TY - JOUR

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

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

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 -