Geodesic
Wave-like Solutions for Nonlocal Reaction-diffusion Equations: a Toy Model
G. Nadin1 ; L. Rossi2 ; L. Ryzhik3 ; B. Perthame1
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
Mathematical modelling of natural phenomena, Tome 8 (2013) no. 3, pp. 33-41

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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 [12]. 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.
DOI : 10.1051/mmnp/20138304

G. Nadin 1 ; L. Rossi 2 ; L. Ryzhik 3 ; B. Perthame 1

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 
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G. Nadin; L. Rossi; L. Ryzhik; B. Perthame. Wave-like Solutions for Nonlocal Reaction-diffusion Equations: a Toy Model. Mathematical modelling of natural phenomena, Tome 8 (2013) no. 3, pp. 33-41. doi: 10.1051/mmnp/20138304

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