On the Stability of Periodic Waves for the Cubic Derivative NLS and the Quintic NLS

We study the periodic cubic derivative nonlinear Schrodinger equation (DNLS) and the (focussing) quintic nonlinear Schrodinger equation (NLS). These are both L-2 critical dispersive models, which exhibit threshold-type behavior, when posed on the line 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 thesewaves for stability in the framework of the cubic DNLS. We provide 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 travelingwaves 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.yy

Automated summary

This paper examines the stability of periodic cubic derivative nonlinear Schrodinger equation and (focussing) quintic nonlinear Schrodinger equation with respect to co-periodic perturbations. By analyzing detailed spectral analysis, the paper provides criteria for stability and identifies a parameter space region that produces spectrally stable waves. Additionally, a description of the stability of all bell-shaped traveling waves is provided for the quintic NLS, showing a two-parameter subfamily of waves compared to DNLS.