On the stability analysis of periodic sine-Gordon traveling waves

TL;DR

This paper analyzes the spectral stability of various types of periodic traveling waves in the sine-Gordon equation, revealing that only subluminal rotational waves are stable while others are exponentially unstable.

Contribution

It provides a corrected proof of the stability properties of sine-Gordon traveling waves, clarifying previous misconceptions in the literature.

Findings

Only subluminal rotational waves are spectrally stable.

Subluminal librational, superluminal librational, and superluminal rotational waves are exponentially unstable.

The paper corrects a previously cited proof by Scott.

Abstract

We study the spectral stability properties of periodic traveling waves in the sine-Gordon equation, including waves of both subluminal and superluminal propagation velocities as well as waves of both librational and rotational types. We prove that only subluminal rotational waves are spectrally stable and establish exponential instability in the other three cases. Our proof corrects a frequently cited one given by Scott.