Authors

A. R. Humphries; B. E. Moore;E. S. Van Vleck

Comments

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Abbreviated Journal Title

SIAM J. Appl. Math.

Keywords

traveling fronts; propagation failure; inhomogeneities; bistable; equation; MULTIPLE IMPULSE SOLUTIONS; TRAVELING-WAVE SOLUTIONS; DISCRETE NAGUMO; EQUATION; NERVE EQUATION; STATIONARY FRONTS; MCKEAN CARICATURE; EXCITABLE MEDIA; PROPAGATION; STABILITY; LATTICE; Mathematics, Applied

Abstract

We consider a bistable differential-difference equation with inhomogeneous diffusion. Employing a piecewise linear nonlinearity, often referred to as McKean's caricature of the cubic, we construct front solutions which correspond, in the case of homogeneous diffusion, to monotone traveling front solutions or, in the case of propagation failure, to stationary front solutions. A general form for these fronts is given for essentially arbitrary inhomogeneous discrete diffusion, and conditions are given for the existence of solutions to the original discrete Nagumo equation. The specific case of one defect is considered in depth, giving a complete understanding of propagation failure and a grasp on changes in wave speed. Insight into the dynamic behavior of these front solutions as a function of the magnitude and relative position of the defects is obtained with the assistance of numerical results.

Journal Title

Siam Journal on Applied Mathematics

Volume

71

Issue/Number

4

Publication Date

1-1-2011

Document Type

Article

Language

English

First Page

1374

Last Page

1400

WOS Identifier

WOS:000294288400022

ISSN

0036-1399

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