Propagation failure of fronts in discrete inhomogeneous media with a sawtooth nonlinearity. Issue 12 (1st December 2016)

• Record Type: Journal Article
• Authors: Lydon, Elizabeth; Moore, Brian E.
• Is Part Of: Journal of difference equations and applications. Volume 22:Issue 12(2016)
• Page Start: 1930
• Page End: 1947
• Publication Date: 2016-12-01
• Subjects: Spatially discrete bistable equation -- sawtooth nonlinearity -- inhomogeneous diffusion -- stationary fronts -- interval of propagation failure; 34K21 -- 39A60 -- 92C20; Difference equations -- Periodicals; 515.625
• DOI: 10.1080/10236198.2016.1255209
• Languages: English
• ISSNs: 1023-6198

Abstract

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.