A Symmetry-Based Method for Constructing Nonlocally Related PDE Systems

TL;DR

This paper introduces a symmetry-based method to construct nonlocally related PDE systems, expanding the toolkit beyond conservation laws and revealing new nonlocal symmetries for various nonlinear equations.

Contribution

It presents a novel approach that systematically derives nonlocally related PDE systems from point symmetries, complementing existing conservation law methods.

Findings

New nonlocal symmetries identified for nonlinear wave equations

Constructed nonlocally related systems for reaction-diffusion equations without conservation laws

Demonstrated the method's applicability to diverse nonlinear PDEs

Abstract

Nonlocally related partial differential equation (PDE) systems are useful in the analysis of a given PDE system. It is known that each local conservation law of a given PDE system systematically yields a nonlocally related system. In this paper, a new and complementary method for constructing nonlocally related systems is introduced. In particular, it is shown that each point symmetry of a given PDE system systematically yields a nonlocally related system. Examples include applications to nonlinear diffusion equations, nonlinear wave equations and nonlinear reaction-diffusion equations. As a consequence, previously unknown nonlocal symmetries are exhibited for two examples of nonlinear wave equations. Moreover, since the considered nonlinear reaction-diffusion equations have no local conservation laws, previous methods do not yield nonlocally related systems for such equations.