Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion

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Abstract

In this work we investigate the phenomena of pattern formation and wave propagation for a reaction–diffusion system with nonlinear diffusion. We show how cross-diffusion destabilizes uniform equilibrium and is responsible for the initiation of spatial patterns. Near marginal stability, through a weakly nonlinear analysis, we are able to predict the shape and the amplitude of the pattern. For the amplitude, in the supercritical and in the subcritical case, we derive the cubic and the quintic Stuart–Landau equation respectively.When the size of the spatial domain is large, and the initial perturbation is localized, the pattern is formed sequentially and invades the whole domain as a traveling wavefront. In this case the amplitude of the pattern is modulated in space and the corresponding evolution is governed by the Ginzburg–Landau equation.
Lingua originaleEnglish
pagine (da-a)1112-1132
Numero di pagine21
RivistaMathematics and Computers in Simulation
Volume82
Stato di pubblicazionePublished - 2012

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Turing Instability
Cross-diffusion
Travelling Fronts
Reaction-diffusion System
Nonlinear systems
Nonlinear Systems
Nonlinear analysis
Wavefronts
Wave propagation
Traveling Wavefronts
Ginzburg-Landau Equation
Quintic
Nonlinear Diffusion
Spatial Pattern
Pattern Formation
Nonlinear Analysis
Wave Propagation
Perturbation
Predict

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Computer Science(all)
  • Numerical Analysis
  • Modelling and Simulation
  • Applied Mathematics

Cita questo

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abstract = "In this work we investigate the phenomena of pattern formation and wave propagation for a reaction–diffusion system with nonlinear diffusion. We show how cross-diffusion destabilizes uniform equilibrium and is responsible for the initiation of spatial patterns. Near marginal stability, through a weakly nonlinear analysis, we are able to predict the shape and the amplitude of the pattern. For the amplitude, in the supercritical and in the subcritical case, we derive the cubic and the quintic Stuart–Landau equation respectively.When the size of the spatial domain is large, and the initial perturbation is localized, the pattern is formed sequentially and invades the whole domain as a traveling wavefront. In this case the amplitude of the pattern is modulated in space and the corresponding evolution is governed by the Ginzburg–Landau equation.",
keywords = "Amplitude equation, Ginzburg–Landau equation, Nonlinear diffusion, Pattern formation, Quintic Stuart–Landau equation",
author = "Gaetana Gambino and Lombardo, {Maria Carmela} and Sammartino, {Marco Maria Luigi}",
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journal = "Mathematics and Computers in Simulation",
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T1 - Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion

AU - Gambino, Gaetana

AU - Lombardo, Maria Carmela

AU - Sammartino, Marco Maria Luigi

PY - 2012

Y1 - 2012

N2 - In this work we investigate the phenomena of pattern formation and wave propagation for a reaction–diffusion system with nonlinear diffusion. We show how cross-diffusion destabilizes uniform equilibrium and is responsible for the initiation of spatial patterns. Near marginal stability, through a weakly nonlinear analysis, we are able to predict the shape and the amplitude of the pattern. For the amplitude, in the supercritical and in the subcritical case, we derive the cubic and the quintic Stuart–Landau equation respectively.When the size of the spatial domain is large, and the initial perturbation is localized, the pattern is formed sequentially and invades the whole domain as a traveling wavefront. In this case the amplitude of the pattern is modulated in space and the corresponding evolution is governed by the Ginzburg–Landau equation.

AB - In this work we investigate the phenomena of pattern formation and wave propagation for a reaction–diffusion system with nonlinear diffusion. We show how cross-diffusion destabilizes uniform equilibrium and is responsible for the initiation of spatial patterns. Near marginal stability, through a weakly nonlinear analysis, we are able to predict the shape and the amplitude of the pattern. For the amplitude, in the supercritical and in the subcritical case, we derive the cubic and the quintic Stuart–Landau equation respectively.When the size of the spatial domain is large, and the initial perturbation is localized, the pattern is formed sequentially and invades the whole domain as a traveling wavefront. In this case the amplitude of the pattern is modulated in space and the corresponding evolution is governed by the Ginzburg–Landau equation.

KW - Amplitude equation

KW - Ginzburg–Landau equation

KW - Nonlinear diffusion

KW - Pattern formation

KW - Quintic Stuart–Landau equation

UR - http://hdl.handle.net/10447/62397

UR - http://dx.doi.org/10.1016/j.matcom.2011.11.004

M3 - Article

VL - 82

SP - 1112

EP - 1132

JO - Mathematics and Computers in Simulation

JF - Mathematics and Computers in Simulation

SN - 0378-4754

ER -