Periodic and quasi-periodic solutions of degenerate modulation equations

Doelman, Arjen; Eckhaus, Wiktor

doi:10.1016/0167-2789(91)90065-h

Cited by 59 publications

(84 citation statements)

References 16 publications

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“…Although the small amplitude plane wave is dynamically unstable (see Doelman and Eckhaus [13], p. 255]), the existence of connections to and from it does indicate how solutions started near this wave might evolve: see Section 7.…”

Section: Fronts and Domain Walls: A O <0mentioning

confidence: 99%

“…In fact for fc 3 ,b 4 #0, as Doelman and Eckhaus [13] observe, one can modify the integrals as follows…”

Section: Integrable Structures Symmetry and Critical Pointsmentioning

confidence: 99%

“…For the general equation (1.2), Saarloos and Hohenberg [30] present a framework for the discussion of front, pulse and domain wall dynamics. Doelman and Eckhaus [13], Theorem 3.4, find some homoclinic domain walls with wave speed c = 0, via Poincare maps and Melnikov integrals, extending and correcting the work of Holmes [24].…”

Section: Background and Definitionsmentioning

confidence: 99%

“…However, it is difficult to construct isolating blocks in our case. Therefore we use an analytical approach, see Guckenheimer [25], Doelman and Eckhaus [13], and Campbell and Holmes [5].…”

Section: Background and Definitionsmentioning

confidence: 99%

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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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Section: Fronts and Domain Walls: A O <0mentioning

confidence: 99%

“…In fact for fc 3 ,b 4 #0, as Doelman and Eckhaus [13] observe, one can modify the integrals as follows…”

Section: Integrable Structures Symmetry and Critical Pointsmentioning

confidence: 99%

Section: Background and Definitionsmentioning

confidence: 99%

“…However, it is difficult to construct isolating blocks in our case. Therefore we use an analytical approach, see Guckenheimer [25], Doelman and Eckhaus [13], and Campbell and Holmes [5].…”

Section: Background and Definitionsmentioning

confidence: 99%

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

Duan

Holmes

1995

Proceedings of the Edinburgh Mathematical Society

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“…Such terms prevent possible runaway of spatially homogeneous states and may be brought into the theory via a systematic expansion that treats the coefficient of the cubic term as a small quantity whose magnitude is linked to the modulation length scale of the pattern. A systematic study of this problem reveals the presence, in general, of two additional terms that come in at the same order in perturbation theory as the quintic term [14,15]. These terms have been computed explicitly from the quadratic-cubic SwiftHohenberg equation [6,16], but are present in related computations going back a number of years [17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Weakly subcritical stationary patterns: Eckhaus instability and homoclinic snaking

Kao

Knobloch

2012

Phys. Rev. E

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The transition from subcritical to supercritical stationary periodic patterns is described by the one-dimensional cubic-quintic Ginzburg-Landau equationwhere A(x, t) represents the pattern amplitude and the coefficients µ, a 1 , a 2 and b are real. The conditions for Eckhaus instability of periodic solutions are determined, and the resulting spatially modulated states are computed. Some of these evolve into spatially localized structures in the vicinity of a Maxwell point, while others resemble defect states. The results are used to shed light on the behavior of localized structures in systems exhibiting homoclinic snaking during the transition from subcriticality to supercriticality.

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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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Periodic and quasi-periodic solutions of degenerate modulation equations

Cited by 59 publications

References 16 publications

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

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

Weakly subcritical stationary patterns: Eckhaus instability and homoclinic snaking

On the Validity of the degenerate Ginzburg—Landau equation

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