On the stability of the dnoidal waves for the Schr\"odinger-KdV system

TL;DR

This paper investigates the spectral stability of dnoidal wave solutions in a periodic Schr"odinger-KdV system, extending known results on soliton stability to periodic traveling waves.

Contribution

It characterizes a two-parameter family of periodic dnoidal waves and proves their spectral stability under co-periodic perturbations, linking to soliton stability results.

Findings

Established spectral stability of dnoidal waves for the Schr"odinger-KdV system.

Connected periodic wave stability to soliton stability as period tends to infinity.

Described a two-parameter family of periodic traveling wave solutions.

Abstract

We study the periodic Schr\"odinger-Korteweg de Vries system. We describe the two-parametetric family of $2T$ periodic traveling waves of dnoidal type. The main objective of the paper is to establish their spectral stability with respect to co-periodic perturbations. In the limit $T\to \infty$, we recover the results of Albert-Angulo, for the stability of the soliton solutions of this system on the real line.