Analysis and classification of nonlinear dispersive evolution equations in the potential representation

TL;DR

This paper introduces a potential representation for traveling solutions of nonlinear dispersive equations, classifies K(n,m) equations, and shows soliton solutions only exist in systems with specific dispersion types.

Contribution

It presents a novel potential representation reducing complex PDEs to simpler ODEs and classifies K(n,m) equations based on point transformations, unifying soliton solutions.

Findings

Solitons exist only in systems with linear or quadratic dispersion.

All K(n,m) equations with soliton solutions are classified into a single equivalence class.

The Korteweg-deVries equation serves as the representative of this class.

Abstract

A potential representation for the subset of traveling solutions of nonlinear dispersive evolution equations is introduced. The procedure involves a reduction of a third order partial differential equation to a first order ordinary differential equation. In this representation it can be shown that solitons and solutions with compact support only exist in systems with linear or quadratic dispersion, respectively. In particular, this article deals with so the called K(n,m) equations. It is shown that these equations can be classified according to a simple point transformation. As a result, all equations that allow for soliton solutions join the same equivalence class with the Korteweg-deVries equation being its representative.