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Linearized Oscillation Theory for a Nonlinear Nonautonomous Difference Equation Read chapter Read first chapter

2020 | OriginalPaper | Chapter

Traveling Waves and Pattern Formation for Spatially Discrete Bistable Reaction-Diffusion Equations

Authors : Hermen Jan Hupkes, Leonardo Morelli, Willem M. Schouten-Straatman, Erik S. Van Vleck

Published in: Difference Equations and Discrete Dynamical Systems with Applications

Publisher: Springer International Publishing

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Abstract

We survey some recent results on traveling waves and pattern formation in spatially discrete bistable reaction-diffusion equations. We start by recalling several classic results concerning the existence, uniqueness and stability of travelling wave solutions to the discrete Nagumo equation with nearest-neighbour interactions, together with the Fredholm theory behind some of the proofs. We subsequently discuss extensions involving wave connections between periodic equilibria, long-range interactions and planar lattices. We show how some of the results can be extended to the two-component discrete FitzHugh–Nagumo equation, which can be analyzed using singular perturbation theory. We conclude by studying the behaviour of the Nagumo equation when discretization schemes are used that involve both space and time, or that are non-uniform but adaptive in space.

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previous chapter Generalized Mandelbrot and Julia Sets in a Family of Planar Angle-Doubling Maps
next chapter A Hilbert Space Approach to Fractional Difference Equations
Footnotes
1
Actually, in order to ensure that the boundary conditions (23) are satisfied one needs to consider perturbations \(v \in W^{1,p}\) for \(1 \le p < \infty \) while taking \(\varPhi \in W^{1, \infty }\).
 
2
Here we use modulo arithmetic on i.
 
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Metadata
Title
Traveling Waves and Pattern Formation for Spatially Discrete Bistable Reaction-Diffusion Equations
Authors
Hermen Jan Hupkes
Leonardo Morelli
Willem M. Schouten-Straatman
Erik S. Van Vleck
Copyright Year
2020
Book
Difference Equations and Discrete Dynamical Systems with Applications

Print ISBN: 978-3-030-35501-2

Electronic ISBN: 978-3-030-35502-9

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-35502-9_3

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