A New Evans Function for Quasi-Periodic Solutions of the Linearised Sine-Gordon Equation

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 the Evans-Krein function theory of [KM2014] 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 the locations of Hamiltonian-Hopf bifurcations therein. 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.

Localized structures on librational and rotational travelling waves in the sine-Gordon equation

“…Although the algebraic method used for librational and rotational waves is similar, the outcomes are different dynamically. This difference is explained by the different types of spectral stability of the travelling periodic waves [7][8][9] (see also [10][11][12] for recent contributions). In the superluminal regime (which is the only regime we are interested in), the librational periodic waves are spectrally unstable and the Floquet-Bloch spectrum forms a figure eight intersecting at the origin.…”

Localized structures on librational and rotational travelling waves in the sine-Gordon equation

“…Although the algebraic method used for librational and rotational waves is similar, the outcomes are different dynamically. This difference is explained by the different types of spectral stability of the travelling periodic waves [15,16,19] (see also [12,13] and [11] for recent contributions). In the superluminal regime (which is the only regime we are interested in), the librational periodic waves are spectrally unstable and the Floquet-Bloch spectrum forms a figure eight intersecting at the origin.…”