Nonlinear self-adjointness and conservation laws

TL;DR

The paper introduces the concept of nonlinear self-adjointness for differential equations, unifying various forms, and demonstrates how equations with this property can be transformed into strictly self-adjoint forms to derive conservation laws.

Contribution

It generalizes the concept of self-adjointness to nonlinear cases, unifies previous notions, and provides a method to obtain conservation laws for a broad class of differential equations.

Findings

All linear equations are nonlinear self-adjoint.

Heat equation can be made strictly self-adjoint by multiplication.

Conservation laws can be constructed for equations with nonlinear 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 strict self-adjointness and quasi self-adjointness introduced earlier by the author. It is shown that the equations possessing the nonlinear self-adjointness can be written equivalently in a strictly self-adjoint form by using appropriate multipliers. All linear equations possess the property of nonlinear self-adjointness, and hence can be rewritten in a nonlinear strictly self-adjoint. For example, the heat equation $u_t - \Delta u = 0$ becomes strictly self-adjoint after multiplying by $u^{-1}.$ Conservation laws associated with symmetries can be constructed for all differential equations and systems having the property of nonlinear self-adjointness.