) are given by fBx2 + Cx + D\ * = a i { C-A j ' ( 3 ' 2 5 a ) r = ajt + aa, (3.25b) c*i (A + 2Bx\ /n n . f] = — [ Q _ £ J U + Q 3 U + A 4 X + a s , (3.25c) Vv = y ( J ~ T A ~ 2 U ) + 2 A 4 U + 2 A 3 U + Q 6 ' ( 3 > 2 5 D ^ where ai, i = 1,2,... ,6 are the group parameters. From the forms of the infinitesimals ( 3.25a-d) admitted by ( 3.21a,b) we see that ( 3.21a,b) induces no potential symmetry of the wave equation (3.2). By comparing the projection of the group ( 3.25a-d) on (x,t)-space with the four-parameter group admitted by the scalar wave equation (3.2), we find that one nontrivial symmetry of (3.2) is lost through the nonlinear system ( 3.21a,b). Case III: c(x) arbitrary. In this case the system ( 3.21a,b) is invariant under a trivial two-parameter Lie group of translations in t and scalings in u and v. From the above results we have the following conclusions: • The use of the nonlinear system ( 3.21a,b) does not lead to potential symmetry of the wave equation (3.2). • A more restrictive class of wave speeds must be used for some point symmetries of ( 3.21a,b) to exist. • A nontrivial symmetry of the wave equation (3.2) is lost. • For the wave speed c(x) = (Ax -j- B)2 the number of parameters of the local Lie group admitted by the wave equation (3.2) is reduced from infinite to some finite Chapter 3. Nonlinear Conserved Forms of the Wave Equation 67 number. Hence, we conclude that the nonlinear system ( 3.21 a,b) is not useful in finding new potential symmetries of the wave equation (3.2). A similar analysis of the other two nonlinear conserved forms ( 3.19) and ( 3.20) could be done using Reid's algorithm. Chapter 3. NonHnear Conserved Forms of the Wave Equation 68 3 . 5 Higher Order Conservation Laws When a linear PDE admits a point symmetry, one can show that it always admits an in-finite sequence of Lie-Backlund symmetries. The infinitesimals of the successive elements of the infinite sequence of symmetries depend on higher order derivatives of the dependent variable. Anderson, Kumei, and Wulfman [12] extended Lie's infinitesimal transformation method to include Lie-Backlund transformations and constructed recursion operators to generate infinite sequences of Lie-Backlund symmetries for linear PDE's. We illustrate here the use of a recursion operator to generate an infinite sequence of Lie-Backlund sym-metries admitted by the linear wave equation (3.2). Then we apply Noether's theorem to construct higher order conservation laws for the linear wave equation (3.2). We have shown that the wave equation (3.2) admits the infinitesimal generator X = u ^ , (3.26) corresponding to translations in t. The recursion operator corresponding to ( 3.26) is R = D«. It has been proved in [1] that if (3.2) admits n = Ku then it also admits n = R f cu, k = 1,2,.... Hence, one can generate an infinite sequence of Lie-Backlund symmetries ad-mitted by (3.2). In particular, (3.2) admits X = D * u ^ , (3.27) for k = 1,2,.... The first-extended infinitesimal generator corresponding to ( 3.27) is X<D = D * u | - + D ^ D f u ^ - + B k + 1 u ^ - . (3.28) du dux out For each k = 2,3, . . . in ( 3.27) we check whether ( 3.27) is a variational symmetry for the Lagrangian L given by ( 3.6). Chapter 3. Nonlinear Conserved Forms of the Wave Equation 69 For k = 2, i.e. X = one can show that ( 3.27) does not correspond to a variational symmetry for L. For k = 3, ( 3.27) and ( 3.28) become correspondingly Ou and X = u t t t — , (3.29) d d d X ( 1 ) = um— + u x t t t - h utttt—. (3.30) ou oux out We have 2 X ( 1 ) L = 2uxuxttt TT-rUtUtttt cl(x) = D t ^ 2uxuxtt - (uxt)2 - -^-j(2ututtt - {utt)2)j . (3.31) Hence, ( 3.29) is a variational symmetry for L. For any u(x) solving (3.2) we have X&L = Dx(2uxuttt) - Dt (^pjutuu^j . (3.32) Equating ( 3.31) and ( 3.32) we obtain a third order conservation law for the wave equation (3.2): Vx(2uxuttt) - D t (2uxuxtt - (uxt)2 + = 0. (3.33) Note that ( 3.33) is a conserved form of the wave equation (3.2). That is, ( 3.33) leads directly to the second order wave equation (3.2). We observe that ( 3.27) is a variational symmetry for L only if k is odd. Hence, only a subsequence of the Lie-Backlund symmetries ( 3.27) corresponds to an infinite sequence of variational symmetries for L. By introducing a potential variable v one can write ( 3.33) as an auxiliary system vt = 2uxuttu (3.34a) vx = 2uxuxtt-(uxt)2+{-^. (3.34b) c'(x) Chapter 3. Nonlinear Conserved Forms of the Wave Equation 70 3.6 Chapter Summary In this chapter we have applied Boyer's formulation of Noether's theorem to construct conservation laws for the wave equation (3.2). We have studied one nonlinear system ( 3.21a,b) in detail and found that it does not lead to potential symmetry of the wave equation (3.2). Through the use of the nonlinear system ( 3.21a,b) a symmetry of the scalar wave equation (3.2) is lost. Hence, we conjecture that nonlinear conserved forms of the wave equation (3.2) are probably uninteresting. We have also found higher order conservation laws for the wave equation (3.2) using a recursion operator corresponding to the time translational symmetry of the wave equation (3.2). It can be shown that there exists an infinite number of higher order conservation laws for the wave equation (3.2). Chapter 4 Linear Conserved Forms of the Wave Equation 4.1 Introduction to Linear Conserved Forms From the last chapter on nonlinear conserved forms of the wave equation utt = c2(x)uxx, (4.1) we see that if an auxiliary system is nonlinear it is likely that it is not useful in finding new symmetries of the wave equation (4.1). In this chapter we will consider some new linear conserved forms (Bluman[13]) of the wave equation (4.1). We begin with a general auxiliary linear system for the wave equation (4.1). From this general linear system we can construct linear conserved forms of the wave equation (4.1). Consider the linear system vx = A(x,t)ut + B(x,t)u, (4.2a) vt = C(x,t)ux + D(x,t)u, (4.2b) where (A, B, C, D) are some functions of x and t. Applying the integrability condition vxt = vtx to (4.2a,b) we have Atut + Autt + Btu + But = Cxux + Cuxx + Dxu + Dux. (4.3) Substituting the wave equation (4.1) into (4.3) and then setting to zero the coefficients of u, ut, ux and uxx we have the following set of PDE's: Bt-Dx = 0, (4.4a) 71 Chapter 4. Linear Conserved Forms of the Wave Equation 72 At + B = 0, (4.4b) Cx + D = 0, (4.4c) c\x)A-C = 0. (4.4d) The linear system (4.2a,b) corresponds to a conserved form of the wave equation (4.1) for any solution (A,B,C, D) of (4.4a-d). From (4.4b-d) we can express A, B and D in terms of C: A = — — , .(4.5a) cl(x) B = ( 4 ' 5 b ) D = -Cx. (4.5c) Then from (4.4a) we see that C must satisfy Ctt = c\x)Cxx, (4.6) i.e. C satisfies the wave equation (4.1)! Hence, for any solution of the wave equation ( 4.6) there exists a linear conserved form of the wave equation (4.1) given by (4.2a,b). Since there exist infinitely many solutions of the PDE ( 4.6), it turns out that there are an infinite number of conserved forms of the wave equation (4.1) of the form (4.2a,b). We can look for some particular solutions of ( 4.6) in order to construct some linear conserved forms given by (4.2a,b). The simplest solution of ( 4.6) is C = 1. From ( 4.5a-c) we have A = —^— and B = D = 0. cl(x) The linear system (4.2a,b) then becomes ut Chapter 4. Linear Conserved Forms of the Wave Equation 73 which is the associated system ( 2.9a,b) considered previously. We construct two linear conserved forms of the wave equation (4.1) by using two simple particular solutions of ( 4.6). Linear System I One particular solution of ( 4.6) is C = x. In this case we have A = x/c2(x), B = 0, D = -l. The system (4.2a,b) then becomes Li{x,t,u,v): xut vx = — , (4.7a) vt = xux — u. (4.7b) Linear System II Another particular solution of ( 4.6) is C = t. In this case we have A = t/c2(x), B = -l/c2(x), D = 0. The system (4.2a,b) then becomes L?{x,t,u, v}: tut — u v* = ! 5 £ p ( 4 - 8 a ) vt = tux. (4.8b) In the next two sections we will give a group classification of each of the linear systems Li{x,t,u,v} and L2{x,t,u,v}. Chapter 4. Linear Conserved Forms of the Wave Equation 74 4.2 Group Classification of vx = xut/c2(x), vt = xux — u. Let the linear system L\{x,t,u,v] given by ( 4.7a,b) admit an infinitesimal generator of the form X = t(x,t)— + T(x,t)— + (f1(x,t)u + g1(x,t)v) — + (f2(x,t)u + g2(x,t)v)—. (4.9) Using Reid's algorithm we classify the linear system Li{x, t,u, v} as follows: Case I: c(x) = Ax2 for arbitrary constant A. In this case the linear system Li{x,t,u,v] admits an infinite-parameter Lie group. (Recall that the scalar wave equation (4.1) admits an infinite-parameter Lie group if and only if c(x) = (Ax + B)2.) We note that the wave speed c(x) in this case is a degenerate form of the wave speed admitted by the scalar wave equation (4.1) in the infinite-group case, since the linear system L\{x, t, u, v} is not invariant under translations in x. Case II: x(xH - 2)2H"' + (5x2(2 - xH)H' + 2xzH* - 10x2H2 + 6xH 4- 12)H" + 4x3(H')3 + (-4x3H2 + 10x2H + 20x)(H')2 + (-4x2H3 + 2xH2 + S6H)H' - 4xH4 + 12H3 = 0, (4.10) where H = c'/c. Equation ( 4.10) is a fourth-order ODE for c(x) and is invariant under a two-parameter Lie group of scalings in x and scalings in c. In this case Lx{x, t, u, v} admits a four-parameter Lie group of point transformations. The standard form of the determining equations is given by dt: x2cc" - x2(c')2 + 2c2 dx x2cc' — 2xc2 (4.11a) Chapter 4. Linear Conserved Forms of the Wave Equation 75 dt dr 2x 2xgu (4.11b) 9u (4.11c) dx c2(xY dr x2c"-2xd + 2c^ Tt = ~ x2d — 2xc <4-lld> dh _ c'(-x3cc"' - 4c2) + 2x2c2c"' + 2x3c(c")2 dx ~ 2xc(xd-2c2)2 * _ c"(-x3(d)2 - 2x2cc') + 4xc(c')2 2xc(xc'- 2c2)2 ( 4 ' l l e ) dh x2cc"+ x2(c')2-4xcd+ 2c2 dgi xcc" — x(c')2 + cd 9u (4-llg) dx xcd — 2c2 dgj_ _ (-x2c2d 4 - 2xc3)d" + 2x2c2(c"Y dt ~ 2x(xc> - 2c)2 _ (~*MC')2 ~ 4xc2c; + 4c3)c" + 2xc(c')3 - 2c2(c')2 2x(xd - 2c)2 (4-llh) x2 h = j^9u (4.11i) —xd 4 - c 92 = £ + (4.11J) xc Integrating ( 4.11a) we have £ = F(x)G(tY (4.12) where F(x) = xcf^2c and (7(£) to be determined. Substituting ( 4.12) into ( 4.11b) we have FC" * - " I T - ( 4 ' 1 3 ) Then by substituting ( 4.12) and ( 4.13) into ( 4.11h) we have (2xc3 - x2c2d)d" + 2x2c2(c")2 + (-x2c(c'Y - 4xc2d + 4c3)c" + 2xc(c')3 - 2c2(c')2 _ 2 (xd - 2c)2 (4-14) _ G"(t) G(t) Chapter 4. Linear Conserved Forms of the Wave Equation 76 where A is a real or imaginary constant. One can show that ( 4.14) is an integral of the classification O D E ( 4.10). Note that ( 4.14) is invariant under only a one-parameter Lie group of scalings. For A ^  0, the solution of ( 4.11a-j) is xc xd — 2c (peXt + qe~x% T = h = 91 = h = 92 = .1fx2cc"-2xcc' + 2c2\. x t _x \ [xd-2cy ) { p e - q e ) + ! (x2cc" + x2(d)2 - Axed + 2c V {xd - 2d)2 A x 2 (peXt + qe~xt) + s, (peXt - qe~X% (4.15a) (4.15b) (4.15c) (4.15d) (4.15e) (peM + qe~M) + s, (4.15f) 2c(xc' - 2c) '-xd + c j x2cc" + x 2 ( c ' ) 2 - 4xcc' + 2c 2 xd - 2c ~ {xd - 2c)2 where {p, q,r, s} are arbitrary constants corresponding to the four parameters of the group. The infinitesimal generators corresponding to ( 4.15a-f) are given by X„ = - A t xc d_ _ j / x 2 c c " - 2 x c c ' + 2 c 2 \ d_ xc'-2cdx V ( x c ' - 2 c ) 2 J dt a fx2cc" + x2{c')2 - Axed + 2d {xd - 2d)2 u + Ac e~M + + d A x 2 2c(xc xc d d_ 2{xd - 2c) u \ du .3? (-xd + c 1x2cc" + x2{c')2 - Axed + 2c2\ 1 d_) •' -2c)U [ xd - 2c (xd - 2c) 2 )V\dv) + A _! f x2cc" - 2xcd + 2c2\ d xd — 2c dx 1 \ (xd — 2d)2 _! (x2cc" + x2(c')2 - Axed + 2 c 2 \ dt Ac d_ du \ (xd - 2d)2 ) 2(xd - 2c) _ A x 2 / - x d + c ^ _ 1 x 2 c c / / -f x 2 ( c ' ) 2 - 4xcc' + 2 c 2 \ 2 c ( x c ' - 2 c ) U + V xd - 2c ~ (xd - 2c)2 ) V dvi' Chapter 4. Linear Conserved Forms of the Wave Equation 77 9 9 X s = u— + v—. ou ov The nonzero commutators of the corresponding Lie algebra are . , xc d (x2cc" -2xcc' + 2c2\ (x2cc" - 2xcc' + 2c2\ • 1 p ' g l ~ A \(xc'-2c)~dx~ [ (xc' - 2c)2 ) + { (xc' - 2c)2 / ' r' [ X r , X p ] — A X P , [ X r , X 9 ] — — A X g . It immediately follows that xc d (x2cc" -2xcc' + 2c2\ (x2cc" - 2xcc' + 2c2\ 2 2 JxT^c)Tx{ (xc'-2c)2 ) + [ (xc'-2c)2 ) = c o n s t = ^- (4-16) The third-order ODE ( 4.16) for c(x) is still invariant under a two-parameter Lie group of scalings in x and scalings in c, and hence it can be reduced to a first-order ODE plus two quadratures. In particular one can let ^ xc1 x2cc" - 2xcc' 4 - 2c2 ~ ~c~' ~ (xd - 2c)2 then ( 4.16) becomes dU UV-2V-U + 1 dv = ^ V 2 • ( 4 - 1 7 ) Note that ( 4.17) is a linear first-order ODE. If U = $(V) solves ( 4.17) then one can reduce the second-order ODE U(x,c,d) = $(x,c,c',c") (4.18) to two quadratures since ( 4.18) is invariant under a two-parameter group of scalings in x and scalings in c. One can show that ( 4.14) and ( 4.16) are two independent integrals of the classification ODE ( 4.10) and both ( 4.16) and ( 4.10) are invariant under the same two-parameter Lie group of scalings in x and scalings in c. Chapter 4. Linear Conserved Forms of the Wave Equation 78 It should be emphasized that the commutator relation preserves all the symmetries admitted by the ODE ( 4.10), since both ( 4.10) and ( 4.16) are invariant under the same two-parameter group. Consequently, the order of ODE ( 4.10) can be reduced by three even though ( 4.10) admits only a two-parameter group! Two of the local point symmetries, X p and X ? , admitted by Li{x, t,u, v] are non-local (potential) symmetries of the scalar wave equation (4.1). Moreover, the potential symmetries of (4.1) obtained through the use of Li{x,t,u,v} are all nontrivial poten-tial symmetries of type I, since the infinitesimals £ and r admitted by Li{x,t,u,v} are essentially different from those admitted by (4.1). Most importantly, these new poten-tial symmetries are beyond those potential symmetries obtained through the use of the associated system S{x,t,u,v} and the cascaded system T2{x,t,u,v,w}. The classification ODE ( 4.10) for the linear system Ly{x,t,u, v} is distinct from all the classification ODE's for the wave equation (4.1) and the auxiliary systems studied previously. Hence, we can find new wave speeds such that the wave equation (4.1) admits new potential symmetries. Moreover, one can apply the method given in [8] to find a common solution set of ODE's (1.33) and ( 4.10). Knowing this common solution set for c(x) one can find new potential symmetries admitted by the wave equation (4.1) through the use of the linear system L\{x, t, u, v}. Case