Abstract
In this thesis we study semi-linear wave equations with spatial inhomogeneity. The spatial inhomogeneity corresponds to a localised spatially dependent scaling of the nonlinear potential term. This thesis will consider the existence of stationary fronts and the dynamics of travelling solutions.
In the first part of the thesis we investigate the existence of stationary fronts in a coupled system of two sine-Gordon equations with a smooth, ``hat-like'' spatial inhomogeneity. 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 lose 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, ``hat-like'' spatial inhomogeneity by a piecewise constant function. With this approximation, we can treat the inhomogeneous sine-Gordon equation as homogeneous sine-Gordon equations in each of the three regions of the piecewise constant function. Doing so we can then construct stationary front solutions explicitly and 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.
In the second part of the thesis we investigate the dynamics of travelling solutions to semi-linear wave equations with a piecewise constant spatial inhomogeneity. We assume the underlying homogeneous semi-linear wave equations possess travelling front solutions which can be interpreted as a two parameter family of wave shapes. We show that travelling solutions to the inhomogeneous semi-linear wave equation can be decomposed as a unique wave shape within the family plus a remainder which lies in some subspace, for some time interval to be determined. Carrying out a numerical investigation we show that this decomposition holds for long time intervals. Finally, given initial data that can be decomposed as a subset of the family of wave shapes with small remainder term, we show the decomposition holds for all time analytically.