Complete Classification of Local Conservation Laws for Generalized Cahn-Hilliard-Kuramoto-Sivashinsky Equation

TL;DR

This paper classifies all local conservation laws for a generalized multidimensional Cahn-Hilliard-Kuramoto-Sivashinsky equation, identifying cases with nontrivial laws and providing explicit forms, revealing the absence of such laws in the original Kuramoto-Sivashinsky equation.

Contribution

It provides a complete classification of local conservation laws for the generalized equation, including explicit forms and the identification of cases with nontrivial laws.

Findings

Kuramoto-Sivashinsky equation admits no nontrivial local conservation laws

Complete list of cases with nontrivial conservation laws for the generalized equation

Explicit forms of all local conservation laws for identified cases

Abstract

In the present paper we consider nonlinear multidimensional Cahn-Hilliard and Kuramoto-Sivashinsky equations that have many important applications in physics and chemistry, and a certain natural generalization of these equations. For an arbitrary number of spatial independent variables we present a complete list of cases when the generalized Cahn-Hilliard-Kuramoto-Sivashinsky equation admits nontrivial local conservation laws of any order, and for each of those cases we give an explicit form of all the local conservation laws of all orders modulo trivial ones admitted by the equation under study. In particular, we show that the original Kuramoto-Sivashinsky equation admits no nontrivial local conservation laws, and find all nontrivial local conservation laws for the Cahn-Hilliard equation.