Soliton resolution and asymptotic stability of $N$-soliton solutions for the defocusing mKdV equation with a non-vanishing background

TL;DR

This paper proves the soliton resolution conjecture and demonstrates the asymptotic stability of N-soliton solutions for the defocusing mKdV equation with a non-vanishing background using advanced analytical techniques.

Contribution

It provides the first rigorous proof of the soliton resolution conjecture and stability for this class of solutions under the specified conditions.

Findings

Confirmed the soliton resolution conjecture for the defocusing mKdV with non-vanishing background.

Established the asymptotic stability of N-soliton solutions.

Applied $ar{ ext{D}}$-nonlinear steepest descent analysis to Riemann-Hilbert problems.

Abstract

We analytically study the large-time asymptotics of the solution of the defocusing modified Korteweg-de Vries (mKdV) equation under a symmetric non-vanishing background, which supports the emergence of solitons. It is demonstrated that the asymptotic expansion of the solution at the large time could verify the renowned soliton resolution conjecture. Moreover, the asymptotic stability of $N$-soliton solution is also exhibited in the present work. We establish our results by performing a $\bar{\partial}$-nonlinear steepest descent analysis to the associated Riemann-Hilbert (RH) problem.