Stationary bistable pulses in discrete inhomogeneous media
Abbreviated Journal Title
J. Differ. Equ. Appl.
spatially discrete; FitzHugh-Nagumo equation; propagation failure; McKean's caricature; inhomogeneous diffusion; MULTIPLE IMPULSE SOLUTIONS; DIFFERENTIAL-DIFFERENCE EQUATIONS; TRAVELING-WAVE SOLUTIONS; FITZHUGH-NAGUMO SYSTEM; NERVE EQUATION; MCKEAN; CARICATURE; COUPLED SYSTEMS; PROPAGATION; STABILITY; DIFFUSION; Mathematics, Applied
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.
Journal of Difference Equations and Applications
"Stationary bistable pulses in discrete inhomogeneous media" (2014). Faculty Bibliography 2010s. 5854.