Boundary value conditions for wave fronts in reaction-diffusion systems

1984 ◽  
Vol 99 (1-2) ◽  
pp. 127-136 ◽  
Author(s):  
J. M. Fraile ◽  
J. Sabina
Keyword(s):  
Wave Front ◽  
Reaction Diffusion ◽  
Propagation Direction ◽  
Dirichlet Problems ◽  
Reaction Diffusion Systems ◽  
New Class ◽  
Diffusion Systems ◽  
Front Solutions ◽  
Distinguished Boundary ◽  
Directional Wave

SynopsisIn this paper, we introduce a new class of solutions of reaction-diffusion systems, termed directional wave front solutions. They have a propagating character and the propagation direction selects some distinguished boundary points on which we can impose boundary conditions. The Neumann and Dirichlet problems on these points are treated here in order to prove some theorems on the existence of directional wave front solutions of small amplitude, and to partially establish their asymptotic behaviour.

scholarly journals Directional wave fronts of reaction-diffusion systems

1986 ◽  
Vol 33 (1) ◽  
pp. 1-20 ◽  
Author(s):  
J. Sabina
Keyword(s):  
Plane Wave ◽  
Natural Extension ◽  
Reaction Diffusion ◽  
Scalar Case ◽  
Wave Fronts ◽  
Reaction Diffusion Systems ◽  
Reaction Term ◽  
Diffusion Systems ◽  
Typical Behaviour ◽  
Directional Wave

In this work, we study types of undulatory solutions, that we term Directional Wave Fronts (DWF), of non scalar reaction diffusion systems. The DWFs are a natural extension of the well known Plane Wave Fronts (PWFs) solutions. However, the DWFs admit a certain type of boundary conditions. In the present work we show, under suitable conditions on the reaction term, that DWFs also exhibit typical behaviour of PWFs: we just prove the existence of heteroclinic, homoclinic and periodic families of DWFs. Essentially, we require the reaction term to be linearly uncoupled. These results are the generalization of a previous work, concerning the scalar case.

A geometrical approach to wave-type solutions of excitable reaction-diffusion systems

1991 ◽  
Vol 433 (1887) ◽  
pp. 151-164 ◽  
Keyword(s):  
Numerical Solution ◽  
Wave Front ◽  
Eikonal Equation ◽  
Reaction Diffusion ◽  
Stationary Wave ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Wave Boundary

We formulate the eikonal equation approximation for travelling waves in excitable reaction-diffusion systems, which have been proposed as models for a large number of biomedical situations. This formulation is particularly suited, in a natural way, to numerical solution by finite difference methods. We show how this solution is independent of the parametric variable used for expressing the eikonal equation, and how a reduction of dimensionality implies a major saving over the time taken to solve the original reaction-diffusion system. Neumann boundary conditions on reactants in the original system translate into a geometric constraint on the wave boundary itself. We show how this leads to geometrically stable stationary wave boundaries in appropriately shaped non-convex domains. This analytical prediction is verified by numerical solution of the eikonal equation on a domain which supports geometrically stable stationary wave boundary configurations. We show how the concepts of geometrical stability and wave-front stability relate to a problem where a bi-stable reaction-diffusion system has a stable stationary wave-front configuration.

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