'Society for Industrial & Applied Mathematics (SIAM)'
Doi
Abstract
This is the published version, also available here: http://dx.doi.org/10.1137/S0036139996312703.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 eiθ, and we 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 we 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. We show that a∗:R→Riscontinuousateachpoint\thetawhere\tan\thetaisirrational,andisdiscontinuouswhere\tan\theta$ is rational or infinite