Bounded solutions of the nonlinear parabolic amplitude equation for plane Poiseuille flow

Holmes, C. A.

doi:10.1098/rspa.1985.0120

Cited by 17 publications

(14 citation statements)

References 18 publications

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“…The method is very classical and has already been applied (in more details than we will present here) to equations of Ginzburg-Landau type by Holmes [20]: we study the bifurcations of unstable periodic solutions of (1.1) of the form…”

Section: Bifurcation Into Traveling Wavesmentioning

confidence: 99%

“…x ~ ~ so that we can set U(x, t) = ff= fi(n, t)ei"~dn, (2.5) we restrict ourselves to the class of periodic perturbations with wavenumber l and use l as a parameter (like in [20]):…”

Section: Bifurcation Into Traveling Wavesmentioning

confidence: 99%

“…Substitution of (4,20) into the expression 2tR(Ys) and use of (4. By performing the perturbation analysis in more detail we are now able to compute and/~ (which are defined in (4.11), see also (4.19)) up to any accuracy.…”

Section: Ubn(x T) = E(t/2)iqeg(~)f(~)-(l+ix) E -I°~tmentioning

confidence: 99%

“…We will not perform the necessary center manifold analysis in great detail; we will not do any explicit computations (for these we refer the reader to [20]). We are primarily interested in showing the existence of traveling quasi-periodic solutions and in deriving their structure (this was not done in [20]). Substitution of (2.1) into (1.…”

Section: Bifurcation Into Traveling Wavesmentioning

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: Bifurcation Into Traveling Wavesmentioning

confidence: 99%

“…x ~ ~ so that we can set U(x, t) = ff= fi(n, t)ei"~dn, (2.5) we restrict ourselves to the class of periodic perturbations with wavenumber l and use l as a parameter (like in [20]):…”

Section: Bifurcation Into Traveling Wavesmentioning

confidence: 99%

Section: Ubn(x T) = E(t/2)iqeg(~)f(~)-(l+ix) E -I°~tmentioning

confidence: 99%

Section: Bifurcation Into Traveling Wavesmentioning

confidence: 99%

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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Solutions of the Ginzburg‐Landau Equation of Interest in Shear Flow Transition

Landamn

1987

Stud Appl Math

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The Ginzburg-Landau equation may be used to describe the weakly nonlinear 2-dimensional evolution of a disturbance in plane Poiseuille flow at Reynolds number near critical. We consider a class of quasisteady solutions of this equation whose spatial variation may be periodic, quasiperiodic, or solitarywave-like. Of particular interest are solutions describing a transition from the laminar solution to finite amplitude states. The existence of these solutions suggests the existence of a similar class of solutions in the Navier-Stokes equations, describing pulses and fronts of instability in the flow.

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Instability, Three-dimensional Effects and Transition in Shear Flows

Stuart

1987

Perspectives in Turbulence Studies

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Bounded solutions of the nonlinear parabolic amplitude equation for plane Poiseuille flow

Abstract: Some analysis is possible, but a more complete picture is obtained from computer results. These show the existence of limit cycles and solutions that appear to lie on two tori in space.Lastly we consider the existence of waves varying slowly in space and time.4 ACKNOWLEDGEMENTS

Cited by 17 publications

References 18 publications

Traveling waves in the complex Ginzburg-Landau equation

Traveling waves in the complex Ginzburg-Landau equation

Solutions of the Ginzburg‐Landau Equation of Interest in Shear Flow Transition

Instability, Three-dimensional Effects and Transition in Shear Flows

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