Stability of Periodic Traveling Waves for Nonlinear Dispersive Equations

TL;DR

This paper investigates the stability properties of periodic traveling waves in nonlinear dispersive equations, establishing conditions for nonlinear stability and identifying scenarios with linear instability via exponential growth.

Contribution

It introduces a variational approach to determine the nonlinear stability of periodic waves in fractional KdV equations, extending previous results to new dispersive regimes.

Findings

Constrained minimizers are nonlinearly stable to period-preserving perturbations.

Linearized equations can admit exponentially growing solutions under certain conditions.

The variational structure is key to analyzing stability in these equations.

Abstract

We consider the stability and instability of periodic travling waves for Korteweg-de Vries type equations with fractional dispersion and other nonlinear dispersive equations. We establish that a constrained minimizer for the related variational problem is nonlinearly stable to period preserving perturbations. We also discuss when the associated linearized equation admits exponentially growing solutions. The proof utilizes the variational structure of the equation.