# Stability of periodic waves for the fractional KdV and NLS equations

## TL;DR

This paper rigorously analyzes the stability of periodic traveling waves in fractional KdV and NLS equations, constructing explicit solutions that are spectrally and orbitally stable for sub-critical dispersion.

## Contribution

It constructs explicit families of stable periodic waves for fractional KdV and NLS equations without prior smoothness assumptions, extending stability results to fractional models.

## Key findings

- Existence of parameterized families of stable traveling waves.
- Proof of spectral and orbital stability for these waves.
- Construction of solutions without smoothness assumptions.

## Abstract

We consider the focusing fractional periodic Korteweg-deVries (fKdV) and fractional periodic nonlinear Schr\"odinger equations (fNLS) equations, with $L^2$ sub-critical dispersion. In particular, this covers the case of the periodic KdV and Benjamin-Ono models. We construct two parameter family of bell-shaped traveling waves for KdV (standing waves for NLS), which are constrained minimizers of the Hamiltonian. We show in particular that for each $\lambda>0$, there is a traveling wave solution to fKdV and fNLS $\varphi: \|\varphi\|_{L^2[-T,T]}^2=\lambda$, which is non-degenerate and spectrally stable, as well as orbitally stable. This is done completely rigorously, without any {\it a priori} assumptions on the smoothness of the waves or the Lagrange multipliers.