Traveling Wavefronts in an Antidiffusion Lattice Nagumo Model

This is the published version, also available here: http://dx.doi.org/10.1137/100819461.We consider a system of lattice Nagumo equations with cubic nonlinearity, but with a negative discrete diffusion coefficient. We are interested in the existence, uniqueness, stability, and nonexistence of the traveling wavefront solutions of this system, and we shall call this problem the antidiffusion lattice Nagumo problem. By rewriting this system as a spatially periodic system with inhomogeneous but positive periodic diffusion coefficients and periodic nonlinearities, we uncover a rich solution behavior that includes many possible connecting orbits in the antidiffusion case. Second, we observe the presence of bistable and monostable dynamics. In the bistable region, we study the phenomenon of propagation of failure while in the monostable region, we compute the minimum wave speed

Solutions of Lattice Differential Equations over Inhomogeneous Media

This thesis investigates one-dimensional spatially-discrete reaction-diffusion equations with a diffusion term that involves nearest-neighbor coupling and with a reaction-term that is a smooth-cubic nonlinearity. Specifically, we consider two nontrivial examples of lattice differential equations (LDEs) on Z that are related to the (homogeneous) lattice Nagumo equation. The LDEs that we consider are used to model natural phenomena defined over an inhomogeneous medium, namely: (1) a lattice Nagumo equation with a negative diffusion coefficient. Such is still a well-posed problem in the LDE setting and has been shown to arise from a discrete model of phase transition for shape memory alloys. This thesis shows that the anti-diffusion lattice Nagumo equation has a period-2 traveling wavefront solution that is stable and unique. Utilizing the concrete expressions for the nonlinearities, we obtain criteria on the (d, a)-parameter plane that guarantee a display of bistable and monostable dynamics. Where there's bistable dynamics, we study the propagation failure phenomenon; where there's monostable dynamics, we compute a minimum wave speed for the traveling waves. (2) a lattice Nagumo equation that has a single diffusion-defect in the middle of Z, which may occur due to deviations in the diffusive property of the medium. This thesis shows that such an equation has a time-global solution which behaves as two fronts coming from the both sides of Z. A key idea for the existence proof is a characterization of the asymptotic behavior of the solutions for negative time in terms of an appropriate super-solution, sub-solution pair

Traveling waves and pattern formation for spatially discrete bistable reaction-diffusion equations (survey)

Analysis and Stochastic

Negative Diffusion and Traveling Waves in High Dimensional Lattice Systems

This is the publisher's version, also available electronically from http://epubs.siam.org/doi/abs/10.1137/120880628We consider bistable reaction diffusion systems posed on rectangular lattices in two or more spatial dimensions. The discrete diffusion term is allowed to have positive spatially periodic coefficients, and the two spatially periodic equilibria are required to be well ordered. We establish the existence of traveling wave solutions to such pure lattice systems that connect the two stable equilibria. In addition, we show that these waves can be approximated by traveling wave solutions to systems that incorporate both local and nonlocal diffusion. In certain special situations our results can also be applied to reaction diffusion systems that include (potentially large) negative coefficients. Indeed, upon splitting the lattice suitably and applying separate coordinate transformations to each sublattice, such systems can sometimes be transformed into a periodic diffusion problem that fits within our framework. In such cases, the resulting traveling structure for the original system has a separate wave profile for each sublattice and connects spatially periodic patterns that need not be well ordered. There is no direct analogue of this procedure that can be applied to reaction diffusion systems with continuous spatial variables