III: c(x) arbitrary. In this case the system L\{x,t,u,v} is invariant only under translations in t and scalings in u and v. In summary we have the following theorems: Theorem 4.1 The linear system L-i{x,t,u,v] is invariant under an infinite-parameter Lie group of point transformations if and only if the wave speed c(x) = Ax2 for arbitrary constant A. Theorem 4.2 The linear system Li{x,t,u,v} is invariant under a four-parameter Lie Chapter 4. Linear Conserved Forms of the Wave Equation 79 group of point transformations if and only if the wave speed c(x) satisfies the fourth-order ODE (4.10). Theorem 4.3 For any wave speed c(x) satisfying the fourth-order ODE ( 4-10), two of the four point symmetries of the linear system Li{x, t, u, v] correspond to potential symmetries of the wave equation (4-1)• Theorem 4.4 The potential symmetries of the wave equation (4-1) obtained through the use of the linear system Li{x,t,u, v} are all nontrivial potential symmetries of type f beyond those obtained by S{x, t, u, v} and T2{x, t, u, v, w}. Theorem 4.5 For any wave speed c(x) simultaneously satisfies (1.33) and ( 4-10) the wave equation (4-1) admits new potential symmetries through Li{x,t,u,v} beyond those obtained by S{x, t, u, v} and T2{x, t, it, v, w}. Theorem 4.6 For any other wave speeds the linear system Li{x,t,u,v} is invariant under a trivial two-parameter Lie group of translations in t and uniform scalings in u and v. Chapter 4. Linear Conserved Forms of the Wave Equation 80 4.3 Group Classification of vx = (tut — u)/c2(x), vt = tux. Let the linear system L2{x,t,u,v] given by ( 4.8a,b) admit an infinitesimal generator of the form d d d X = t(x,t)— + T{x,t)— + {f1(x,t)u + g1(x,t)v) — ox Ot ou d + (h(x,t)u + g2(x,t)v)—. (4.19) ov Using Reid's algorithm we classify the linear system L2{x,t,u,v) as follows: Case I: The general solution of ( 4.20) is c(x) = (Ax + B)2/3, (4.21) for arbitrary constants A and B. In this case the linear system L2{x, t, u, v} admits a four-parameter Lie group of point transformations. (We recall that for the wave speed given by ( 4.21) the associated wave equation R{x,t,v] given by ( 2.10) is invariant under an infinite-parameter group.) One can show that the infinitesimals {£, r} for the independent variables admitted by the linear system L2{x,t,u,v} are the same as those admitted by the associated wave equation R{x,t,v] and the cascaded system T2{x, t, u, v, w}. Case II: cc'c'" - 2c(c")2 + (c'fc" = 0. (4.22) The general solution of ( 4.22) is c(x) = (Ax + B)c, (4.23) Chapter 4. Linear Conserved Forms of the Wave Equation 81 for any constant C ^ 0, | . In this case the linear system L2{x,t,u,v) admits a two-parameter Lie group of point transformations with the corresponding infinitesimal gen-erators given by Ci Ci Q X, = (Ax + - A(C - l)tjt - A(2C - l)v—, (4.24a) X 2 = u± + v±. (4.24b) (We recall that for the wave speed given by ( 4.23) with C = 2 the wave equation (4.1) is invariant under an infinite-parameter group, and for the case where C ^ 0,2 the wave equation (4.1) is invariant under a four-parameter group.) We see that none of the local symmetries ( 4.24a,b) admitted by the linear system L2{x, t,u,v] corresponds to a potential symmetry of the wave equation (4.1). Case III: c(x) arbitrary. In this case the linear system L2{x,t,u,v} admits only a one-parameter Lie group of scalings in both u and v. (We recall that for arbitrary wave speed the wave equation (4.1) and its related systems studied previously are invariant under scalings in the dependent variables and translations in t.) The translational invariance in t is