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.

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.