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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 $e^{i\theta}$, and explore the relation between the wave speed $c$, the angle $\theta$, 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 $a=a^*(\theta)$ depends on $\theta$, where $a^*(\theta)$ is defined as the value of the parameter $a$ at which propagation failure (that is, wave speed c=0) occurs. The authors show that $a^*:\bbfR\to\bbfR$ is continuous at each point $\theta$ where $\tan\theta$ is irrational, and is discontinuous where $\tan\theta$ is rational or infinite.

34G20Nonlinear ODE in abstract spaces
34A35ODE of infinite order
35K57Reaction-diffusion equations
74J99Waves (solid mechanics)
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