Journal of Differential Equations 166, 107131 (2000) doi:10.1006Âjdeq.2000.3786, available online at http:ÂÂwww.idealibrary.com on Symmetry Groups of Linear Partial Differential Equations and Representation Theory: The Laplace and Axially Symmetric Wave Equations Mark Craddock1 School of Finance and Economics, University of Technology Sydney, P.O. Box 123, Broadway, New South Wales 2007, Australia Received June 28, 1999; revised October 25, 1999 View metadata,We examinecitation and the similar Lie point papers symmetry at core.ac.uk groups of two important equations of brought to you by CORE mathematics and mathematical physics. We establish that the action of the provided by Elsevier - Publisher Connector symmetry groups are in fact equivalent to principal series representations of the underlying group. Some applications are given. 2000 Academic Press Key Words: symmetry groups; PDEs; harmonic analysis. 1. INTRODUCTION The origins of the theory of Lie groups are to be found in Lie's attempt to develop a method of integrating differential equations using group properties. Lie's analysis led him to consider one parameter groups of transformations of PDEs which preserve solutions. These groups usually turn out to be what are now known as Lie groups. A symmetry of a differential equation is an automorphism of the solution space. In other words a map which takes a solution of the differential equa- tion and transforms it to another solution. Most work on symmetries of differential equations involves the study of groups of symmetries, though one can also consider symmetries which do not form groups. For example, if we have a solution u(x, t) of the heat equation uxx=ut , then the map Ä {a : u(x, t) u(x+a, t) is a symmetry, since u(x+a, t) is also a solution. We regard {a as defining a one parameter group. There are many more symmetries of the heat equation which can be obtained by Lie's prolonga- tion algorithm. 1 The author thanks Professor A. H. Dooley and the Australian Research Council. Some of this work first appeared in the author's Ph.D. thesis. E-mail: Mark.CraddockÄuts.edu. 107 0022-0396Â00 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved. 108 MARK CRADDOCK Briefly, the prolongation algorithm works as follows. Consider a PDE in n variables L(x, D:) u(x)= f(x), (1.1) where |:| D:= , :1 :n x1 }}}xn : a multindex, and a vector field n v= : !k (x, u) +,(x, u) . (1.2) k=1 xk u The transformations generated by (1.2) are Lie point symmetries of (1.1) if and only if the following condition holds. prnv[L(x, D:) u(x)&f ]=0 (1.3) whenever u is a solution of (1.1). prnv is called the nth prolongation of v (hence prolongation algorithm). It is defined in such a way as to ensure that the group of transformations it generates take u and its nth derivatives to the nth derivatives of u~, where u~is the function u transformed under the group generated by v. Essentially we just insist that the chain rule hold. The rigorous justification of this method and an explicit formula for prnv is contained in Olver's book [9]. Olver [9] provides probably the best modern account of Lie's theory and its many applications. It also has numerous examples. The important point to keep in mind is that computation of Lie point symmetries is in practice, an entirely straightforward procedure. Indeed many computer packages exist for determining point symmetries. The symmetries that Lie's prolongation algorithm produces are frequently quite complex and unexpected. As such they yield valuable infor- mation about the PDE in question. However, it has usually been thought that they can only be defined as local groups of symmetries and do not lie within the realm of representation theory. The purpose of this paper, and its predecessor is to demonstrate that there is a strong connection between the group of point symmetries that Lie's method produces and the representation theory of the group in which the symmetries are contained. In fact Lie's prolongation algorithm very often yields representations of the underlying groups. In a previous paper [4] we studied the symmetry groups of three impor- tant linear partial differential equations of mathematical physics. These were the Schro dinger, heat and a particular FokkerPlanck equation. SYMMETRY GROUPS 109 In the case of the Schrodinger equation we showed that the group of point symmetries is in fact a semi direct product of the double cover of SL(2, R) and the three dimensional Heisenberg