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Dynamics of dissipative structures in reaction-diffusion equations

Author
Organization
Abstract
The authors investigate the dynamics of dissipative structures in a reaction-diffusion system. They propose an analytical theory for the behavior of dissipative structures and show that a dissipative structure (DS) that is stable in a one-dimensional homogeneous medium can be induced to drift by slow variation of the diffusion coefficients, or by curvature of the DS in higher dimensions. In one spatial dimension, this motion can be in the direction of increasing or decreasing diffusion coefficient, depending on properties of the DS which can be determined analytically. In two and three dimensions this drift is proportional to the sum of the curvature of the DS and the gradient of the diffusion coefficient of the medium. The analysis of this motion uses standard ideas from perturbation theory to find an equation of motion for the location of the DS. Numerical simulations in one and two dimensions show good quantitative agreement with the theoretical results.
Keywords
CURVATURE, PATTERN, EXCITABLE MEDIA, DISSIPATIVE STRUCTURES, NONLINEAR WAVES, REACTION-DIFFUSION

Citation

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Chicago
Panfilov, Alexander, and JP Keener. 1995. “Dynamics of Dissipative Structures in Reaction-diffusion Equations.” Siam Journal on Applied Mathematics 55 (1): 205–219.
APA
Panfilov, A., & Keener, J. (1995). Dynamics of dissipative structures in reaction-diffusion equations. SIAM JOURNAL ON APPLIED MATHEMATICS, 55(1), 205–219.
Vancouver
1.
Panfilov A, Keener J. Dynamics of dissipative structures in reaction-diffusion equations. SIAM JOURNAL ON APPLIED MATHEMATICS. 1995;55(1):205–19.
MLA
Panfilov, Alexander, and JP Keener. “Dynamics of Dissipative Structures in Reaction-diffusion Equations.” SIAM JOURNAL ON APPLIED MATHEMATICS 55.1 (1995): 205–219. Print.
@article{8608638,
  abstract     = {The authors investigate the dynamics of dissipative structures in a reaction-diffusion system. They propose an analytical theory for the behavior of dissipative structures and show that a dissipative structure (DS) that is stable in a one-dimensional homogeneous medium can be induced to drift by slow variation of the diffusion coefficients, or by curvature of the DS in higher dimensions. In one spatial dimension, this motion can be in the direction of increasing or decreasing diffusion coefficient, depending on properties of the DS which can be determined analytically. In two and three dimensions this drift is proportional to the sum of the curvature of the DS and the gradient of the diffusion coefficient of the medium. 
The analysis of this motion uses standard ideas from perturbation theory to find an equation of motion for the location of the DS. Numerical simulations in one and two dimensions show good quantitative agreement with the theoretical results.},
  author       = {Panfilov, Alexander and Keener, JP},
  issn         = {0036-1399},
  journal      = {SIAM JOURNAL ON APPLIED MATHEMATICS},
  keywords     = {CURVATURE,PATTERN,EXCITABLE MEDIA,DISSIPATIVE STRUCTURES,NONLINEAR WAVES,REACTION-DIFFUSION},
  language     = {eng},
  number       = {1},
  pages        = {205--219},
  title        = {Dynamics of dissipative structures in reaction-diffusion equations},
  url          = {http://dx.doi.org/10.1137/S0036139992229101},
  volume       = {55},
  year         = {1995},
}

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