Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations

Saarloos, Wim van; Hohenberg, P. C.

doi:10.1016/0167-2789(92)90175-m

Cited by 545 publications

(620 citation statements)

References 94 publications

(163 reference statements)

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“…(1) has a variety of localized solutions. These are stationary solitons, sources, sinks, moving solitons and fronts with fixed velocity [13,14]. A multiplicity of solutions can exist simultaneously.…”

Section: Master Equationmentioning

confidence: 99%

Exploding soliton and front solutions of the complex cubic–quintic Ginzburg–Landau equation

Soto-Crespo

Akhmediev

2005

Mathematics and Computers in Simulation

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We present a study of exploding soliton and front solutions of the complex cubic-quintic Ginzburg-Landau (CGLE) equation. We show that exploding fronts occur in a region of the parameter space close to that where exploding solitons exist. Explosions occur when eigenvalues in the linear stability analysis for the ground-state stationary solitons have positive real parts. We also study transition from exploding fronts to exploding solitons and observed extremely asymmetric soliton explosions.

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Section: Master Equationmentioning

confidence: 99%

Exploding soliton and front solutions of the complex cubic–quintic Ginzburg–Landau equation

Soto-Crespo

Akhmediev

2005

Mathematics and Computers in Simulation

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“…These results only exist for patterns which are much less complicated than most of the ones constructed in this paper. Here, we only refer to [28] and the recent paper [16], in which the approach is also geometrical, and the references given there.…”

Section: Discussionmentioning

confidence: 99%

“…These patterns are also described by the singularly perturbed system, but this system also governs a very complicated set of 'localised' patterns, corresponding to heteroclinic and homoclinic solutions. These types of patterns are important in the dynamics of the uncoupled Ginzburg-Landau equation (see for instance [28]). …”

Section: Discussionmentioning

confidence: 99%

“…Such localised structures can be stable in the uncoupled Ginzburg-Landau equation (see for instance [28] for a survey). However, none of these patterns can be described by the nonlocal reduction (1.6).…”

Section: 5)mentioning

confidence: 99%

“…These localised structures are again very important for understanding the dynamics of the solutions of the PDE. We refer to [28] and the references given there for an extensive discussion of the existence and stability of these solutions in the single Ginzburg-Landau equation.…”

Section: Heteroclinic and Homoclinic Orbitsmentioning

confidence: 99%

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Singularly perturbed and nonlocal modulation equations for systems with interacting instability mechanisms

Doelman

Rottschäfer

1997

J Nonlinear Sci

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Summary. Two related systems of coupled modulation equations are studied and compared in this paper. The modulation equations are derived for a certain class of basic systems which are subject to two distinct, interacting, destabilising mechanisms. We assume that, near criticality, the ratio of the widths of the unstable wavenumber-intervals of the two (weakly) unstable modes is small-as, for instance, can be the case in doublelayer convection. Based on these assumptions we first derive a singularly perturbed modulation equation and then a modulation equation with a nonlocal term. The reduction of the singularly perturbed system to the nonlocal system can be interpreted as a limit in which the width of the smallest unstable interval vanishes. We study and compare the behaviour of the stationary solutions of both systems. It is found that spatially periodic stationary solutions of the nonlocal system exist under the same conditions as spatially periodic stationary solutions of the singularly perturbed system. Moreover, these solutions can be interpreted as representing the same quasi-periodic patterns in the underlying basic system. Thus, the 'Landau reduction' to the nonlocal system has no significant influence on the stationary quasi-periodic patterns. However, a large variety of intricate heteroclinic and homoclinic connections is found for the singularly perturbed system. These orbits all correspond to so-called 'localised structures' in the underlying system: They connect simple periodic patterns at x → ±∞. None of these patterns can be described by the nonlocal system. So, one may conclude that the reduction to the nonlocal system destroys a rich and important set of patterns.

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On the Validity of the degenerate Ginzburg—Landau equation

Shepeleva

1997

Math. Meth. Appl. Sci.

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The Ginzburg–Landau equation which describes nonlinear modulation of the amplitude of the basic pattern does not give a good approximation when the Landau constant (which describes the influence of the nonlinearity) is small. In this paper a derivation of the so‐called degenerate (or generalized) Ginzburg–Landau (dGL)‐equation is given. It turns out that one can understand the dGL‐equation as an example of a normal form of a co‐dimension two bifurcation for parabolic PDEs. The main body of the paper is devoted to the proof of the validity of the dGL as an equation whose solution approximate the solution of the original problem. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

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Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations

Cited by 545 publications

References 94 publications

Exploding soliton and front solutions of the complex cubic–quintic Ginzburg–Landau equation

Exploding soliton and front solutions of the complex cubic–quintic Ginzburg–Landau equation

Singularly perturbed and nonlocal modulation equations for systems with interacting instability mechanisms

On the Validity of the degenerate Ginzburg—Landau equation

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