Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

Kolokolnikov, Théodore; Ward, Michael J.; Wei, Juncheng

doi:10.1007/s00332-008-9024-z

Cited by 79 publications

(201 citation statements)

References 54 publications

(77 reference statements)

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“…It would also be interesting to study the evolution of spots in a two-dimensional singularly perturbed Brusselator model. In [19], spot replication was studied for a singularly perturbed Schnakenberg model on a unit square and unit disk.…”

Section: Resultsmentioning

confidence: 99%

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

Tzou¹,

Bayliss

Matkowsky

et al. 2011

Eur. J. Appl. Math

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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.

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Section: Resultsmentioning

confidence: 99%

“…[10,11]) and Schnakenberg (e.g. [19,30]) models, have been of recent interest. In one dimension, behaviours such as slow pulse evolution, pulse splitting, and pulse oscillations, have been predicted analytically and confirmed numerically.…”

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

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“…22 and only below another critical temperature that they call the "micelle disassociation temperature," which is higher than the CMT. This paper uses density functional theory to study energy-minimizing equilibria and dynamics in a region of parameter space where the homogeneous state is stable.…”

Section: Tionsmentioning

confidence: 99%

“…Self-replication in higher dimensions. The self-replication phenomenon in higher dimensions has also been studied in the literature (e.g., [22]). We conclude our discussion of dynamical phenomenon by demonstrating how this unfolds for the present model, using arguments analogous to those discussed in section 4.2.…”

Section: Filamentary Instability Of Cylindrical Micelles and Bilayer mentioning

confidence: 99%

Spatially Localized Structures in Diblock Copolymer Mixtures

Glasner¹

2010

SIAM J. Appl. Math.

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Abstract. Above the critical temperature for the order-disorder transition, diblock copolymer melts have been observed to exhibit localized structures that exist within the homogeneous mixture. This paper uses an Ohta-Kawasaki-type density functional to explore this regime. Spatially localized peak-shaped equilibria are studied in one, two, and three dimensions, corresponding to amphiphilic bilayers, cylindrical micelles, and spherical micelles, respectively. A combination of rigorous estimates, asymptotic analysis, and numerical computation is used to characterize solutions and the regime where they exist. The interaction of superpositions of these solutions is studied by a perturbation analysis and shows how steady multipeak configurations can be achieved. Evidence is found for a secondary bifurcation slightly below the spinodal instability threshold, beyond which self-replication phenomena are observed. Dynamics in two dimensions are also illustrated, suggesting other mechanisms for instability and growth.Key words. diblock copolymer, micelle, density functional theory, self-replication AMS subject classifications. 82D60, 35K55DOI. 10.1137/080743913 1. Introduction. Diblock copolymers are linear chain molecules composed of two distinct subchains. At low temperatures, repulsion between subchains can result in phase segregation, but macroscopic segregation cannot occur because of covalent bonding between subchains. This incomplete segregation can give rise to a variety of stable microstructures (see, e.g., [1,11,14,24]), which are typically characterized as spatially periodic patterns whose domain boundaries closely resemble minimal surfaces, such as spheres, cylinders, and gyroids [39].Outside of the regime where mixtures are unstable to spinodal decomposition, the situation for copolymers is more complicated than simple binary systems, such as those described by the Cahn-Hilliard theory [5]. In addition to a stable homogeneous state, there are other observed behaviors. Physical experiments have identified inhomogeneous states above the critical temperature, such as sparse arrangements of micellar structures [35,36,43] and anisotropic lamellar grains [15]. This paper seeks to investigate this regime and its dynamics using density functional theory and the corresponding gradient flow dynamics.Theoretical work on the disordered regime has focused on spherical micelles. Dormidontova and Lodge [9] use Semenov's strong segregation theory [37] to predict the concentration of micelle assemblies. Their theory predicts that the number of micelles increases rapidly below a "critical micelle temperature" (CMT), which is greater than the temperature at which a homogeneous mixture is spinodally unstable. Wang, Wang, and Yang [42] use self-consistent field theory calculations to find the density profile of individual spherical micelles and evaluate their excess free energy. They find that micelles exist (within the mean field approximation) for volume frac-

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“…With C D 0, it has been advocated by T. Kolokolnikov et al [13] as a simple model for spot-pattern formation and spot-splitting in the following asymptotic regime…”

Section: Schnakenberg Systemmentioning

confidence: 98%

Linear Instability, Turing Instability and Pattern Formation

Perthame

2015

Lecture Notes on Mathematical Modelling in the Life Sciences

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Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

Cited by 79 publications

References 54 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

Spatially Localized Structures in Diblock Copolymer Mixtures

Linear Instability, Turing Instability and Pattern Formation

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