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PAC

Information Center for Mathematical Science

PAC

Wave-like solutions for nonlocal reaction-diffusion equations: a toy model
Author Lenya Ryzhik (Stanford University)
Homepage Url http://math.stanford.edu/~ryzhik/
Coauthors Gregoire Nadin, Luca Rossi, Benoit Perthame
Abstract Traveling waves for the nonlocal Fisher Equation can exhibit much more complex behaviours 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 [11]. This is proved for monotonic waves [1, 5] in the case where the speed is far from minimal. 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 connect monotonically 0 and 1, can connect these two states but being non monotonic, and also they can connect 0 to a wavetrain around the Turing unstable state 1. These exist in a regime where time dynamics converge to another object observed in [2, 7]: a wave that connects 0 to a pulsating wave around 1.
Abstract Url http://math.stanford.edu/~ryzhik/nprr-toy.pdf