`Relativistic' propagation of instability fronts in nonlinear Klein-Gordon equation dynamics

TL;DR

This paper analyzes the propagation of instability fronts in nonlinear Klein-Gordon equations using Whitham modulation theory, revealing that these fronts move at maximal group velocity in self-similar regimes.

Contribution

It applies the Whitham method to derive self-similar solutions for instability front propagation in nonlinear Klein-Gordon systems, highlighting maximal group velocity behavior.

Findings

Instability fronts propagate with maximal group velocity.

Self-similar solutions describe the long-time behavior of instability regions.

The approach is demonstrated with specific nonlinear examples.

Abstract

We consider propagation of instability fronts in conservative nonlinear wave systems by the Whitham method. Whitham modulation equations for periodic solutions of the generalized Klein-Gordon equation are solved in the limit of asymptotically large times, when the size of the instability wave region is much greater than the size of the initial localized disturbance, so the solution reaches the self-similar regime. The general self-similar solution is illustrated by two typical examples of the nonlinearity function. It is shown that in these models the instability fronts propagate with maximal group velocity.