Stability of elliptic function solutions for the focusing modified KdV equation

TL;DR

This paper investigates the spectral and orbital stability of elliptic function solutions for the focusing mKdV equation, providing stability criteria, constructing breather solutions, and analyzing their dynamic behavior.

Contribution

It introduces a comprehensive stability analysis framework for elliptic solutions of the focusing mKdV equation using theta functions and the MSW method, including stability conditions and breather construction.

Findings

Spectral stability condition for elliptic solutions derived.

Orbital stability established under spectrum stability.

Breather solutions exhibit stable or unstable dynamics depending on parameters.

Abstract

We study the spectral and orbital stability of elliptic function solutions for the focusing modified Korteweg-de Vries (mKdV) equation and construct the corresponding breather solutions to exhibit the stable or unstable dynamic behavior. The elliptic function solutions of the mKdV equation and related fundamental solutions of the Lax pair are exactly represented by theta functions. Based on the `modified squared wavefunction' (MSW) method, we construct all linear independent solutions of the linearized mKdV equation and then provide a necessary and sufficient condition of the spectral stability for elliptic function solutions with respect to subharmonic perturbations. In the case of spectrum stability, the orbital stability of elliptic function solutions is established in a suitable Hilbert space. Using Darboux-B\"acklund transformation, we construct breather solutions to exhibit unstable or stable dynamic behavior. Through analyzing the asymptotic behavior, we find that the breather solution under the cn-type solution background is equivalent to the elliptic function solution adding a small perturbation as $t\to\pm\infty$.