Loading...
Traveling Wave Solutions for Systems of ODEs on a Two-Dimensional Spatial Lattice
Van Vleck, Erik S. Mallet-Paret, John Cahn, John W.
Van Vleck, Erik S.
Mallet-Paret, John
Cahn, John W.
Citations
Altmetric:
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.
Date
1999-03-05
Journal Title
Journal ISSN
Volume Title
Publisher
Society for Industrial and Applied Mathematics
Collections
Archive Status
This item contains archived web content.
Files
Research Projects
Organizational Units
Journal Issue
Keywords
traveling waves, propagation failure, anisotropy, Allen--Cahn equation
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.