Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation

Grégoire Nadin (1) , Benoît Perthame (1, 2) , Min Tang (2)

1 LJLL - Laboratoire Jacques-Louis Lions
2 BANG - Nonlinear Analysis for Biology and Geophysical flows

This note investigates the properties of the traveling waves solutions of the nonlocal Fisher equation. The existence of such solutions has been proved recently in \cite{BNPR} but their asymptotic behavior was still unclear. We use here a new numerical approximation of these traveling waves which shows that some traveling waves connect the two homogeneous steady states $0$ and $1$, which is a striking fact since $0$ is dynamically unstable and $1$ is unstable in the sense of Turing.

Domaines: Formation de Structures et Solitons [nlin.PS]; Equations aux dérivées partielles [math.AP]

Grégoire Nadin, Benoît Perthame, Min Tang. Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation. 2010. ⟨hal-00537986⟩