Stability of Transition Front Solutions in Multidimensional Cahn–Hilliard Systems

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.

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Travelling fronts in a food-limited population model with time delay

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500001530 ◽

2002 ◽

Vol 132 (1) ◽

pp. 75-89 ◽

Cited By ~ 33

Author(s):

S. A. Gourley ◽

M. A. J. Chaplain

Keyword(s):

Population Model ◽

Time Delays ◽

Front Solutions ◽

Heteroclinic Connection ◽

Non Local ◽

Spatial Terms ◽

Travelling Fronts ◽

Manifold Theory ◽

And Diffusion ◽

Existence Question

In this paper we study travelling front solutions of a certain food-limited population model incorporating time-delays and diffusion. Special attention is paid to the modelling of the time delays to incorporate associated non-local spatial terms which account for the drift of individuals to their present position from their possible positions at previous times. For a particular class of delay kernels, existence of travelling front solutions connecting the two spatially uniform steady states is established for sufficiently small delays. The approach is to reformulate the problem as an existence question for a heteroclinic connection in R4. The problem is then tackled using dynamical systems techniques, in particular, Fenichel's invariant manifold theory. For larger delays, numerical simulations reveal changes in the front's profile which develops a prominent hump.

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

Physical Review E ◽

10.1103/physreve.60.5231 ◽

1999 ◽

Vol 60 (5) ◽

pp. 5231-5243 ◽

Cited By ~ 46

Author(s):

Vicenç Méndez ◽

Joaquim Fort ◽

Jordi Farjas

Keyword(s):

Wave Front ◽

Reaction Diffusion ◽

Reaction Diffusion Equations ◽

Diffusion Equations ◽

Front Solutions

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Propagation of transition front in bi-stable nondegenerate chains: Model dependence and universality

Journal of the Mechanics and Physics of Solids ◽

10.1016/j.jmps.2017.04.011 ◽

2017 ◽

Vol 104 ◽

pp. 144-156 ◽

Cited By ~ 7

Author(s):

I.B. Shiroky ◽

O.V. Gendelman

Keyword(s):

Transition Front ◽

Model Dependence

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FRONT STRUCTURES IN A REAL GINZBURG-LANDAU EQUATION COUPLED TO A MEAN FIELD

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127494001040 ◽

1994 ◽

Vol 04 (05) ◽

pp. 1343-1346 ◽

Cited By ~ 6

Author(s):

HENAR HERRERO ◽

HERMANN RIECKE

Keyword(s):

Analytical Study ◽

Landau Equation ◽

Mean Field ◽

Travelling Wave ◽

Opposite Orientation ◽

Ginzburg Landau ◽

Separation Ratio ◽

Ginzburg Landau Equation ◽

Front Solutions ◽

Wave Trains

Localized travelling wave trains or pulses have been observed in various experiments in binary mixture convection. For strongly negative separation ratio, these pulse structures can be described as two interacting fronts of opposite orientation. An analytical study of the front solutions in a real Ginzburg-Landau equation coupled to a mean field is presented here as a first approach to the pulse solution. The additional mean field becomes important when the mass diffusion in the mixture is small as is the case in liquids.

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