Stability of Small Periodic Waves in Fractional KdV Type Equations

TL;DR

This paper investigates the spectral stability of small periodic traveling waves in fractional KdV-type equations, developing a nonlocal Floquet theory to analyze how dispersion and nonlinearity influence stability.

Contribution

It introduces a nonlocal Floquet-like spectral analysis method for fractional KdV equations, linking nonlinearity power and dispersion symbol to stability outcomes.

Findings

Spectral stability depends on the relationship between nonlinearity and dispersion.

Developed a nonlocal Floquet theory suitable for fractional dispersive operators.

Derived criteria for stability and instability of small periodic waves.

Abstract

We consider the effects of varying dispersion and nonlinearity on the stability of periodic traveling wave solutions of nonlinear PDE of KdV-type, including generalized KdV and Benjamin-Ono equations. In this investigation, we consider the spectral stability of such solutions that arise as small perturbations of an equilibrium state. A key feature of our analysis is the development of a nonlocal Floquet-like theory that is suitable to analyze the $L^2(\RM)$ spectrum of the associated linearized operators. Using spectral perturbation theory then, we derive a relationship between the power of the nonlinearity and the symbol of the fractional dispersive operator that determines the spectral stability and instability to arbitrary small localized perturbations.