Finite Wavetrains in a One-Dimensional Medium

1983 ◽  
Vol 38 (6) ◽  
pp. 648-667
Author(s):  
C. Kahlert ◽  
O. E. Rössler
Keyword(s):  
Finite Number ◽  
Present Method ◽  
Piecewise Linear ◽  
Reaction Diffusion ◽  
Standard Technique ◽  
Diffusion Type ◽  
One Dimensional ◽  
Reaction Diffusion Systems ◽  
Wave Solutions ◽  
Diffusion Systems

A modified Rinzel-Keller equation of reaction-diffusion type is investigated analytically. Both on the ring and on the linear fiber, the possible existence of wavetrains containing an arbitrary finite number of impulses is demonstrated. The longest wavetrain shown explicitly consists of ten pulses. Unlike other approaches, the present method varies many parameters simultaneously. An implicit algebraic equation is formulated which contains all possible wave solutions of the system. This equation is then solved with a standard technique (Powell's algorithm). The results obtained extend the findings of other authors. The method can be applied, after appropriate modification, to other piecewise linear systems, that is, to other prototype reaction-diffusion systems

Stable Kink-Antikink Pairs in Bistable Reaction-Diffusion Systems with Strong Nonlocalities

1995 ◽  
Vol 50 (12) ◽  
pp. 1128-1134 ◽  
Author(s):  
Thomas Christen
Keyword(s):  
Driving Force ◽  
Piecewise Linear ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
One Dimensional ◽  
Reaction Diffusion Systems ◽  
Uniform State ◽  
Diffusion Systems ◽  
Globally Stable ◽  
External Driving

Abstract We investigate the statics, nucleation, and dynamics of stable kink-antikink pairs (KAP) in a one-dimensional, one-component reaction-diffusion equation with a piecewise linear nonlinearity. The stabilization of the KAP is due to the presence of a strongly nonlocal inhibitor. We find a saddle-node bifurcation of a metastable KAP with a separation proportional to In L, where L is the length of the sample. The KAP becomes globally stable at a characteristic separation proportional to √L. The nucleation of a KAP from the metastable uniform state differs from the case without nonlocality mainly by a change of the activation energy induced by the nonlocality. Furthermore, we investigate the dynamics of the stable KAP in the presence of an external driving force and a diluted density of pointlike impurities; in particular, we derive expressions for the mobility and the average elongation of the KAP.

scholarly journals Stabilizing a homoclinic stripe

2018 ◽  
Vol 376 (2135) ◽  
pp. 20180110 ◽  
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)’.

scholarly journals Collision patterns on mollusc shells

1997 ◽  
Vol 1 (1) ◽  
pp. 57-76 ◽  
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.

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

2004 ◽  
Vol 18 (14) ◽  
pp. 2047-2055 ◽  
Author(s):  
AMIR AGHAMOHAMMADI ◽  
MOHAMMAD KHORRAMI
Keyword(s):  
Reaction Diffusion ◽  
Solvable Models ◽  
Empty Interval ◽  
Interval Method ◽  
One Dimensional ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Exactly Solvable ◽  
Necessary And Sufficient ◽  
Time Evolution Equation

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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