Stability and instability of traveling wave solutions to nonlinear wave equations

TL;DR

This paper investigates the stability and instability of plane wave solutions in semilinear wave equations satisfying the null condition, identifying conditions for global stability and linear instability through geometric analysis.

Contribution

It introduces a specific condition that ensures global nonlinear stability and demonstrates linear instability when the condition is not satisfied, using geometric optics methods.

Findings

Global nonlinear stability under certain geometric conditions

Linear instability when conditions are not met

Use of geometric optics ansatz for instability analysis

Abstract

In this paper, we study the stability and instability of plane wave solutions to semilinear systems of wave equations satisfying the null condition. We identify a condition which allows us to prove the global nonlinear asymptotic stability of the plane wave. The proof of global stability requires us to analyze the geometry of the interaction between the background plane wave and the perturbation. When this condition is not met, we are able to prove linear instability assuming an additional genericity condition. The linear instability is shown using a geometric optics ansatz.