Determination of approximate nonlinear self-adjointenss and approximate conservation law

TL;DR

This paper explores the relationship between approximate nonlinear self-adjointness and conservation laws in perturbed PDEs, providing a method to classify equations and generate new approximate conservation laws.

Contribution

It introduces a simple approach to determine approximate nonlinear self-adjointness and derives explicit formulas for approximate conservation laws based on unperturbed PDEs.

Findings

Classified perturbed wave equations as approximately nonlinear self-adjoint.

Derived explicit formulas for approximate conservation laws.

Showed that approximate nonlinear self-adjointness can produce new conservation laws.

Abstract

Approximate nonlinear self-adjointness is an effective method to construct approximate conservation law of perturbed partial differential equations (PDEs). In this paper, we study the relations between approximate nonlinear self-adjointness of perturbed PDEs and nonlinear self-adjointness of the corresponding unperturbed PDEs, and consequently provide a simple approach to discriminate approximate nonlinear self-adjointness of perturbed PDEs. Moreover, a succinct approximate conservation law formula by virtue of the known conservation law of the unperturbed PDEs is given in an explicit form. As an application, we classify a class of perturbed wave equations to be approximate nonlinear self-adjointness and construct the general approximate conservation laws formulae. The specific examples demonstrate that approximate nonlinear self-adjointness can generate new approximate conservation laws.