Existence of stationary fronts in a coupled system of two inhomogeneous sine-Gordon equations

Speaker: Jacob Brooks (Surrey)

Abstract: In this talk 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. Numerically, we find the uncoupled inhomogeneous sine-Gordon equation has stable stationary fronts. These front solutions persist in the coupled system. Carrying out further numerical investigation it is found that stable fronts bifurcate from these inhomogeneous sine-Gordon fronts provided the coupling between the two inhomogeneous sine-Gordon equations is strong enough. In order to analytically study the emerging 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 of the inhomogeneous sine-Gordon fronts. To complete the argument, we use geometric singular perturbation theory to prove transverse fronts for a piecewise constant inhomogeneity persist for the smooth "hat-like" spatial inhomogeneity.