On the stability of periodic waves for the cubic derivative NLS and the quintic NLS

TL;DR

This paper investigates the stability of periodic bell-shaped solutions in cubic derivative NLS and quintic NLS equations, providing explicit stability criteria based on spectral analysis and identifying stable and unstable wave regions.

Contribution

It offers a detailed spectral stability analysis of periodic solutions for both equations, including explicit stability criteria and the characterization of stable and unstable parameter regions.

Findings

Identified stable regions for cubic derivative NLS waves.

Provided explicit stability criteria depending on spectral properties.

Complete stability classification for bell-shaped traveling waves in quintic NLS.

Abstract

We study the periodic cubic derivative non-linear Schr\"odinger equation (dNLS) and the (focussing) quintic non-linear Schr\"odinger equation (NLS). These are both $L^2$ critical dispersive models, which exhibit threshold type behavior, when posed on the line ${\mathbb R}$. We describe the (three parameter) family of non-vanishing bell-shaped solutions for the periodic problem, in closed form. The main objective of the paper is to study their stability with respect to co-periodic perturbations. We analyze these waves for stability in the framework of the cubic DNLS. We provide a criteria for stability, depending on the sign of a scalar quantity. The proof relies on an instability index count, which in turn critically depends on a detailed spectral analysis of a self-adjoint matrix Hill operator. We exhibit a region in parameter space, which produces spectrally stable waves. We also provide an explicit description of the stability of all bell-shaped traveling waves for the quintic NLS, which turns out to be a two parameter subfamily of the one exhibited for DNLS. We give a complete description of their stability - as it turns out some are spectrally stable, while other are spectrally unstable, with respect to co-periodic perturbations.