Waves and patterns in reaction  diffusion systems. Belousov  Zhabotinsky reaction in water-in-oil microemulsions

Vanag, Vladimir K.

doi:10.3367/ufnr.0174.200409d.0991

Cited by 40 publications

(31 citation statements)

References 139 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In this work, we consider the slow evolution of pulses in the one-dimensional activator-inhibitor Brusselator model (see, e.g. [28] and references therein), long a paradigm of non-linear analysis.…”

Section: Introductionmentioning

confidence: 99%

Stationary and slowly moving localised pulses in a singularly perturbed Brusselator model

Tzou¹,

Bayliss

Matkowsky

et al. 2011

Eur. J. Appl. Math

View full text Add to dashboard Cite

Recent attention has focused on deriving localised pulse solutions to various systems of reaction-diffusion equations. In this paper, we consider the evolution of localised pulses in the Brusselator activator-inhibitor model, long considered a paradigm for the study of nonlinear equations, in a finite one-dimensional domain with feed of the inhibitor through the boundary and global feed of the activator. We employ the method of matched asymptotic expansions in the limit of small activator diffusivity and small activator and inhibitor feeds. The disparity of diffusion lengths between the activator and inhibitor leads to pulse-type solutions in which the activator is localised while the inhibitor varies on an O(1) length scale. In the asymptotic limit considered, the pulses become spikes described by Dirac delta functions and evolve slowly in time until equilibrium is reached. Such quasi-equilibrium solutions with N activator pulses are constructed and a differential-algebraic system of equations (DAE) is derived, characterising the slow evolution of the locations and the amplitudes of the pulses. We find excellent agreement for the pulse evolution between the asymptotic theory and the results of numerical computations. An algebraic system for the equilibrium pulse amplitudes and locations is derived from the equilibrium points of the DAE system. Both symmetric equilibria, corresponding to a common pulse amplitude, and asymmetric pulse equilibria, for which the pulse amplitudes are different, are constructed. We find that for a positive boundary feed rate, pulse spacing of symmetric equilibria is no longer uniform, and that for sufficiently large boundary flux, pulses at the edges of the pattern may collide with and remain fixed at the boundary. Lastly, stability of the equilibrium solutions is analysed through linearisation of the DAE, which, in contrast to previous approaches, provides a quick way to calculate the small eigenvalues governing weak translation-type instabilities of equilibrium pulse patterns.

show abstract

Section: Introductionmentioning

confidence: 99%

Stationary and slowly moving localised pulses in a singularly perturbed Brusselator model

Tzou¹,

Bayliss

Matkowsky

et al. 2011

Eur. J. Appl. Math

View full text Add to dashboard Cite

show abstract

“…For example, such structures are observed in planar gas-discharge systems with a high ohmic barrier (Fig. 1a) [1][2][3], granular materials [4] and optical systems [5,6], in modified Belousov-Zhabotinsky reactions [7][8][9][10][11], in catalytic reduction of NO with hydrogen on a Rh(110) surface [12] and oxidation of CO on Pt(110) (Fig. 1b) [13].…”

Section: Introductionmentioning

confidence: 95%

Polymorphic and regular localized activity structures in a two-dimensional two-component reaction–diffusion lattice with complex threshold excitation

Nekorkin

Dmitrichev

Bilbault

et al. 2010

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

“…Recently, several new types of autowaves have been discovered, which have significant application value. A whole bunch of new types of autowaves is described in the works of V. K. Vanag [14][15][16]. These are, for example, perforated spiral autowaves, segment waves, accelerating waves, packet waves, etc.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Concave Spiral Waves in Active Media Transferring Energy

Mazurov¹

2019

EPJ Web Conf.

View full text Add to dashboard Cite

Spiral concave autowaves are widely implemented in physics, chemistry, hydrodynamics, meteorology and other fields. A mathematical model of spiral concave autowaves based on the Fitzhugh-Nagumo equation and modified axiomatic models are presented. The existence of spiral concave autowaves transferring energy was predicted via computational experiments. Applications of spiral concave autowaves carrying energy in hydrodynamics, generation of tornadoes, breaking waves, and tsunamis and examples of such autowaves in biology and medicine are reviewed and the importance of concave spiral autowaves transferring energy is emphasized.

show abstract

Waves and patterns in reaction diffusion systems. Belousov Zhabotinsky reaction in water-in-oil microemulsions

Cited by 40 publications

References 139 publications

Stationary and slowly moving localised pulses in a singularly perturbed Brusselator model

Stationary and slowly moving localised pulses in a singularly perturbed Brusselator model

Polymorphic and regular localized activity structures in a two-dimensional two-component reaction–diffusion lattice with complex threshold excitation

Nonlinear Concave Spiral Waves in Active Media Transferring Energy

Contact Info

Product

Resources

About

Waves and patterns in reaction  diffusion systems. Belousov  Zhabotinsky reaction in water-in-oil microemulsions

Cited by 40 publications

References 139 publications

Stationary and slowly moving localised pulses in a singularly perturbed Brusselator model

Stationary and slowly moving localised pulses in a singularly perturbed Brusselator model

Polymorphic and regular localized activity structures in a two-dimensional two-component reaction–diffusion lattice with complex threshold excitation

Nonlinear Concave Spiral Waves in Active Media Transferring Energy

Contact Info

Product

Resources

About

Waves and patterns in reaction diffusion systems. Belousov Zhabotinsky reaction in water-in-oil microemulsions