Effective dynamics of double solitons for perturbed mKdV

TL;DR

This paper demonstrates that double soliton solutions to the perturbed mKdV equation closely follow an effective dynamics derived from Hamilton's equations, linking integrability and stability analysis.

Contribution

It introduces an effective dynamical system for double solitons in the perturbed mKdV, combining algebraic integrability with stability methods.

Findings

Double solitons remain close to the effective dynamics in $H^2$ norm.

The effective dynamics are derived from Hamilton's equations on the soliton manifold.

The approach bridges integrability and perturbation stability analysis.

Abstract

We show that an interacting double soliton solution to the perturbed mKdV equation is close in $H^2$ to a double soliton following an effective dynamics obtained as Hamilton's equations for the restriction of the mKdV Hamiltonian to the submanifold of solitons. The interplay between algebraic aspects of complete integrability of the unperturbed equation and the analytic ideas related to soliton stability is central in the proof.