Conservation laws of partial differential equations: Symmetry, adjoint symmetry and nonlinear self-adjointness

TL;DR

This paper explores the relationship between symmetries, adjoint symmetries, and nonlinear self-adjointness in PDEs, providing a unified framework for constructing conservation laws and illustrating it with examples.

Contribution

It establishes that adjoint symmetries are differential substitutions of nonlinear self-adjointness, linking symmetries directly to conservation laws in PDEs.

Findings

Any adjoint symmetry is a differential substitution of nonlinear self-adjointness.

Each symmetry corresponds to a conservation law if the PDE system is nonlinearly self-adjoint.

Differential substitutions include conservation law multipliers as a subset.

Abstract

Nonlinear self-adjointness method for constructing conservation laws of partial differential equations (PDEs) is further studied. We show that any adjoint symmetry of PDEs is a differential substitution of nonlinear self-adjointness and vice versa. Consequently, each symmetry of PDEs corresponds to a conservation law via a formula if the system of PDEs is nonlinearly self-adjoint with differential substitution. As a byproduct, we find that the set of differential substitutions includes the set of conservation law multipliers as a subset. The results are illustrated by three typical examples.