Localized structures generated by subcritical instabilities

Thual, Olivier; Fauve, S.

doi:10.1051/jphys:0198800490110182900

Cited by 400 publications

(181 citation statements)

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“…Many different stable localized structures such as pulses (Thual & Fauve 1988;Deissler & Brand 1994;Afanasjev et al 1996) and holes (Sakaguchi 1991;Descalzi & Brand 2005; have been observed, as well as coexistence between those structures (Hayase et al 2004;Descalzi et al 2006a) and their interactions (Descalzi et al 2006b. However, this paper will focus only on the particular case of pulses.…”

Section: Introductionmentioning

confidence: 99%

“…From the pioneering results of Thual & Fauve (1988), many theoretical and numerical studies of pulses have been carried out (Fauve & Thual 1990;Malomed & Nepomnyashchy 1990;Hakim & Pomeau 1991). We can remark on the contribution of van Saarloos & Hohenberg (1990): they established that there is a well-defined range of parameters where pulses can be found, and also, out of that range, the system shows fronts.…”

Section: Introductionmentioning

confidence: 99%

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Transition from pulses to fronts in the cubic–quintic complex Ginzburg–Landau equation

Gutiérrez

Escaff

Descalzi

2009

Phil. Trans. R. Soc. A.

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The cubic-quintic complex Ginzburg-Landau is the amplitude equation for systems in the vicinity of an oscillatory sub-critical bifurcation (Andronov-Hopf), and it shows different localized structures. For pulse-type localized structures, we review an approximation scheme that enables us to compute some properties of the structures, like their existence range. From that scheme, we obtain conditions for the existence of pulses in the upper limit of a control parameter. When we study the width of pulses in that limit, the analytical expression shows that it is related to the transition between pulses and fronts. This fact is consistent with numerical simulations.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Transition from pulses to fronts in the cubic–quintic complex Ginzburg–Landau equation

Gutiérrez

Escaff

Descalzi

2009

Phil. Trans. R. Soc. A.

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“…[4]. Stable axisymmetric solitons with an internal vorticity ("spin") have been found too in the isotropic CQ CGL equation [5].…”

Section: Introductionmentioning

confidence: 99%

Two-dimensional solitary pulses in driven diffractive–diffusive complex Ginzburg–Landau equations

Sakaguchi

Malomed

2002

Physica D: Nonlinear Phenomena

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Two models of driven optical cavities, based on two-dimensional GinzburgLandau equations, are introduced. The models include loss, the Kerr nonlinearity, diffraction in one transverse direction, and a combination of diffusion and dispersion in the other one (which is, actually, a temporal direction). Each model is driven either parametrically or directly by an external field. By means of direct simulations, stable completely localized pulses are found (in the directly driven model, they are built on top of a nonzero flat background). These solitary pulses correspond to spatio-temporal solitons ("light bullets") in the optical cavities. Basic results are presented in a compact form as stability regions in a full three-dimensional parameter space of either model. The stability region is bounded by two surfaces; beyond the left one, any two-dimensional (2D) pulse decays to zero, while quasi-1D pulses, representing spatial solitons in the optical cavity, are found beyond the right boundary. The spatial solitons are found to be stable both inside the stability region of the 2D pulses (hence, bistability takes place in this region in the two models) and beyond the right boundary of this region (although they are not stable everywhere). Unlike the spatial solitons, their quasi-1D counterparts in the form of purely temporal solitons are always subject to modulational instability, which splits them into an array of 2D pulses, that further coalesce into two final pulses. A uniform nonzero state in the parametrically driven model is also modulationally unstable, which leads to formation of many 2D pulses that subsequently merge into few ones.

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“…Hence, short LTWs show a strong lateral variation of their properties. The shape of their amplitudes superficially resemble the pulse solutions of the complex GinzburgLandau equation [29,30,32,59,73,74,75].…”

Section: Short Ltwsmentioning

confidence: 99%

“…Stable pulse solutions of the complex Ginzburg-Landau equation (CGLE) were proposed as a model for confined binary mixture convection [73]. The nonlinear interaction between the local amplitude and frequency seems to be the essential localization mechanism in this approximation.…”

Section: E Comparison With Ltw Modelsmentioning

confidence: 99%

Traveling wave fronts and localized traveling wave convection in binary fluid mixtures

Jung

Lücke

2005

Phys. Rev. E

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Nonlinear fronts between spatially extended traveling wave convection (TW) and quiescent fluid and spatially localized traveling waves (LTWs) are investigated in quantitative detail in the bistable regime of binary fluid mixtures heated from below. A finite-difference method is used to solve the full hydrodynamic field equations in a vertical cross section of the layer perpendicular to the convection roll axes. Results are presented for ethanol-water parameters with several strongly negative separation ratios where TW solutions bifurcate subcritically. Fronts and LTWs are compared with each other and similarities and differences are elucidated. Phase propagation out of the quiescent fluid into the convective structure entails a unique selection of the latter while fronts and interfaces where the phase moves into the quiescent state behave differently. Interpretations of various experimental observations are suggested.

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Localized structures generated by subcritical instabilities

Cited by 400 publications

References 11 publications

Transition from pulses to fronts in the cubic–quintic complex Ginzburg–Landau equation

Transition from pulses to fronts in the cubic–quintic complex Ginzburg–Landau equation

Two-dimensional solitary pulses in driven diffractive–diffusive complex Ginzburg–Landau equations

Traveling wave fronts and localized traveling wave convection in binary fluid mixtures

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