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Front Solutions for Bistable Differential-Difference Equations with Inhomogeneous Diffusion
Humphries, A. R. Moore, Brian E. Van Vleck, Erik S.
Humphries, A. R.
Moore, Brian E.
Van Vleck, Erik S.
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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.
Description
This is the published version, also available here: http://dx.doi.org/10.1137/100807156.
Date
2011-08-09
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Society for Industrial and Applied Mathematics
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Keywords
traveling fronts, propagation failure, inhomogeneities, bistable equation
Citation
Humphries, A. R., Moore, Brian E., Van Vleck, Erik. "Front Solutions for Bistable Differential-Difference Equations with Inhomogeneous Diffusion." (2011) SIAM J. Appl. Math., 71(4), 1374–1400. (27 pages). http://dx.doi.org/10.1137/100807156.