Wave-like solutions for nonlocal reaction-diffusion equations: a toy model

Abstract : 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 \cite{NPT}. This is proved in \cite{AlfaroCoville, FangZhao} 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 \cite{BNPR, GVA}: a wave that connects $0$ to a pulsating wave around $1$.
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Grégoire Nadin, Luca Rossi, Leonid Ryzhik, Benoît Perthame. Wave-like solutions for nonlocal reaction-diffusion equations: a toy model. Mathematical Modelling of Natural Phenomena, EDP Sciences, 2013, 8 (3), pp.33-41. ⟨10.1051/mmnp/20127102⟩. ⟨hal-00923692⟩

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