Dynamical Hamiltonian-Hopf instabilities of periodic traveling waves in Klein-Gordon equations

TL;DR

This paper investigates the spectral stability of periodic traveling waves in Klein-Gordon equations, introducing the concept of dynamical Hamiltonian-Hopf instabilities and providing criteria for their detection.

Contribution

It defines dynamical Hamiltonian-Hopf instabilities and links their occurrence to the discriminant of an associated Hill's equation, offering new stability analysis tools.

Findings

Dynamical Hamiltonian-Hopf instabilities are characterized as accumulation points of unstable spectrum.

Criteria for the existence of these instabilities are derived from the discriminant of Hill's equation.

The results simplify the detection of instabilities in the spectral analysis of Klein-Gordon waves.

Abstract

We study the unstable spectrum close to the imaginary axis for the linearization of the nonlinear Klein-Gordon equation about a periodic traveling wave in a co-moving frame. We define dynamical Hamiltonian-Hopf instabilities as points in the stable spectrum that are accumulation points for unstable spectrum, and show how they can be determined from the knowledge of the discriminant of an associated Hill's equation. This result allows us to give simple criteria for the existence of dynamical Hamiltonian-Hopf instabilities in terms of instability indices previously shown to be useful in stability analysis of periodic traveling waves.