Determination of approximate non-linear self-adjointness and approximate conservation law

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.

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

“…Three different types of examples illustrate our results. In addition, the presented results, after proper arrangements, can be applied to study approximate nonlinear self-adjointness of perturbed PDEs [8,23,24].…”

Conservation laws of the generalized short pulse equation

“…[16,17] Approximate nonlinear selfadjointness and approximate conservation laws were studied in Refs. [17]- [19].…”