Propagation Failure Of Fronts In Discrete Inhomogeneous Media With A Sawtooth Nonlinearity

Keywords

inhomogeneous diffusion; interval of propagation failure; sawtooth nonlinearity; Spatially discrete bistable equation; stationary fronts

Abstract

Exact, steady-state, single-front solutions are constructed for a spatially discrete bistable equation with a piecewise linear reaction term, known as a sawtooth nonlinearity. These solutions are obtained by solving second-order difference equations with variable coefficients, which are linear under certain assumptions on the expected solutions. An algorithmic procedure for constructing solutions in general, for both homogeneous and inhomogeneous diffusion, is obtained using a combination of Jacobi-Operator theory and the Sherman–Morrison formula. The existence of solutions for the difference equation, implies propagation failure of fronts for the corresponding differential-difference equation. The interval of propagation failure, which is the range of values of the detuning parameter that render stationary fronts, is studied in detail for the case of a single defect in the medium of propagation. Explicit formulae reveal precise relationships between parameter values that cause traveling fronts to fail to propagate when the interface reaches the inhomogeneities in the medium. These explicit formulae are also compared to numerical computations using the proposed algorithmic approach, which provides a check of its computational usefulness and illustrates its capabilities for problems with more complicated choices of parameter values.

Publication Date

12-1-2016

Publication Title

Journal of Difference Equations and Applications

Volume

22

Issue

12

Number of Pages

1930-1947

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1080/10236198.2016.1255209

Socpus ID

84994844638 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84994844638

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