Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice. (English) Zbl 0917.34052
Summary: The authors consider infinite 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, they obtain traveling wave solutions in each direction , and explore the relation between the wave speed , the angle , and the detuning parameter of the nonlinearity. Of particular interest is the phenomenon of “propagation failure”, and the authors study how the critical value depends on , where is defined as the value of the parameter at which propagation failure (that is, wave speed c=0) occurs. The authors show that is continuous at each point where is irrational, and is discontinuous where is rational or infinite.
|34G20||Nonlinear ODE in abstract spaces|
|34A35||ODE of infinite order|
|74J99||Waves (solid mechanics)|