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arXiv · 1812.02688

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

Jacob Brooks·Gianne Derks·David J. B. Lloyd
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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.

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Jacob Brooks, Gianne Derks, David J. B. Lloyd. 2018-12-06. Existence of stationary fronts in a system of two coupled wave equations with spatial inhomogeneity. https://doi.org/10.1088/1361-6544%2Fab2ca5

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