A cta Applicandae Mathematicae 8, (1987), 107-147 (~) 1987 by D. Reidel Publishing Company 107 Coupled Systems of Nonlinear Wave Equations and Finite-Dimensional Lie Algebras I P. J. VASSILIOU Department o[ Mathematics University College, University o[ New South Wales, Australian De[ence Force Academy, Campbell, ACT, Australia, 2601 and Department o[ Applied Mathematics, University o[ Sydney, Sydney, NSW, Australia, 2006 (Received: 3 September 1985; revised: 7 October 1986) Abstract. We study coupled systems of nonlinear wave equations from the point of view of their formal Darboux integrability. By making use of Vessiot's geometric theory of differential equations, it is possible to associate to each system of nonlinear wave equations a module of vector fields on the second-order jet bundle - the Vessiot distribution. By imposing certain conditions of the structure of the Vessiot distributions, we identify the so-called separable Vessiot distributions. By expressing the separable Vessiot distributions in a basis of singular vector fields, we show that there are, at most, 27 equivalence classes of such distributions. Of these, 14 classes are associated with Darboux integrable nonlinear systems. We take one of these Darboux integrable classes and show that it is in cor- respondence with the class of six-dimensional simply transitive Lie algebras. Finally, this later result is used to reduce the problem of constructing exact general solutions of the nonlinear wave equations understudy to the integration of Lie systems. These systems were first discovered by Sophus Lie as the most general class of ordinary differential equations which admit nonlinear superposition principles. AMS (MOS) subject classification (1980). 58G16. Key words. Nonlinear wave equations, Darboux integrable, Vessiot distribution, Lie algebras, Lie systems. 1. Introduction In recent years, there has been considerable interest in the explicit integration of systems of 1+ 1-dimensional nonlinear wave equations [1, 9-12]. In [1], Bar- bashov et al. exploit the differential geometric context to obtain explicit general solutions in terms of arbitrary functions. In [9-12] Leznov et al. make use of the representation theory of the associated Lie algebras to obtain general solutions of the nonlinear equations. From these works and many others, it seems that we are seeing a revival of the 19th Century research in partial differential equations as exemplified by E. Goursat's [4] well-known book Lecons sur l'intdgration des dquations aux ddriv~es partieIles du second ordre, written at the close of that century and collecting together most of the important known techniques for solving nonlinear partial differential equations. In the present century we see this program carried on and, in a sense, brought to completion in the writings of l~lie Cartan in his theory of Pfaff systems [3]. For a discussion of the modern developments in these and