The harmonically forced complex Ginzburg-Landau equation.
Item
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Title
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The harmonically forced complex Ginzburg-Landau equation.
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Identifier
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AAI9618118
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identifier
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9618118
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Creator
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Wielaard, Dingeman Jacob.
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Contributor
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Adviser: George Triantafyllou
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Date
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1996
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Language
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English
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Publisher
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City University of New York.
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Subject
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Physics, General
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Abstract
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The complex Ginzburg-Landau equation {dollar}\partial\sb{lcub}t{rcub}\Psi-\sigma\Psi+\lambda\partial{lcub}\sbsp{lcub}x{rcub}{lcub}2{rcub}{rcub}\Psi+ \beta\mid\Psi\mid\sp2\Psi=F(x,t){dollar} is studied as a model equation regarding the accomplishment of spatio-temporal order in extended systems (systems of infinite length) as a result of a harmonic external forcing {dollar}F(x,t)= F(x)e\sp{lcub}iwt{rcub}.{dollar} Two types of spatial behaviour of the forcing are considered: a spatially localized forcing of Gaussian shape, {dollar}F(x)=F\sb0e\sp{lcub}-K\sp2x\sp2{rcub}{dollar}, and a spatially extended forcing in the form of a plane wave, {dollar}F(x)=F\sb0e\sp{lcub}iKt{rcub}{dollar}.;For the spatially independent case the lock-in region in the phase diagram in {dollar}F\sb0{lcub}-{rcub}\omega{dollar} space is obtained analytically. Stable locked-in states ({dollar}{lcub}\sim{rcub}e\sp{lcub}iwt{rcub}{dollar}) become possible for sufficiently large forcing strength. The only other stable state observed is a quasi-periodic state. Coexistence of two locked states as well as a locked and a quasi-periodic state is possible.;For a spatially dependent linear growth rate of the form {dollar}Re\sigma(x) = \sigma\sb{lcub}\infty{rcub}+\sigma\sb0e\sp{lcub}-Q\sp2x\sp2{rcub},{dollar} where {dollar}\sigma\sb0 > {lcub}-{rcub}\sigma\sb{lcub}\infty{rcub} > 0,{dollar} the system size is effectively determined by the width of the growth rate (Q). In small systems the behaviour shows great similarity with the spatially independent case, although some differences are observed as function of the location. As the system size is increased, the similarity with the spatially independent case remains present for where it concerns phase-locked states. For the other states the behaviour depends strongly on the location in the system. Local organization becomes possible with a small forcing (around resonance). States occur which show phase-locked, quasi-periodic and chaotic behaviour in different spatial regions of the system.;For the situation with constant coefficients ({dollar}Re\beta > 0{dollar} and {dollar}Re\sigma > 0{dollar}) and a plane wave forcing, the region of stability for phase-locked states (plane waves) is constructed in the {dollar}F\sb0{dollar}-{dollar}\omega{dollar}-K space. Stability requires in all cases {dollar}Re\lambda > 0{dollar}. In the Benjamin-Feir unstable regime for the unforced equation, stable forced plane wave states become possible for sufficiently strong forcing with {dollar}k\sbsp{lcub}min{rcub}{lcub}2{rcub} (F\sb0) < K\sp2 < K\sbsp{lcub}max{rcub}{lcub}2{rcub} (F\sb0),{dollar} where {dollar}K\sbsp{lcub}min{rcub}{lcub}2{rcub} = 0{dollar} for most parameter choices. It is found that the classification of the different spatio-temporal chaotic states in the forced system in terms of amplitude chaos and phase chaos is insufficient, a more elaborate description is needed.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
