Stationary bistable pulses in discrete inhomogeneous media

Abstract

A second-order difference equation with boundary conditions at infinity is solved, and solutions are analysed in terms of problem parameters. The equation describes stationary pulse solutions of differential-difference equations with a nonlinearity known as McKean's caricature of the cubic. The method of solution reduces the nonlinear problem to a linear inhomogeneous problem under certain conditions. The most important feature of the problem is that coefficients of the difference terms are allowed to vary on a finite interval, leading to changes in solution shapes, as well as changes in parameter values that are acceptable for generating solutions to the problem. Formulas for multiple-pulse solutions are derived, while 1-pulse solutions are considered in detail, and the range of parameter values that allow for the existence of stationary pulses is determined. Numerical methods applied to a spatially discrete FitzHugh–Nagumo equation demonstrate the solution stability and the relationship between the existence of stationary pulses and propagation failure of travelling waves.

Keywords::

• spatially discrete
• FitzHugh–Nagumo equation
• propagation failure
• McKean's caricature
• inhomogeneous diffusion

Keywords::

• 39A70
• 34K21