lost since the right-hand sides of L2{x, t, u, v] depend explicitly on t. From the above group classification of the linear system L2{x, t, u, v) we see that the size of the group admitted by the wave equation (4.1) is reduced when (4.1) is embedded in the linear system L2{x,t,u,v}. Moreover, the use of L2{x,t,u,v} leads to a more restrictive class of wave speeds c(x) admitted by the wave equation (4.1). Chapter 4. Linear Conserved Forms of the Wave Equation 82 4.4 Chapter Summary In this chapter we have studied linear conserved forms of the wave equation (4.1). We have considered a general linear system given by (4.2a,b) with arbitrary coefficients satisfying a system of PDE's given by (4.4a-d). It turns out that there exist infinitely many linear conserved forms of the wave equation (4.1). Two new linear systems have been analysed in detail. The linear system I given by Li{x, t, u, v] admits an infinite-parameter Lie group of point transformations if and only if the wave speed c(x) = Ax2 for arbitrary constant A, and it admits a four-parameter local Lie group of point transformations if and only if the wave speed c(x) satisfies the fourth-order ODE given by ( 4.10) which is distinct from the other classification ODE's found previously. For any wave speed c(x) satisfying ( 4.10) two of the four local symmetries admitted by Li{x,t,u,v} correspond to nontrivial type I potential symmetries of the wave equation (4.1). Most importantly, these new potential symmetries are beyond those obtained by S{x, t, u, v} and T2{x, t, u, v, w}. Moreover, for any wave speed c(x) simultaneously satisfying (1.33) and ( 4.10) there exist new potential symmetries of the wave equation (4.1) through the linear system Ly{x, t, u, v}. For any other wave speeds the linear system L\{x,t,u,v} admits only a trivial two-parameter Lie group of scalings in the dependent variables and translations in t. The linear system II given by L2{x,t,u, v} admits a four-parameter local Lie group of point transformations if and only if the wave speed c(x) = (Ax + B)2^3 for arbitrary constants A and B, in which case one can show that the infinitesimals for the independent variables admitted by L2{x, t, u, v} are the same as those admitted by the associated wave equation R{x,t,v}. For the case c(x) — (Ax 4- B)c where C ^ 0, | , the linear system L2{x,t,u,v} admits a two-parameter local Lie group of point transformations which correspond to local symmetries of the wave equation (4.1). For any other wave speeds Chapter 4. Linear Conserved Forms of the Wave Equation 83 c(x) the linear system L2{x,t,u,v} admits only a trivial one-parameter Lie group of scalings in the dependent variables. By allowing the right-hand sides of the linear system to depend explicitly on t, the translational symmetry in t is lost. Chapter 5 Discussion 5.1 Conclusions In this thesis we have significantly extend the classes of potential symmetries admitted by the wave equation R. The results obtained lead to a better understanding of the limits in the construction of potential symmetries for differential equations. In Chapter 1 we have given a summary of the invariance properties of the wave equation R{x, t,u} c2(x)uxx — utt = 0, (5.1) and its associated system S{x,t,u, v} vt = ux, (5.2a) vx = ut/c2(x). (5.2b) A complete group classification of both R{x,t,u} and S{x,t,u,v} has been done in recent papers [2],[6]. By replacing R{x,t,u] with S{x,t,u,v] one can enlarge the class of symmetries admitted by the wave equation R{x, t, u}. More importantly, one can find a symmetry group acting in a space beyond the space which contains the independent variables, the dependent variables, and their derivatives up to some finite order. The new classes of symmetries obtained through the use of one or more auxiliary variables, or potentials, in a conserved form are defined as potential symmetries. These potential symmetries lead to construction of new invariant solutions of the wave equation R{x, t, u}. 