Spatial structure of time-periodic solutions of the Ginzburg-Landau equation
type: Item
instance of: scholarly article
title: Spatial structure of time-periodic solutions of the Ginzburg-Landau equation
zbMATH Open document ID: 0618.35003
DOI: 10.1016/0167-2789(86)90114-4
published in: Physica D
publication date: 1986-01-01T00:00:00Z
review text: We study solutions of the form \(A(x,t)=a(x)e^{i\omega t}\) of the complex Ginzburg-Landau equation \(iA_ t+{\bar \alpha}A_{xx}={\bar \beta}A-{\bar \gamma}| A|^ 2A\), near those of the integrable limit: the nonlinear Schrödinger equation \(iA_ t+{\bar \alpha}A_{xx}=-{\bar \gamma}| A|^ 2A,{\bar \alpha},{\bar \gamma}\) real. We show that spatially periodic and quasiperiodic solutions as well as heteroclinic orbits to them exist, in addition to the spatially decaying (homoclinic) solutions found by \textit{L. M. Hocking} and \textit{K. Stewartson} [Proc. R. Soc. Lond., Ser. A 326, 289- 313 (1972; Zbl 0228.76074)]. We speculate briefly on spatially chaotic solutions.
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 4002587
zbMATH Keywords: complex Ginzburg-Landau equation
zbMATH Keywords: integrable limit
zbMATH Keywords: nonlinear Schrödinger equation
zbMATH Keywords: spatially periodic and quasiperiodic solutions
zbMATH Keywords: heteroclinic orbits
zbMATH Keywords: spatially decaying (homoclinic) solutions
zbMATH Keywords: spatially chaotic solutions
author: Philip Holmes
MaRDI profile type: Publication
full work available at URL: 0167-2789(86)90114-4
OpenAlex ID: W2020411560
cites work: Q4151274
cites work: Q3223594
cites work: Dynamics of Perturbed Wavetrain Solutions to the Ginzburg-Landau Equation
cites work: Instabilities of the Ginzburg-Landau equation: periodic solutions
cites work: Instabilities of the Ginzburg-Landau equation. II. Secondary bifurcation
cites work: Periodic solutions of the Ginzburg-Landau equation
cites work: Q3028597
cites work: Reduction of symplectic manifolds with symmetry
cites work: Subordinate Šil'nikov bifurcations near some singularities of vector fields having low codimension
cites work: Q5509205
cites work: On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance
cites work: Q5536828
cites work: A CONTRIBUTION TO THE PROBLEM OF THE STRUCTURE OF AN EXTENDED NEIGHBORHOOD OF A ROUGH EQUILIBRIUM STATE OF SADDLE-FOCUS TYPE
cites work: On the existence of separatrix loops in four-dimensional systems similar to the integrable Hamiltonian systems
cites work: Periodic nonlinear waves on a half-line
description: scientific article; zbMATH DE number 4002587