Modulated Wave Trains in Lattice Differential Systems

The existence of weak sinks in mixed parabolic-lattice systems on the real line is established for systems that incorporate discrete coupling on an underlying lattice in addition to continuous diffusion. Sinks can be thought of as interfaces that separate two spatially periodic structures with different wave numbers: the corresponding modulated wave train is time periodic in the frame that moves with the speed of the interface. In this paper, the existence of weak sinks is proved that connect wave trains with almost identical wave number. The main difficulty is the global coupling between points on the underlying lattice, since its presence turns the equation solved by sinks into an ill-posed functional differential equation of mixed type.