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Wave-like Solutions for Nonlocal Reaction-diffusion Equations:a Toy Model

Published online by Cambridge University Press:  12 June 2013

G. Nadin ,
L. Rossi ,
L. Ryzhik  and
B. Perthame
G. Nadin*
Affiliation:
Laboratoire Jacques-Louis Lions, UPMC Univ. Paris 6 and CNRS UMR 7598, F-75005, Paris
L. Rossi
Affiliation:
Dipartimento di Matematica, Università degli Studi di Padova
L. Ryzhik
Affiliation:
Department of Mathematics, Stanford University, Stanford CA 94305
B. Perthame
Affiliation:
Laboratoire Jacques-Louis Lions, UPMC Univ. Paris 6 and CNRS UMR 7598, F-75005, Paris
*
⋆ Corresponding author. E-mail: [email protected]
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Abstract

Traveling waves for the nonlocal Fisher Equation can exhibit much more complex behaviourthan for the usual Fisher equation. A striking numerical observation is that a travelingwave with minimal speed can connect a dynamically unstable steady state 0 to a Turingunstable steady state 1, see [12]. This is provedin [1, 6] inthe case where the speed is far from minimal, where we expect the wave to be monotone.

Here we introduce a simplified nonlocal Fisher equation for which we can build simpleanalytical traveling wave solutions that exhibit various behaviours. These travelingwaves, with minimal speed or not, can (i) connect monotonically 0 and 1, (ii) connectthese two states non-monotonically, and (iii) connect 0 to a wavetrain around 1. Thelatter exist in a regime where time dynamics converges to another object observed in[3, 8]: awave that connects 0 to a pulsating wave around 1.

Keywords

traveling wavewavetrainsnonlocal elliptic equations
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 3: Front Propagation , 2013 , pp. 33 - 41
Copyright
© EDP Sciences, 2013

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