Sufficient Conditions for Orbital Stability of Periodic Traveling Waves

TL;DR

This paper establishes sufficient conditions for the orbital stability of periodic traveling waves in nonlinear dispersive evolution equations, extending solitary wave stability theory to periodic waves without relying on wave parametrization.

Contribution

It introduces a parametrization-independent stability criterion for periodic waves, extending solitary wave stability theory to a broader class of evolution equations.

Findings

The stability criterion for solitary waves applies to periodic waves.

Positivity of the Hessian matrix entries ensures stability.

The method applies to various nonlinear dispersive models.

Abstract

The present paper deals with sufficient conditions for orbital stability of periodic waves of a general class of evolution equations supporting nonlinear dispersive waves. Our method can be seen as an extension to spatially periodic waves of the theory of solitary waves recently developed in \cite{st}. Firstly, our main result do not depend on the parametrization of the periodic wave itself. Secondly, motived by the well known orbital stability criterion for solitary waves, we show that the same criterion holds for periodic waves. In addition, we show that the positiveness of the principal entries of the Hessian matrix related to the "energy surface function" are also sufficient to obtain the stability. Consequently, we can establish the orbital stability of periodic waves for several nonlinear dispersive models. We believe our method can be applied in a wide class of evolution equations; in particular it can be extended to regularized dispersive wave equations.