Traveling wave front and stability as planar wave of reaction diffusion equations with nonlocal delays

2012 ◽  
Vol 64 (4) ◽  
pp. 1005-1023
Author(s):  
Guangying Lv ◽  
Mingxin Wang
Keyword(s):  
Wave Front ◽  
Traveling Wave ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Traveling Wave Front ◽  
Planar Wave

scholarly journals Speed of wave-front solutions to hyperbolic reaction-diffusion equations

1999 ◽  
Vol 60 (5) ◽  
pp. 5231-5243 ◽  
Author(s):  
Vicenç Méndez ◽  
Joaquim Fort ◽  
Jordi Farjas
Keyword(s):  
Wave Front ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Front Solutions

TRANSITION WAVES THAT LEAVE BEHIND REGULAR OR IRREGULAR SPATIOTEMPORAL OSCILLATIONS IN A SYSTEM OF THREE REACTION–DIFFUSION EQUATIONS

1993 ◽  
Vol 03 (05) ◽  
pp. 1269-1279 ◽  
Author(s):  
JONATHAN A. SHERRATT
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Wave Front ◽  
Plane Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Spatiotemporal Oscillations ◽  
Transition Wave ◽  
Periodic Plane

Transition waves are widespread in the biological and chemical sciences, and have often been successfully modelled using reaction–diffusion systems. I consider a particular system of three reaction–diffusion equations, and I show that transition waves can destabilise as the kinetic ordinary differential equations pass through a Hopf bifurcation, giving rise to either regular or irregular spatiotemporal oscillations behind the advancing transition wave front. In the case of regular oscillations, I show that these are periodic plane waves that are induced by the way in which the transition wave front approaches its terminal steady state. Further, I show that irregular oscillations arise when these periodic plane waves are unstable as reaction–diffusion solutions. The resulting behavior is not related to any chaos in the kinetic ordinary differential equations.

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