A New Evans Function for Quasi-Periodic Solutions of the Linearised Sine-Gordon Equation
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A New Evans Function for Quasi-Periodic Solutions of the Linearised Sine-Gordon Equation W. A. Clarke1 · R. Marangell1 Received: 18 May 2020 / Accepted: 16 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
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 EvansKrein function theory of Kollár and Miller (SIAM Rev 56(1):73–123, 2014) 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 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. Keywords Evans function · Periodic travelling waves · Spectral stability · Non-linear Klein–Gordon equation · Krein signature · Hamiltonian–Hopf bifurcations Mathematics Subject Classification 35B10 · 35C07 · 35B35 · 35P05 · 47A75
1 Introduction We consider the sine-Gordon equation, u tt − u x x + sin(u) = 0,
(1)
Communicated by Peter Miller.
B
R. Marangell [email protected] W. A. Clarke [email protected]
1
School of Mathematics and Statistics, University of Sydney, Sydney, Australia
123
Journal of Nonlinear Science
where u(x, t) : R × [0, +∞) −→ R. This equation has been used in the modelling of a number of different physical and biological systems. For example, equation (1) models the electrodynamics of a long Josephson junction arising in the theory of superconductors (Barone and Paternò 1982; Derks et al. 2012). Solitary wave solutions to equation (1) have been used to describe the dynamics of DNA as it interacts with RNA-polymerase (Derks and Gaeta 2011). More recent research has seen sineGordon solitons used as scalar gravitational fields in the theory of general relativity, the solutions of which are solitonic stars and black holes (Cadoni et al. 2019). Equation (1) can also be derived from classical mechanics applied to a mechanical transmission line in which pendula are coupled to their nearest neighbours by springs obeying Hooke’s law (Knobel 2000). See Barone et al. (1971) for an extensive list of applications of the sine-Gordon equation. In this paper, we focus on the problem of spectral stability of periodic solutions to the sine-Gordon equation. In Scott (1969), Scott correctly attributed spectral stability and instability to various types of periodic wavetrains; however, this was rigorously proved only recently in Jones et al. (2013) in which the authors related the sine-Gordon equation to a Hill’s equation using Floquet theory and known results on Hil
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