Space-modulated Stability and Averaged Dynamics

TL;DR

This paper reviews recent advances in the nonlinear and linearized stability analysis of spectrally stable periodic waves in parabolic systems and cnoidal waves of the Korteweg-de Vries equation, aiming to develop a dispersive theory.

Contribution

It summarizes a comprehensive new theory on the nonlinear dynamics near stable periodic waves and introduces parallel linearized results for KdV cnoidal waves.

Findings

Unified theory for nonlinear stability of periodic waves

Parallel linearized stability results for KdV cnoidal waves

Foundations for future dispersive stability theory

Abstract

In this brief note we give a brief overview of the comprehensive theory, recently obtained by the author jointly with Johnson, Noble and Zumbrun, that describes the nonlinear dynamics about spectrally stable periodic waves of parabolic systems and announce parallel results for the linearized dynamics near cnoidal waves of the Korteweg-de Vries equation. The latter are expected to contribute to the development of a dispersive theory, still to come.