Abstract
We 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, we obtain traveling wave solutions in each direction $e^{i\theta}$, and we 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 we 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. We show that $a^*:\Bbb{R}\to\Bbb{R} is continuous at each point $\theta$ where $\tan\theta$ is irrational, and is discontinuous where $\tan\theta$ is rational or infinite.
Description
This is the published version, also available here: http://dx.doi.org/10.1137/S0036139996312703.
Citation
Van Vleck, Erik., Mallet-Paret, John., Cahn, John W. "Traveling Wave Solutions for Systems of ODEs on a Two-Dimensional Spatial Lattice." (1999) SIAM J. Appl. Math., 59(2), 455–493. (39 pages). http://dx.doi.org/10.1137/S0036139996312703.