Pattern Forming Dynamical Instabilities of Bose–einstein Condensates

Kevrekidis, P. G.; Frantzeskakis, D. J.

doi:10.1142/s0217984904006809

Cited by 179 publications

(162 citation statements)

References 112 publications

(169 reference statements)

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“…The discrete nonlinear Schrödinger (DNLS) model constitutes a ubiquitous example of a nonlinear dynamical lattice with a wide range of applications, extending from the nonlinear optics of fabricated AlGaAs waveguide arrays as in [1][2][3], to the atomic physics of Bose-Einstein condensates in sufficiently deep optical lattices analyzed in [4][5][6][7]. Partly also due to these applications, the DNLS has been a focal point of numerous mathematical/computational investigations in its own right, a number of which has been summarized in [8][9][10][11][12][13] and is related to models used in numerous other settings including micromechanical cantilever arrays [14] and DNA breathing dynamics [15], among others.…”

Section: Introductionmentioning

confidence: 99%

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

2012

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We discuss the existence of breathers and lower bounds on their power, in nonlinear Schrödinger lattices with nonlinear hopping. Our methods extend from a simple variational approach to fixed point arguments, deriving lower bounds for the power which can serve as a threshold for the existence of breather solutions. Qualitatively, the theoretical results justify non-existence of breathers below the prescribed lower bounds of the power which depend on the dimension, the parameters of the lattice as well as of the frequency of breathers. In the case of supercritical power nonlinearities we investigate the interplay of these estimates with the optimal constant of the discrete interpolation inequality. Improvements of the general estimates, taking into account the localization of the true breather solutions are derived. Numerical studies in the one dimensional lattice corroborate the theoretical bounds and illustrate that in certain parameter regimes of physical significance, the estimates can serve as accurate predictors of the breather power and its dependence on the various system parameters.

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Section: Introductionmentioning

confidence: 99%

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

2012

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“…In particular, as can be seen in Fig. 4, there is a branch of solutions bifurcating from zero (amplitude) for γ > √ 2k 2 − E 2 , denoted by u (3) , which is the solid line in Fig. 4.…”

Section: Trimermentioning

confidence: 83%

“…The evolution of the three distinct branches of solutions, namely, the chiefly unstable one, u (1) , the chiefly stable one, u (2) , and finally the one persisting past the linearly unstable limit, u (3) , is shown, respectively, in Figs. 5, 6, and 7.…”

Section: Trimermentioning

confidence: 99%

“…The motivation for such studies stems from a variety of physical settings including, among others, the themes of optical beam dynamics in coupled waveguide arrays or optically induced photonic lattices in photorefractive crystals [2], the temporal evolution of Bose-Einstein condensates (BECs) in optical lattices [3], and the DNA double-strand denaturation in biophysics [4]. One of the common focal points among all of these areas has been the intense study of the existence, stability, and dynamical properties of their nonlinear (often localized in the form of solitary waves) solutions, which are of principal interest and experimental observability within various applications; see [5] for a relevant recent review.…”

Section: Introductionmentioning

confidence: 99%

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PT-symmetric oligomers: Analytical solutions, linear stability, and nonlinear dynamics

2011

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In the present work we focus on the case of (few-site) configurations respecting the parity-time (PT ) symmetry, i.e., with a spatially odd gain-loss profile. We examine the case of such "oligomers" with not only two sites, as in earlier works, but also the cases of three and four sites. While in the former case of recent experimental interest the picture of existing stationary solutions and their stability is fairly straightforward, the latter cases reveal a considerable additional complexity of solutions, including ones that exist past the linear PT -symmetry breaking point in the case of the trimer, and symmetry-breaking bifurcations, as well as more complex, even asymmetric solutions in the case of the quadrimer with nontrivial properties in their linear stability and in their nonlinear dynamics. The linearization around the obtained solutions and their dynamical evolution, when unstable, are discussed.

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“…Modulational instability is responsible for the formation of bright matter-wave solitons [40][41][42], as was analyzed by various theoretical works (see, e.g., Refs. [87][88][89] and the reviews [90,91]). …”

Section: Small-amplitude Linear Excitationsmentioning

confidence: 99%

Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

2008

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Abstract. The aim of the present review is to introduce the reader to some of the physical notions and of the mathematical methods that are relevant to the study of nonlinear waves in Bose-Einstein Condensates (BECs). Upon introducing the general framework, we discuss the prototypical models that are relevant to this setting for different dimensions and different potentials confining the atoms. We analyze some of the model properties and explore their typical wave solutions (plane wave solutions, bright, dark, gap solitons, as well as vortices). We then offer a collection of mathematical methods that can be used to understand the existence, stability and dynamics of nonlinear waves in such BECs, either directly or starting from different types of limits (e.g., the linear or the nonlinear limit, or the discrete limit of the corresponding equation). Finally, we consider some special topics involving more recent developments, and experimental setups in which there is still considerable need for developing mathematical as well as computational tools.PACS numbers: 03.75. Kk, 03.75.Lm, 05.45 • EP: Ermakov-Pinney (Equation)• GP: Gross-Pitaevskii (Equation)• KdV: Korteweg-de Vries (Equation)• LS: Lyapunov-Schmidt (Technique)• MT: Magnetic Trap• NLS: Nonlinear Schrödinger (Equation)• NPSE: Non-polynomial Schrödinger Equation IntroductionThe phenomenon of Bose-Einstein condensation is a quantum phase transition originally predicted by Bose [1] and Einstein [2,3] in 1924. In particular, it was shown that below a critical transition temperature T c , a finite fraction of particles of a boson gas (i.e., whose particles obey the Bose statistics) condenses into the same quantum state, known as the Bose-Einstein condensate (BEC). Although BoseEinstein condensation is known to be a fundamental phenomenon, connected, e.g., to superfluidity in liquid helium and superconductivity in metals (see, e.g., Ref.[4]), BECs were experimentally realized 70 years after their theoretical prediction: this major achievement took place in 1995, when different species of dilute alkali vapors confined in a magnetic trap (MT) were cooled down to extremely low temperatures [5][6][7], and has already been recognized through the 2001 Nobel prize in Physics [8,9]. This first unambiguous manifestation of a macroscopic quantum state in a manybody system sparked an explosion of activity, as reflected by the publication of several thousand papers related to BECs since then. Nowadays there exist more than fifty experimental BEC groups around the world, while an enormous amount of theoretical work has followed and driven the experimental efforts, with an impressive impact on many branches of Physics.From a theoretical standpoint, and for experimentally relevant conditions, the static and dynamical properties of a BEC can be described by means of an effective mean-field equation known as the Gross-Pitaevskii (GP) equation [10,11]. This is a variant of the famous nonlinear Schrödinger (NLS) equation [12] (incorporating an external potential used to confine th...

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Pattern Forming Dynamical Instabilities of Bose–einstein Condensates

Cited by 179 publications

References 112 publications

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

PT-symmetric oligomers: Analytical solutions, linear stability, and nonlinear dynamics

Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

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