Nonlinear self-adjointness in constructing conservation laws

TL;DR

This paper introduces the concept of nonlinear self-adjointness in differential equations, unifying various notions and enabling the construction of conservation laws for a broad class of equations, including linear and nonlinear cases.

Contribution

It generalizes the concept of self-adjointness to nonlinear equations, allowing systematic derivation of conservation laws for diverse differential systems.

Findings

Includes all linear equations as special cases.

Enables construction of conservation laws for nonlinear and linear equations.

Unifies previous notions of self-adjointness.

Abstract

The general concept of nonlinear self-adjointness of differential equations is introduced. It includes the linear self-adjointness as a particular case. Moreover, it embraces the previous notions of self-adjoint and quasi self-adjoint nonlinear equations. The class of nonlinearly self-adjoint equations includes, in particular, all linear equations. Conservation laws associated with symmetries can be constructed for all nonlinearly self-adjoint differential equations and systems. The number of equations in systems can be different from the number of dependent variables.