Orbital Stability of Periodic Traveling-Wave Solutions for a Dispersive Equation

TL;DR

This paper proves the orbital stability of periodic traveling-wave solutions for a broad class of dispersive equations, using spectral analysis and the Implicit Function Theorem, with applications to the Kawahara equation.

Contribution

It introduces a general method for establishing orbital stability of periodic waves in dispersive equations, applicable to various models including the Kawahara equation.

Findings

Proved orbital stability of dnoidal waves for the Kawahara equation.

Established conditions on the linearized operator for stability.

Used the Implicit Function Theorem to ensure smooth solution dependence.

Abstract

In this paper we establish the orbital stability of periodic traveling waves for a general class of dispersive equations. We use the Implicit Function Theorem to guarantee the existence of smooth solutions depending of the corresponding wave speed. Essentially, our method establishes that if the linearized operator has only one negative eigenvalue which is simple and zero is a simple eigenvalue the orbital stability is determined provided that a convenient condition about the average of the wave is satisfied. We use our approach to prove the orbital stability of periodic dnoidal waves associated with the Kawahara equation.