Stability of Lie Groups of Nonlinear Hyperbolic Equations

Abstract

Nonlinear theories, started in the 1960's, provided for the first time in the literature global generalized solutions for arbitrary continuous nonlinear partial differential equations (PDEs). These theories have now been extended to Lie symmetry groups for classical and global generalized solutions of nonlinear PDEs. My work in this thesis is on stability analysis of Lie groups of nonlinear PDEs. Nonlinear algebraic theory of generalized solutions for large classes of nonlinear PDEs was originated by Elemer E. Rosinger ' who published the first two papers on the subject in 1966 and 1968. He has since developed the theory further, culminating in the publication of four research monographs (1978, 1980, 1984, 1990). In these monographs the algebraic theory, complete with applications in the study of nonlinear PDEs, is well presented. Some of the major results obtained by Rosinger in this line of research include: · the solution of the celebrated 1954 impossibilty result of L. Schwartz? regarding the multiplication of distributions (1966); · the characterization of all possible nonlinear algebraic theories of generalized functions (1980); · the global solution of arbitrary nonlinear analytic PDEs (1987); · the algebraic characterization of the solvability of large classes of nonlinear PDEs (1990). J.F. Colornbeau+" of Lions, France, is the other main protagonist of this field. He independently developed an algebraic nonlinear theory of generalized functions, first published in 1984. Although Colombeau's theory is considered to be the most complete, very powerful and so far the most widely studied of the various possible algebraic methods, it is of narrower applicability due to the fact that it is merely a particular case of the whole class of such possible theories already developed fully and characterized completely by Rosinger in 1980 (see Colombeau's review in Bull. AMS vol. 20, no. 1, January 1989, pp.