Soliton dynamics for a non-Hamiltonian perturbation of mKdV

TL;DR

This paper investigates the behavior of solitons in a non-Hamiltonian perturbed mKdV equation, demonstrating their stability and parameter evolution over specific time scales despite the perturbation.

Contribution

It introduces a novel analysis of soliton dynamics under non-Hamiltonian perturbations, showing stability and parameter evolution on multiple time scales.

Findings

Solutions stay close to solitons over long time scales.

Soliton parameters follow specific ODEs.

Perturbation influences soliton position significantly.

Abstract

We study the dynamics of soliton solutions to the perturbed mKdV equation $\partial_t u = \partial_x(-\partial_x^2 u -2u^3) + \epsilon V u$, where $V\in \mathcal{C}^1_b(\mathbb{R})$, $0<\epsilon\ll 1$. This type of perturbation is non-Hamiltonian. Nevertheless, via symplectic considerations, we show that solutions remain $O(\epsilon \la t\ra^{1/2})$ close to a soliton on an $O(\epsilon^{-1})$ time scale. Furthermore, we show that the soliton parameters can be chosen to evolve according to specific exact ODEs on the shorter, but still dynamically relevant, time scale $O(\epsilon^{-1/2})$. Over this time scale, the perturbation can impart an O(1) influence on the soliton position.