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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 e iθ , and explore the relation between the wave speed c, the angle θ, and the detuning parameter a of the nonlinearity. Of particular interest is the phenomenon of “propagation failure”, and the authors study how the critical value a=a * (θ) depends on θ, where a * (θ) is defined as the value of the parameter a at which propagation failure (that is, wave speed c=0) occurs. The authors show that a * : is continuous at each point θ where tanθ is irrational, and is discontinuous where tanθ is rational or infinite.
MSC:
34G20Nonlinear ODE in abstract spaces
34A35ODE of infinite order
35K57Reaction-diffusion equations
74J99Waves (solid mechanics)