Dynamical systems theory for nonlinear evolution equations

TL;DR

This paper applies dynamical systems theory to analyze nonlinear evolution equations, revealing conditions for stable solutions like solitons and compactons, and exploring how solutions vary with parameters.

Contribution

It introduces a Hamiltonian framework for Rosenau and Hymann equations and analyzes their phase space to understand solution stability and parameter effects.

Findings

Equations can support both compacton and soliton solutions.

Parameter variation can transform solution types in certain equations.

Some equations lack stable points, representing constant acceleration motion.

Abstract

We observe that the fully nonlinear evolution equations of Rosenau and Hymann, often abbreviated as $K(n,\,m)$ equations, can be reduced to Hamiltonian form only on a zero-energy hypersurface belonging to some potential function associated with the equations. We treat the resulting Hamiltonian equations by the dynamical systems theory and present a phase-space analysis of their stable points. The results of our study demonstrate that the equations can, in general, support both compacton and soliton solutions. For the $K(2,\,2)$ and $K(3,\,3)$ cases one type of solutions can be obtained from the other by continuously varying a parameter of the equations. This is not true for the $K(3,\,2)$ equation for which the parameter can take only negative values. The $K(2,\,3)$ equation does not have any stable point and, in the language of mechanics, represents a particle moving with constant acceleration.