Soliton dynamics for a non-Hamiltonian perturbation of mKdV

Abstract

We study the dynamics of soliton solutions to the perturbed mKdV equation ${\partial }_{t}u={\partial }_x(-{\partial }_x^{2}u-2u^3)+\epsilon Vu$, where $V\in {\mathcal{C}}_b^1(\mathbb{R})$, $0<\epsilon \ll 1$. This type of perturbation is non-Hamiltonian. Nevertheless, via symplectic considerations, we show that solutions remain $O(ε\la t\ra^{1/2})$ 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.

Cited by in corpus (1)

• On approximate controllability of generalized KdV solitons