Travelling Waves In 2D Lattice Systems

In this paper we discuss the existence of travelling wave solutions in the two dimensional lattice systems. We study the dissipative systems and the conservative systems with linear coupling or sinusoidal coupling. Using the symmetry of Z2 lat-tice, we reduce the travelling waves in two-dimensional lattice to a one-dimensinonal problem. For the dissipative systems, the existence of travelling wave solution is con-verted into a fixed point problem for an operator equation. Applying the Schauder fixed point theorem, we obtain the existence conclusions of travelling wave solutions. For the conservative system, a travelling wave solution is converted to a periodic solu-tion of an advance-delay differential equation. We define a C1 functional in the given space. If this functional has a critical point in the given space, then this critical point corresponds to the periodic solution of the conservative system. Finally, appling the saddle-point theorem, we show the existence of critical points, leading to the existence conclusions of travelling wave solutions.