Generalized symmetry reduction of nonlinear differential equations

Abstract

We study the application of generalized symmetry for reducing nonlinear partial differential equations. We construct the ansatzes for dependent variable u which reduce the scalar partial differential equation with two independent variables to systems of ordinary differential equations. The operators of Lie-Bäcklund symmetry of the second order ordinary differential equation are used. We apply the method to nonlinear evolutionary equations and find solutions which cannot be obtained in the framework of classical Lie approach. The method is also applicable to partial differential equations which are not restricted to evolution type ones. We construct the solution of nonlinear hyperbolic equation depending on an arbitrary smooth function on one variable. We study also the correlation between the dimension of symmetry Lie algebra and possibility of constructing non-invariant solutions to the equation under study.