  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.
