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.