A domain - wall between single - mode and bimodal statesG

We examine a model equation describing spatial patterns in a class of physical systems where instabilities to travelling waves occur. The spatial patterns are modelled by a system of two second order ordinary diierential equations, in which the cross-coupling coeecient is spatially dependent. The system has two clearly distinct types of stationary states, of which the stability depends on the cross-coupling coeecient. Under mild assumptions on the cross-coupling coeecient, we apply a variational method to prove the existence of a heteroclinic orbit between both types of states, corresponding to a domain-wall in the physical picture. This solution is found as a minimiser of a Lyapunov functional and the variational structure is exploited to obtain detailed information about the shape of the solution. In the case of a constant cross-coupling coeecient we nd heteroclinic solutions connecting stationary states of the same type.