group. Moreover G=H3 < SL(2, R)isaglobal group of symmetries, and if we denote the symmetry operator that Lie's prolongation algorithm gives by _, then _ is a genuine irreducible representation of G. (We refer to _ as the Lie representation of G). Further there exists a unitary representation 2 [T1 , L (R)] of G, and a unitary operator A (built essentially from the fundamental solution of the Schrodinger equation) such that (_(g) Af )(x, t)=(AT1(g) f )(x, t) (1.4) for all g # G and f # L2(R). For the FokkerPlanck and heat equations analagous results hold, although _ is no longer unitary and the representation space is now a space of distributions rather than L2(R). Analogous results hold for the n-dimen- sional versions of the heat and Schro dinger equations, and Schro dinger's equation for several important nonzero potentials. In this paper we present similar results for two different equations. Namely the axially symmetric wave equation and the Laplace equation on R2. These equations are respectively hyperbolic and elliptic. For the axially symmetric wave equation we have complete results. For Laplace's equation we have only partial results. The essential reason for the incompleteness of the results for the Laplace equation on R2 is that the group of point sym- metries is infinite dimensional, whereas the groups for the wave equation, heat equation, etc. are finite dimensional. Nevertheless the results we have suggest strongly that the phenomenon we are describing is in fact of a general nature. Namely that for linear PDEs the group of point symmetries that Lie's algorithm produces is equivalent to the action of a representation of the underlying group on some locally convex topological vector space. This suggests that one may exploit the machinery of representation theory to study solutions of PDEs in a very direct manner. We will present some results which utilise the connection between the Lie symmetries and a representation of the underlying group, to illustrate the basic ideas. 2. THE AXIALLY SYMMETRIC WAVE EQUATION As our first, and most complete case, we consider the axially symmetric wave equation 2u 1 u 2u + = . (2.1) x2 x x t2 110 MARK CRADDOCK Equation (2.1) is a hyperbolic PDE which arises in a number of areas of mathematical physics. It has a four dimensional Lie algebra of symmetries which may readily be computed. We state the result and refer the reader to Bluman and Kumei [1] for the details. Theorem 2.1. The symmetry algebra of (2.1) is spanned by the vector fields v = (2.2) 1 t 1 v =x +t & u (2.3) 2 x t 2 u v =2xt +(x2+t2) &tu (2.4) 3 x t u v =u (2.5) 4 u v =:(x, t) , (2.6) : u where : is any solution of (2.1). We will be concerned with the vector fields [v1 , v2 , v3 , v4]. The infinite dimensional Lie algebra generated by the v: reflects the fact that the equation is linear and one can add solutions together to form new solu- tions. The next result is an elementary calculation with commutator tables. Lemma 2.2. [v1 , v2 , v3 , v4] generate a copy of the Lie algebra of sl2(R)ÄR. Proof. Fix the following basis for sl2 , 1 0 0 2 0 0 &1 X1= , X2= 1 , X3= (2.7) \1 0+ \0 &2+ \0 0+ and let X4 be any basis for the one dimensional Lie algebra of R. If one constructs the commutator tables for [X1 , X2 , X3 , X4] and [v1 , v2 , v3 , v4], we see that there is a Lie algebra isomorphism which send vk to Xk , k=1, ..., 4. K SYMMETRY GROUPS 111 Corollary 2.3. The Lie group of symmetries spanned by v1 , ..., v4 is locally isomorphic to SL(2, R)_R. Our aim is to show that the symmetries generated by the above vector fields is in fact equivalent to an action of a representation of the principal series of SL(2, R). In order to show this we need to know what the Lie point symmetries do to solutions of (2.1). This is the content of the next result. The action of the symmetries on solutions we refer to as the Lie representation. Proposition 2.4. The Lie representation {, obtained from exponentiating v1 , ..., v4 is given by {(exp =v1) u(x, t)=u(x, t&=) &(1Â2) = &= &= {(exp =v2) u(x, t)=e u(e x, e t) 1 {(exp =v3) u(x, t)= - 1+2=t&=2(x2&t2) x t&=(x2&t2) _u , \1+2=t&=2(x2&t2) 1+2=t&=2(x2&t2)+ = {(exp =v4) u(x, t)=e u(x, t), where u is any solution of (2.1). The proof is once more straightforward. The content of this result is that if we have a solution of the axially symmetric wave equation, and we trans- form it according to Propostion 2.4, we will still have a solution, at least if = is not too large.