The orbital stability of the periodic traveling wave solutions to the defocusing complex modified Korteweg-de Vries equation

TL;DR

This paper investigates the orbital stability of elliptic periodic solutions to the defocusing complex modified Korteweg-de Vries (cmKdV) equation, especially those oscillating around zero, using integrability and conserved quantities.

Contribution

It extends stability analysis of the defocusing cmKdV equation to solutions oscillating around zero, establishing spectral and orbital stability through conserved quantities.

Findings

Spectral stability of elliptic solutions proven.

A Lyapunov functional constructed from conserved quantities.

Orbital stability established for solutions oscillating around zero.

Abstract

The stability of the elliptic solutions to the defocusing complex modified Korteweg-de Vries (cmKdV) equation is studied. The orbital stability of the cmKdV equation was established in [19] when the periodic orbits do not oscillate around zero. In this paper, we study the periodic solutions corresponding to the case that the orbits oscillate around zero. Using the integrability of the defocusing cmKdV equation, we prove the spectral stability of the elliptic solutions. We show that one special linear combination of the first five conserved quantities produces a Lyapunov functional, which implies that the elliptic solutions are orbitally stable with respect to the subharmonic perturbations.