Algebraic method of linearization of the fully nonlinear second order PDEs

Abstract

In this paper the group theoretical methods have been applied to study linearization and integrability of the fully nonlinear second order partial differential equations (PDEs) with two and three independent variables. The procedure of using Lie symmetry groups for formulating sufficient conditions of linearization of studied equations is described and also explicit algorithms for constructing their exact solutions are demonstrated. Two specific five-dimensional Lie algebras sufficient in determining the nonsingular change of variables which represents an invertible linearization mapping for fully nonlinear PDEs are studied. The criteria for linearization are formulated in a pure algebraic way based on the dimension and structure of the symmetry Lie algebra of the studied equation. It is well known that linear and integrable PDEs admit infinite-dimensional Lie algebra of symmetry, nevertheless in presented results it is shown that only five-dimensional Lie algebra provides linearization of PDEs. Algebraic methods have the advantage of being applicable to arbitrary equation including fully nonlinear PDE and they yield results like linearization of PDEs as well as construction of broad classes of explicit solutions depending on arbitrary functions which cannot be obtained by other approaches.