Evolution equations: Frobenius integrability, conservation laws and travelling waves

TL;DR

This paper explores the integrability and conservation laws of evolution equations with travelling wave solutions, providing new insights into their structure and applying these results to the Korteweg de Vries equation.

Contribution

It introduces a new powerful result explaining the integrability of certain evolution equations and discusses local conservation laws in a general framework.

Findings

New integrability criteria for evolution equations with travelling waves

Application of results to the Korteweg de Vries equation demonstrating practical relevance

Enhanced understanding of conservation laws in evolution equations

Abstract

We give new results concerning the Frobenius integrability and solution of evolution equations admitting travelling wave solutions. In particular, we give a powerful result which explains the extraordinary integrability of some of these equations. We also discuss "local" conservations laws for evolution equations in general and demonstrate all the results for the Korteweg de Vries equation.