Traveling Wave Solutions for Systems of ODEs on a Two-Dimensional Spatial Lattice

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 $e^{i\theta}$, 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

Travelling wavefronts in nonlocal diffusion equations with nonlocal delay effects

This paper deals with the existence, monotonicity, uniqueness and asymptotic behaviour of travelling wavefronts for a class of temporally delayed, spatially nonlocal diffusion equations

Heteroclinic orbits in plane dynamical systems

summary:We consider general second order boundary value problems on the whole line of the type $u^{\prime \prime }=h(t,u,u^{\prime })$, $u(-\infty )=0, u(+\infty )=1$, for which we provide existence, non-existence, multiplicity results. The solutions we find can be reviewed as heteroclinic orbits in the $(u,u^{\prime })$ plane dynamical system