Dissipative Solitons in Reaction-Diffusion Systems

Purwins, H.‐G.; Bödeker, H. U.; Liehr, Andreas W.

doi:10.1007/10928028_11

Cited by 33 publications

(61 citation statements)

References 57 publications

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“…[40]. By multiple scale theory for small perturbations of the order ε [38,40,45], the following equation of motion (EOM) for the position φ(t) of the TW in the presence of a boundary-induced advection term is obtaineḋ (10) with constant…”

Section: B Projection Methods -Multiple Scale Analysismentioning

confidence: 99%

Front propagation in channels with spatially modulated cross section

2015

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The problem of front propagation in a three-dimensional channel with spatially varying crosssection is reduced to an equivalent reaction-diffusion-advection equation with boundary-induced advection term. Treating the advection term as a weak perturbation, an equation of motion for the front position is derived. We analyze channels whose cross-sections vary periodically with L along the propagation direction of the front. Taking the Schlögl model as representative example, we calculate analytically the nonlinear dependence of the front velocity on the ratio L/l where l denotes the intrinsic front width. Our analytical results agree well with the results obtained by numerical simulations. In particular, the peculiarity of boundary-induced propagation failure for a finite range of L/l values is predicted by analytical calculations. Lastly, we demonstrate that the front velocity is determined by the suppressed diffusivity of the reactants for L l.

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Section: B Projection Methods -Multiple Scale Analysismentioning

confidence: 99%

Front propagation in channels with spatially modulated cross section

2015

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“…1, this is one of the foundation stones of the new concept. More specifically, the name "autosoliton" came naturally from the theory of reaction-diffusion systems [14]. Usually, it had to be supplemented with the notions of "activator" and "inhibitor" to be understood in its full complexity.…”

Section: Introductionmentioning

confidence: 99%

Three Sources and Three Component Parts of the Concept of Dissipative Solitons

Akhmediev

Ankiewicz

2008

Lecture Notes in Physics

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We explain the notion of dissipative solitons within a historical perspective. We show that the ideas of the theory of dissipative solitons emerge from several fields, including classical soliton theory, nonlinear dynamics, with its theory of bifurcations, and Prigogine's concept of self-organization. A new notion, emerging from this three-part foundation, allows us to build the novel concept of the dissipative soliton. We also show that reductions to lower dimensional systems have to be done carefully and should always include a comparison of the results with numerical simulations of the original equations.

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“…More formally, they are expressed as semilinear parabolic partial differential equations (PDEs; see, e.g., [55]). The evolution of the state vector u(x, t) describing the concentration of the different reagents is determined by anisotropic diffusion as well as local reactions:…”

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confidence: 99%

Ricci flow and nonlinear reaction–diffusion systems in biology, chemistry, and physics

Ivancevic

Ivancevic²

2010

Nonlinear Dyn

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This paper proposes the Ricci-flow equation from Riemannian geometry as a general geometric framework for various nonlinear reaction-diffusion systems (and related dissipative solitons) in mathematical biology. More precisely, we propose a conjecture that any kind of reaction-diffusion processes in biology, chemistry, and physics can be modeled by the combined geometric-diffusion system. In order to demonstrate the validity of this hypothesis, we review a number of popular nonlinear reaction-diffusion systems and try to show that they can all be subsumed by the presented geometric framework of the Ricci flow.

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Dissipative Solitons in Reaction-Diffusion Systems

Cited by 33 publications

References 57 publications

Front propagation in channels with spatially modulated cross section

Front propagation in channels with spatially modulated cross section

Three Sources and Three Component Parts of the Concept of Dissipative Solitons

Ricci flow and nonlinear reaction–diffusion systems in biology, chemistry, and physics

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