New classes of nonlinearly self-adjoint evolution equations of third- and fifth-order

TL;DR

This paper classifies new nonlinearly self-adjoint third- and fifth-order evolution equations with time-dependent coefficients and derives associated conservation laws, expanding the understanding of their mathematical structure.

Contribution

It provides a novel classification of nonlinearly self-adjoint equations of third and fifth order, including new subclasses and conservation laws, based on Ibragimov's framework.

Findings

Five subclasses of fifth-order nonlinearly self-adjoint equations identified.

Four subclasses of third-order nonlinearly self-adjoint equations identified.

Conservation laws established for selected equations.

Abstract

In a recent communication Nail Ibragimov introduced the concept of nonlinearly self-adjoint differential equation [N. H. Ibragimov, Nonlinear self-adjointness and conservation laws, J. Phys. A: Math. Theor., vol. 44, 432002, 8 pp., (2011)]. In the present communication a nonlinear self-adjoint classification of a general class of fifth-order evolution equation with time dependent coefficients is presented. As a result five subclasses of nonlinearly self-adjoint equations of fifth-order and four subclasses of nonlinearly self-adjoint equations of third-order are obtained. From the Ibragimov's theorem on conservation laws [N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., vol. 333, 311--328, (2007)] conservation laws for some of these equations are established.