EXACTLY SOLVABLE MODELS THROUGH THE GENERALIZED EMPTY INTERVAL METHOD: MULTI-SPECIES AND MORE-THAN-TWO-SITE INTERACTIONS

Multi-species reaction-diffusion systems, with more-than-two-site interaction on a one-dimensional lattice are considered. Necessary and sufficient constraints on the interaction rates are obtained, that guarantee the closure of the time evolution equation for [Formula: see text], the expectation value of the product of certain linear combination of the number operators on n consecutive sites at time t.

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Exactly solvable models through the empty-interval method

Physical Review E ◽

10.1103/physreve.64.056116 ◽

2001 ◽

Vol 64 (5) ◽

Cited By ~ 13

Author(s):

M. Alimohammadi ◽

M. Khorrami ◽

A. Aghamohammadi

Keyword(s):

Exactly Solvable Models ◽

Solvable Models ◽

Empty Interval ◽

Interval Method ◽

Exactly Solvable

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Stabilizing a homoclinic stripe

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2018.0110 ◽

2018 ◽

Vol 376 (2135) ◽

pp. 20180110 ◽

Cited By ~ 4

Author(s):

Theodore Kolokolnikov ◽

Michael Ward ◽

Justin Tzou ◽

Juncheng Wei

Keyword(s):

Reaction Diffusion ◽

Dissipative Structures ◽

Vegetation Patterns ◽

Theme Issue ◽

One Dimensional ◽

Reaction Diffusion Systems ◽

Diffusion Systems ◽

Schnakenberg Model ◽

Break Up ◽

Variable Substrate

For a large class of reaction–diffusion systems with large diffusivity ratio, it is well known that a two-dimensional stripe (whose cross-section is a one-dimensional homoclinic spike) is unstable and breaks up into spots. Here, we study two effects that can stabilize such a homoclinic stripe. First, we consider the addition of anisotropy to the model. For the Schnakenberg model, we show that (an infinite) stripe can be stabilized if the fast-diffusing variable (substrate) is sufficiently anisotropic. Two types of instability thresholds are derived: zigzag (or bending) and break-up instabilities. The instability boundaries subdivide parameter space into three distinct zones: stable stripe, unstable stripe due to bending and unstable due to break-up instability. Numerical experiments indicate that the break-up instability is supercritical leading to a ‘spotted-stripe’ solution. Finally, we perform a similar analysis for the Klausmeier model of vegetation patterns on a steep hill, and examine transition from spots to stripes. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.

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Collision patterns on mollusc shells

Discrete Dynamics in Nature and Society ◽

10.1155/s1026022697000071 ◽

1997 ◽

Vol 1 (1) ◽

pp. 57-76 ◽

Cited By ~ 2

Author(s):

P. J. Plath ◽

J. K. Plath ◽

J. Schwietering

Keyword(s):

Reaction Diffusion ◽

Growth Front ◽

Cell System ◽

One Dimensional ◽

Reaction Diffusion Systems ◽

Diffusion Systems ◽

Different Types ◽

Classical Waves ◽

Cellular Behaviour ◽

Mollusc Shells

On mollusc shells one can find famous patterns. Some of them show a great resemblance to the soliton patterns in one-dimensional systems. Other look like Sierpinsky triangles or exhibit very irregular patterns. Meinhardt has shown that those patterns can be well described by reaction–diffusion systems [1]. However, such a description neglects the discrete character of the cell system at the growth front of the mollusc shell.We have therefore developed a one-dimensional cellular vector automaton model which takes into account the cellular behaviour of the system [2]. The state of the mathematical cell is defined by a vector with two components. We looked for the most simple transformation rules in order to develop quite different types of waves: classical waves, chemical waves and different types of solitons. Our attention was focussed on the properties of the system created through the collision of two waves.

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