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Existence of stationary fronts in a system of two coupled wave equations with spatial inhomogeneity

Brooks, Jacob, Derks, Gianne and Lloyd, David J.B. (2019) Existence of stationary fronts in a system of two coupled wave equations with spatial inhomogeneity Nonlinearity, 32 (11). pp. 4147-4187.

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Abstract

We investigate the existence of stationary fronts in a coupled system of two sine-Gordon equations with a smooth, “hat-like” spatial inhomogeneity. The spatial inhomogeneity corresponds to a spatially dependent scaling of the sine-Gordon potential term. The uncoupled inhomogeneous sine-Gordon equation has stable stationary front solutions that persist in the coupled system. Carrying out a numerical investigation it is found that these inhomogeneous sine-Gordon fronts loose stability, provided the coupling between the two inhomogeneous sine-Gordon equations is strong enough, with new stable fronts bifurcating. In order to analytically study the bifurcating fronts, we first approximate the smooth spatial inhomogeneity by a piecewise constant function. With this approximation, we prove analytically the existence of a pitchfork bifurcation. To complete the argument, we prove that transverse fronts for a piecewise constant inhomogeneity persist for the smooth “hat-like” spatial inhomogeneity by introducing a fast-slow structure and using geometric singular perturbation theory.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
NameEmailORCID
Brooks, Jacobjacob.brooks@surrey.ac.uk
Derks, GianneG.Derks@surrey.ac.uk
Lloyd, David J.B.D.J.Lloyd@surrey.ac.uk
Date : 26 September 2019
DOI : 10.1088/1361-6544/ab2ca5
Copyright Disclaimer : Copyright 2019 IOP Publishing. This is the accepted version of the following article: Kostianko, Anna, Titi, Edriss and Zelik, Sergey (2018) Large dispersion, averaging and attractors: three 1D paradigms Nonlinearity, 31 (12), R317, which has been published in final form at http://iopscience.iop.org/article/10.1088/1361-6544/aae175/meta
Uncontrolled Keywords : Coupled inhomogeneous wave equations; Bifurcation analysis; Sine-Gordon front; Existence; Geometric singular perturbation theory; Mathematics Subject Classification numbers: 37J20, 35L71, 34C37
Depositing User : Clive Harris
Date Deposited : 01 Jul 2019 10:35
Last Modified : 17 Oct 2019 07:43
URI: http://epubs.surrey.ac.uk/id/eprint/852186

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