Abstract
We consider infiinite systems of ODEs on the two-dimensional integer lattice, given by a bistable scalar ODE at each point, with a nearest neighbor coupling between lattice points. For a class of ideal nonlinearities, we obtain travelling wave solutions in each direction, and we explore the relation between the wave speed c, the angle, and the detuning parameter of the nonlinearity. Of particular interest is the phenomenon of propagation failure, and we study how the critical value depends on, where is defined as the value of the parameter at which propagation failure occurs. In the study of crystal growth in material science, there is often an underlying spatial lattice which plays an important rule in the evolution and dynamics of the system. For example, wave motion in a lattice may exhibit a velocity which is direction dependent; in addition, the lattice may cause a freezing or pinning of the wave motion, which does not occur for such motion in a continum. Systems of differential equations with an inderlying lattice structure occur in mathematical models in many different scientific disciplines. Besides material science, we mention biology, and pattern recognition.