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Zbl 0917.34052
Cahn, John W.; Mallet-Paret, John; van Vleck, Erik S.
Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice.
(English)
[J] SIAM J. Appl. Math. 59, No.2, 455-493 (1999). ISSN 0036-1399; ISSN 1095-712X/e

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.
MSC 2000:
*34G20 Nonlinear ODE in abstract spaces
34A35 ODE of infinite order
35K57 Reaction-diffusion equations
74J99 Waves

Keywords: traveling waves; propagation failure; anisotropy; Allen-Cahn equation

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