Wave-like Solutions for Nonlocal Reaction-diffusion Equations : a Toy Model

Content Provider	Semantic Scholar
Author	Nadina, Grégoire Rossib, Luca Ryzhikc, Lenya Perthamea, Benoit March
Copyright Year	2012
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 [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.
File Format	PDF HTM / HTML
Alternate Webpage(s)	http://math.stanford.edu/~ryzhik/nprr-toy.pdf
Alternate Webpage(s)	http://hal.upmc.fr/hal-00923692/document
Alternate Webpage(s)	http://hal.upmc.fr/docs/00/92/36/92/PDF/wavetrainFinal.pdf
Alternate Webpage(s)	https://www.mmnp-journal.org/articles/mmnp/pdf/2013/03/mmnp201383p33.pdf
Alternate Webpage(s)	https://hal.sorbonne-universite.fr/hal-00923692/document
Language	English
Access Restriction	Open
Subject Keyword	Aharonov–Bohm effect Control theory Nonlocal Lagrangian Numerical analysis Pulse (signal processing) Solutions Steady state Turing Unstable Medical Device Problem Wave packet monotone travel
Content Type	Text
Resource Type	Article