Traveling waves in the complex Ginzburg-Landau equation
type: Item
instance of: scholarly article
title: Traveling waves in the complex Ginzburg-Landau equation
zbMATH Open document ID: 0805.35050
DOI: 10.1007/BF02429865
published in: Journal of Nonlinear Science
publication date: 1994-05-30T00:00:00Z
review text: Travelling wave solutions of the one-dimensional Ginzburg-Landau equation \[ u_ t = u + (1 + i \alpha) u_{xx} - (1 + i \beta) | u |^ 2u, \quad x \in \mathbb{R} \tag{1} \] are studied. The equation is assumed to be weakly complex, that is, \(\alpha = \varepsilon a\), \(\beta = \varepsilon b\) with \(0<\varepsilon \ll 1\). For \(\varepsilon = 0\), the equation for travelling waves reduces to a 3-dimensional integrable system of ODEs. Using this fact, the author constructs certain travelling waves of (1) by continuation from the ``real case'' \(\varepsilon = 0\). In particular, two-parameter families of spatially quasiperiodic solutions are obtained this way.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B60
Mathematics Subject Classification ID: 34C37
zbMATH DE Number: 563611
zbMATH Keywords: pattern formation
zbMATH Keywords: modular equations
zbMATH Keywords: perturbed integrable systems
zbMATH Keywords: one- dimensional Ginzburg-Landau equation
zbMATH Keywords: travelling waves
zbMATH Keywords: two-parameter families of spatially quasiperiodic solutions
author: Arjen Doelman
reviewed by: Peter Poláčik
MaRDI profile type: Publication
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cites work: On the possibility of soft and hard turbulence in the complex Ginzburg- Landau equation
cites work: Slowly varying fully nonlinear wavetrains in the Ginzburg-Landau equation
cites work: Applications of centre manifold theory
cites work: The time dependent amplitude equation for the Swift-Hohenberg problem
cites work: Non-linear wave-number interaction in near-critical two-dimensional flows
cites work: Slow time-periodic solutions of the Ginzburg-Landau equation
cites work: Periodic and quasi-periodic solutions of degenerate modulation equations
cites work: Low-dimensional behaviour in the complex Ginzburg-Landau equation
cites work: Q5330852
cites work: Geometric singular perturbation theory for ordinary differential equations
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: On the validity of the Ginzburg-Landau equation
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cites work: Spatial structure of time-periodic solutions of the Ginzburg-Landau equation
cites work: Slowly Varying Waves and Shock Structures in Reaction-Diffusion Equations
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cites work: Stability of plane wave solutions of complex Ginzburg-Landau equations
cites work: Q4059421
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cites work: Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations
cites work: A non-linear instability theory for a wave system in plane Poiseuille flow
cites work: Global bifurcations and chaos. Analytical methods
description: scientific article; zbMATH DE number 563611