R. Carretero-González

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

The aim of this review is to introduce the reader to some of the physical notions and 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 analyse 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.

Simulation of the effect of wind speedup in the formation of transverse dune fields

A computer simulation model for transverse-dune-field dynamics, corresponding to a uni-directional wind regime, is developed. In a previous formulation, two distinct problems were found regarding the cross-sectional dune shape, namely theerosionintheleeofdunesandthesteepnessofthewindwardslopes.Thefirstproblemissolvedbyintroducingnoerosion inshadowzones.Thesecondissueisovercomebyintroducingawindspeedup(shearvelocityincrease)factor,whichcanbe accountedforbyaddingatermtotheoriginaltransportlength,whichisproportionaltothesurfaceheight.Byincorporating these features we are able to model dunes whose individual shape and collective patterns are similar to those observed in nature.Moreoverweshowhowtheintroductionofanon-linearshear-velocity-increasetermleadstothereductionofdune height,andthismayresultinanequilibriumdunefieldconfiguration.Thisisthoughttobebecausethenon-linearincreaseof thetransportlengthmakesthesandtrappingefficiencylowerthanunity,evenforhigherdunes,sothattheincomingandthe outgoingsandfluxareinbalance.Tofullydescribetheinter-dunemorphologymoreprecisedynamicsintheleeofthedune must be incorporated. Copyright # 2000 John Wiley & Sons, Ltd.

Modelling desert dune fields based on discrete dynamics

A mathematical formulation is developed to model the dynamics of sand dunes. The physical processes display strong non-linearity that has been taken into account in the model. When assessing the success of such a model in capturing physical features we monitor morphology, dune growth, dune migration and spatial patterns within a dune field. Following recent advances, the proposed model is based on a discrete lattice dynamics approach with new features taken into account which reflect physically observed mechanisms.

Scaling and interleaving of subsystem Lyapunov exponents for spatio-temporal systems.

The computation of the entire Lyapunov spectrum for extended dynamical systems is a very time consuming task. If the system is in a chaotic spatio-temporal regime it is possible to approximately reconstruct the Lyapunov spectrum from the spectrum of a subsystem by a suitable rescaling in a very cost effective way. We compute the Lyapunov spectrum for the subsystem by truncating the original Jacobian without modifying the original dynamics and thus taking into account only a portion of the information of the entire system. In doing so we notice that the Lyapunov spectra for consecutive subsystem sizes are interleaved and we discuss the possible ways in which this may arise. We also present a new rescaling method, which gives a significantly better fit to the original Lyapunov spectrum. We evaluate the performance of our rescaling method by comparing it to the conventional rescaling (dividing by the relative subsystem volume) for one- and two-dimensional lattices in spatio-temporal chaotic regimes. Finally, we use the new rescaling to approximate quantities derived from the Lyapunov spectrum (largest Lyapunov exponent, Lyapunov dimension, and Kolmogorov-Sinai entropy), finding better convergence as the subsystem size is increased than with conventional rescaling. (c) 1999 American Institute of Physics.

Nonequilibrium dynamics and superfluid ring excitations in binary bose-einstein condensates

We revisit a classic study [D. S. Hall, Phys. Rev. Lett. 81, 1539 (1998)10.1103/PhysRevLett.81.1539] of interpenetrating Bose-Einstein condensates in the hyperfine states |F=1,m{f}=-1 identical with |1 and |F=2,m{f}=+1 identical with |2 of 87Rb and observe striking new nonequilibrium component separation dynamics in the form of oscillating ringlike structures. The process of component separation is not significantly damped, a finding that also contrasts sharply with earlier experimental work, allowing a clean first look at a collective excitation of a binary superfluid. We further demonstrate extraordinary quantitative agreement between theoretical and experimental results using a multicomponent mean-field model with key additional features: the inclusion of atomic losses and the careful characterization of trap potentials (at the level of a fraction of a percent).

Bright-dark soliton complexes in spinor Bose-Einstein condensates

We consider vector solitons of mixed bright-dark types in quasi-one-dimensional spinor

Multistable solitons in the cubic–quintic discrete nonlinear Schrödinger equation

(F=1)

Guiding-center dynamics of vortex dipoles in Bose-Einstein condensates

Bose-Einstein condensates. Using a multiscale expansion technique, we reduce the corresponding nonintegrable system of three coupled Gross-Pitaevskii equations (GPEs) to an integrable Yajima-Oikawa system. In this way, we obtain approximate solutions for small-amplitude vector solitons of dark-dark-bright and bright-bright-dark types, in terms of the

Multipole-mode solitons in Bessel optical lattices

{m}_{F}=+1,\ensuremath{-}1,0

Dynamics of a few corotating vortices in Bose-Einstein condensates.

spinor components, respectively. By means of numerical simulations of the full GPE system, we demonstrate that these states indeed feature soliton properties, i.e., they propagate undistorted and undergo quasielastic collisions. It is also shown that in the presence of a parabolic trap the bright component(s) is (are) guided by the dark one(s) and, as a result, the small-amplitude vector soliton as a whole performs quasiharmonic oscillations. The oscillation frequency is found as a function of the spin-dependent interaction strength for both small-amplitude and large-amplitude solitons.