84 Chapter 5. Discussion 85 Furthermore, these invariant solutions can be superposed to solve general initial value problems posed for the wave equation R{x,t,u}. An important equation related to the wave equation R{x,t,u} is R{x,t,v) [c2(x)vx)x = vtu (5.3) which can be obtained by applying the integrability condition uxt = utxto S{x, t,u, v}. In Chapter 2 we have investigated further the invariance properties of the wave equa-tion R{x,t,u}, the associated system S{x,t,u,v), and the associated wave equation R{x,t,v} by considering three cascaded systems given by Ti{x,t,u,v, 4>): <f>x = v, (5.4a) (j>t = u, (5.4b) vx = ut/c2(x); (5.4c) T2{a;, t, u, v, w): wx = u/c2(x), (5.5a) wt = v, (5.5b) vt = ux; (5.5c) T3{x,t,u,v,w,<j>}: 4>x = v, (5.6a) 4>t = (5.6b) wx = u/c2(x), (5.6c) wt = v. (5.6d) Chapter 5. Discussion 86 We have shown that a point symmetry is admitted by T\{x, t, u, v, tb} if and only if a point symmetry is admitted by R{x,t,cb}; a point symmetry is admitted by T2{x, t, u, v, w} if and only if a point symmetry is admitted by R{x, t, w}; and a point symmetry is admitted by T3{x, t, u, v, w, <f>] if and only if a point symmetry is admitted by S{x, t, tb, w}. We have obtained new nontrivial potential symmetries for R{x,t,v} through the use of Ti{x, t, u, v, </>}, and new nontrivial potential symmetries for R{x,t,u} through the use of T2{x,t,u,v,w}. Most importantly, these new potential symmetries are beyond those obtained by the use of S{x,t,u, v}. Moreover, we have found nontrivial potential symmetries for S{x,t,u,v} through the use of T\{x, t, u, v, cb} or T2{x, 2, u, v, w}. The cascaded system T3{x, t, u , v, w, tb} appears to be not useful in obtaining new symmetries for the wave equations R{x,t,u} and R{x,t,v}. In Chapter 3 we have constructed nonlinear conserved forms of the wave equation R{x,t,u} by an application of Noether's theorem. We have determined which of the point symmetries admitted by R{x,t,u} is a variational symmetry for a Lagrangian L for the wave equation R{x,t,u}. We have found three variational symmetries for L. For specific wave speeds c(x) satisfying a fifth-order O D E , three conservation laws corresponding to these variational symmetries are constructed. One conservation law corresponds to the nonlinear system *>x = -^-r(ut)2 + (ux)2, (5.7a) C {X) vt = 2uxut. (5.7b) A group analysis on the nonlinear system (5.7a,b) reveals that some point symmetries of R{x,t,u} are lost for some given wave speeds. Moreover, the use of (5.7a,b) does not lead to the construction of a potential symmetry of R{x,t,u}. We have used a recursion operator corresponding to the time translational invariance of R{x, t, u} to construct higher order conservation laws. We have found that there exist Chapter 5. Discussion 87 infinitely many higher order conservation laws for the wave equation R{x,t,u). In Chapter 4 we have studied linear conserved forms of the wave equation R{x, t, u}. We have considered a general linear system given by (4.2a,b) with arbitrary coefficients satisfying a system of PDE's given by (4.4a-d). By choosing particular solutions of (4.4a-d) we have constructed two linear conserved forms for R{x,t,u}: Li{x,t,u,v}: _ xut cz(x) vt — xux — u and L2{x,t,u,v}: tut — u V x = ~d*~(xT vt = tux. We have shown that the linear system Li{x, t, u, v} is invariant under a four-parameter local Lie group of transformations if and only if the wave speed c(x) satisfies a fourth-order ODE given by ( 4.10) which is distinct from all other classification ODE's obtained previously. For any wave speed c(x) satisfying ( 4.10) two of the four local symme-tries admitted by Li{x, t, u, v} correspond to nontrivial type II potential symmetries of R{x,t,u}. Most importantly, these new potential symmetries are beyond those obtained through S{x, t,u,v} and T2{x,t,u,v,w}. Furthermore, for some wave speeds which are common solutions of ( 4.10) and the classification ODE's obtained previously, we can find new potential symmetries of R{x,t,u}. The linear system L2{x, t, u, v} does not lead to new potential symmetries of the wave equation R{x,t, u}. The classification ODE's for L2{x, t, u, v] are of lower order and thus the allowed wave speeds are very restrictive. (5.8a) (5.8b) (5.9a) (5.9b) Chapter 5. Discussion 88 5.2 Future Research Here we propose some open problems which can be pursued in the near future. 1. Examine the invariance properties of higher order conserved forms constructed by the use of a recursion operator and Noether's theorem. 2. Examine other linear systems and linear combinations of such systems. For exam-ple, (x + t)ut — u V x = ~c\x~) ' vt = (x + t)ux - u. 3. Construct new classes of invariant solutions for the wave equations R{x,t,u} and R{x,t,v} using the new potential symmetries obtained. 4. Analyse qualitatively the classification ODE's ( 2.32) and ( 4.10) for the wave speed c(x). 5. Extend methods to other equations. Bibliography [1] Bluman, G.W. and Kumei, S. Symmetries and Differential Equations. Appl. Math. Sci. No. 81, Springer-Verlag, New York, 1989. [2] Bluman, G.W., Kumei, S., and Reid, G.J. New classes of symmetries for partial differential equations. J. Math. Phys. 29, pp. 806-811, 1988; Erratum, J. Math. Phys. 29, p. 2320, 1988. [3] Kersten, P.H.M. Infinitesimal Symmetries: a Computational Approach. CWI Tract No. 34, Centrum voor Wiskunde en Informatica, Amsterdam, 1987. [4] Schwartz, F. Symmetries of differential equations: from Sophus Lie to computer algebra. SI AM Rev. 30, pp. 450-481, 1988. [5] Reid, G.J. Finding symmetries of differential equations without integrating deter-mining equations, submitted to J. Math. Phys. [6] Bluman, G.W. and Kumei, S. On invariance properties of the wave equation. J. Math. Phys. 28, pp. 307-318, 1987. [7] Ovsiannikov, L.V. Group Analysis of Differential Equations. Academic Press, New York, 1982. [8] Bluman, G.W. and Kumei, S. Use of group analysis in solving overdetermined sys-tems of ordinary differential equations. J. Math. Anal. Appl. 138, pp. 95-105, 1989. [9] Bluman, G.W. and Kumei, S. Exact solutions for wave equations of two-layered media with smooth transition. J. Math. Phys. 29, pp. 86-96, 1988. 89 Bibliography 90 [10] Noether, E . Invariant Variationsprobleme. Nachr. Konig. Gesell. Wissen. Gottingen, Math.-Phys. KL, pp. 235-257, 1918. [11] Miiller, E .A. and Matschat, K. Uber das Auffinden von Ahnlichkeitslosungen par-tieller Differentialgleichungssyteme unter Benutzung von Transformationsgruppen, mit Anwendungen auf Probleme der Stromungsphysik. Miszellaneen der Ange-wandten Mechanik, Berlin, pp. 190-222, 1962. [12] Anderson, R.L. , Kumei, S., and Wulfman, C E . Generalization of the concept of invariance of differential equations. Phys. Rev. Lett. 28, pp. 988-991, 1972. [13] Bluman, G.W. Private communications. 

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