A new Evans function for quasi-periodic solutions of the linearised sine-Gordon equation

TL;DR

This paper introduces a novel Evans function for analyzing quasi-periodic solutions of the linearised sine-Gordon equation, enabling efficient spectral analysis and bifurcation tracking, with potential applications to nonlinear Klein-Gordon equations.

Contribution

A new Evans function formulation based on Hill's equation for quasi-periodic solutions, facilitating spectral analysis and bifurcation detection in sine-Gordon and Klein-Gordon equations.

Findings

Computed spectra of periodic travelling waves.

Tracked Hamiltonian-Hopf bifurcations via Krein signatures.

Demonstrated applicability to nonlinear Klein-Gordon equations.

Abstract

We construct a new Evans function for quasi-periodic solutions to the linearisation of the sine-Gordon equation about a periodic travelling wave. This Evans function is written in terms of fundamental solutions to a Hill's equation. Applying Evans-Krein function theory to our Evans function, we provide a new method for computing the Krein signatures of simple characteristic values of the linearised sine-Gordon equation. By varying the Floquet exponent parametrising the quasi-periodic solutions, we compute the linearised spectra of periodic travelling wave solutions of the sine-Gordon equation and and track dynamical Hamiltonian-Hopf bifurcations via the Krein signature. Finally, we show that our new Evans function can be readily applied to the general case of the nonlinear Klein-Gordon equation with a non-periodic potential.