Math. Model. Nat. Phenom.
Volume 8, Number 3, 2013Front Propagation
|Page(s)||33 - 41|
|Published online||12 June 2013|
Wave-like Solutions for Nonlocal Reaction-diffusion Equations: a Toy Model
1 Laboratoire Jacques-Louis Lions, UPMC
Univ. Paris 6 and CNRS UMR 7598, F-75005, Paris
2 Dipartimento di Matematica, Università degli Studi di Padova
3 Department of Mathematics, Stanford University, Stanford CA 94305
⋆ Corresponding author. E-mail: email@example.com
Traveling waves for the nonlocal Fisher Equation can exhibit much more complex behaviour than for the usual Fisher equation. A striking numerical observation is that a traveling wave with minimal speed can connect a dynamically unstable steady state 0 to a Turing unstable steady state 1, see . This is proved in [1, 6] in the 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 simple analytical traveling wave solutions that exhibit various behaviours. These traveling waves, with minimal speed or not, can (i) connect monotonically 0 and 1, (ii) connect these two states non-monotonically, and (iii) connect 0 to a wavetrain around 1. The latter exist in a regime where time dynamics converges to another object observed in [3, 8]: a wave that connects 0 to a pulsating wave around 1.
Mathematics Subject Classification: 34K13 / 35B36 / 35C07 / 35K57
Key words: traveling wave / wavetrains / nonlocal elliptic equations
© EDP Sciences, 2013
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