Slow time-periodic solutions of the Ginzburg-Landau equation

Doelman, Arjen

doi:10.1016/0167-2789(89)90060-2

Cited by 33 publications

(56 citation statements)

References 14 publications

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“…(The Poincar6 map has no fixed points for parameter combinations outside this region.) This region, F, does not depend on the coefficients c~ = ca and/3 = Eb of Equation We remark here that this result is a significant extension of the work done in [12]: in that paper it was shown that none of the quasi-periodic solutions of the integrable limit, except for a family of degenerate periodic solutions, survive the perturbation. However, in [12] we considered v = 0.…”

Section: U(x T) = Gt(x + Vt)e -Iw'supporting

confidence: 60%

“…Furthermore, we discuss briefly the relation between this paper and earlier work on the weakly complex Ginzburg-Landau equation [12].…”

Section: U(x T) = Gt(x + Vt)e -Iw'mentioning

confidence: 99%

“…Our methods are based on an O(E) approximation of the Poincar6 map associated with the perturbed integrable o.d.e. ([21], [12]). The proof of this result boils down to a detailed analysis of the approximation of the Poincar~ map P. We express this approximation in a linear combination of two types of complete elliptic expressions (Section 3.3).…”

Section: U(x T) = Gt(x + Vt)e -Iw'mentioning

confidence: 99%

“…This paper belongs to the second group. The main aim of papers studying (1.1) on the real axis has been an exploration of the "universum of solutions" of (1.1) by some kind of reduction to a simpler system (Holmes [20] (using a center manifold reduction), Holmes [21], Doelman [12] (both near an integrable limit), and Bernoff [5] (asymptotic analysis of slowly varying solutions)).…”

Section: Introductionmentioning

confidence: 99%

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Traveling waves in the complex Ginzburg-Landau equation

Doelman

1993

J Nonlinear Sci

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Summary. In this paper we consider a modulation (or amplitude) equation that appears in the nonlinear stability analysis of reversible or nearly reversible systems. This equation is the complex Ginzburg-Landau equation with coefficients with small imaginary parts. We regard this equation as a perturbation of the real Ginzburg-Landau equation and study the persistence of the properties of the stationary solutions of the real equation under this perturbation. First we show that it is necessary to consider a two-parameter family of traveling solutions with wave speed v and (temporal) frequency ~o; these solutions are the natural continuations of the stationary solutions of the real equation. We show that there exists a two-parameter family of traveling quasiperiodic solutions that can be regarded as a direct continuation of the two-parameter family of spatially quasi-periodic solutions of the integrable stationary real GinzburgLandau equation. We explicitly determine a region in the (wave speed v, frequency ~)-parameter space in which the weakly complex Ginzburg-Landau equation has traveling quasi-periodic solutions. There are two different one-parameter families of heteroclinic solutions in the weakly complex case. One of them consists of slowly varying plane waves; the other is directly related to the analytical solutions due to Bekki & Nozaki [3]. These solutions correspond to traveling localized structures that connect two different periodic patterns. The connections correspond to a one-parameter family of heteroclinic cycles in an o.d.e, reduction. This family of cycles is obtained by determining the limit behaviour of the traveling quasi-periodic solutions as the period of the amplitude goes to oo. Therefore, the heteroclinic cycles merge into the stationary homoclinic solution of the real Ginzburg-Landau equation in the limit in which the imaginary terms disappear.

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Section: U(x T) = Gt(x + Vt)e -Iw'supporting

confidence: 60%

“…Furthermore, we discuss briefly the relation between this paper and earlier work on the weakly complex Ginzburg-Landau equation [12].…”

Section: U(x T) = Gt(x + Vt)e -Iw'mentioning

confidence: 99%

Section: U(x T) = Gt(x + Vt)e -Iw'mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Traveling waves in the complex Ginzburg-Landau equation

Doelman

1993

J Nonlinear Sci

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“…See Holmes [24], Sirovich and Newton [34], Landman [27], Bernoff [3], Doelman [11,12], Schopf and Kramer [31] and references therein, for example. Recently,…”

Section: Background and Definitionsmentioning

confidence: 99%

Fronts, domain walls and pulses in a generalized Ginzburg-Landau equation

Duan

Holmes

1995

Proceedings of the Edinburgh Mathematical Society

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We discuss the existence and non-existence of front, domain wall and pulse type traveling wave solutions of a Ginzburg-Landau equation with cubic terms containing spatial derivatives and a fifth order term, in both subcritical and supercritical cases. Our results appear to be the first rigorous existence and non-existence proofs for the full equation with all possible terms derived from second order perturbation theory present.1991 Mathematics subject classification: 58F39, 34C27, 34C35, 34C37, 35Q55. Background and definitionsOne way to investigate the dynamics of a pattern formation system modeled by a partial differential equation (PDE) of evolution type, with a single space variable, is via the traveling frame reduction. Introducing the traveling coordinate z = x -ct with wave speed c, we get a boundary value problem for a system of ordinary differential equations (ODE). The critical points of these ODEs correspond to the plane waves or the zero amplitude (trivial) wave of the original PDE. The connections, or transitions, between critical points carry important dynamical information. These connections are a subclass (i.e., uniformly translating structures) of the so-called "coherent structures" (e.g. Saarloos and Hohenberg [29,30]). A heteroclinic connection between the zero amplitude wave and a plane wave is called a front. We call a heteroclinic connection between two different plane waves a heteroclinic domain wall. We call a homoclinic connection from a plane wave to itself a homoclinic domain wall. A homoclinic orbit from the zero amplitude wave to itself is called a pulse. For example, in binary fluid convection (see below), these connections correspond to the "boundary layer" between the coexisting conductive state and periodic, confined convective states.Several

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Multipulse-Like Orbits for a Singularly Perturbed Nearly Integrable System

2002

Journal of Differential Equations

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Slow time-periodic solutions of the Ginzburg-Landau equation

Cited by 33 publications

References 14 publications

Traveling waves in the complex Ginzburg-Landau equation

Traveling waves in the complex Ginzburg-Landau equation

Fronts, domain walls and pulses in a generalized Ginzburg-Landau equation

Multipulse-Like Orbits for a Singularly Perturbed Nearly Integrable System

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