We predict a dynamical classical superfluid-insulator transition in a Bose-Einstein condensate trapped in an optical and a magnetic potential. In the tight-binding limit, this system realizes an array of weakly coupled condensates driven by an external harmonic field. For small displacements of the parabolic trap about the equilibrium position, the condensates coherently oscillate in the array. For large displacements, the condensates remain localized on the side of the harmonic trap with a randomization of the relative phases. The superfluid-insulator transition is due to a discrete modulational instability, occurring when the condensate center of mass velocity is larger than a critical value.

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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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).

Statistical mechanics of the discrete nonlinear Schrödinger equation is studied by means of analytical and numerical techniques. The lower bound of the Hamiltonian permits the construction of standard Gibbsian equilibrium measures for positive temperatures. Beyond the line of T = ∞, we identify a phase transition, through a discontinuity in the partition function. The phase transition is demonstrated to manifest itself in the creation of breather-like localized excitations. Interrelation between the statistical mechanics and the nonlinear dynamics of the system is explored numerically in both regimes.The pioneering studies of Fermi, Pasta and Ulam [1] (FPU) showed that energy exchange between coupled systems may be suppressed in the presence of nonlinearity; instead a type of behavior that severely contrasts equipartition among the linear modes is observed. The question of whether equipartition of excitation energy always appears is a contemporary issue in various fields of physics. Many manifestations of nonequilibrium and non-equipartition phenomena equivalent to the dynamical behavior of systems with few degrees of freedom contrasting statistical mechanics expectations have been observed. Some of these phenomena, and therefore the absence of immediate equipartition expressed in terms of self-trapping of energy, play an important role for optical storage patterns in nonlinear fibers, condensed matter physics, and biophysics [2].A particularity of discrete nonlinear systems is their ability to sustain strong localization of energy [3]. This is accomplished via intrinsic localized modes (breathers) which are modes that remain stable for extremely long times. So far it is a largely unaddressed problem how to handle and describe these excitations in a statistical mechanics framework although it has been argued that breathers may act as virtual bottlenecks [4] delaying the thermalization process.In this work, we develop a statistical understanding of the dynamics, including the breathers, in a discrete nonlinear Schrödinger (DNLS) equation. The DNLS equation plays a significant role in several branches nonlinear physics as a simple physical model because it may approximate many of the above mentioned nonlinear systems. We study analytically and numerically the thermalization of the lattice for T ≥ 0. We identify the regime in phase space wherein regular statistical mechanics considerations apply, and hence, thermalization is observed numerically and explored analytically using regular, grand-canonical, Gibbsian equilibrium measures.However, the nonlinear dynamics of the problem renders permissible the realization of regimes of phase space which would formally correspond to "negative temperatures" [5] in the sense of statistical mechanics. The novel feature of these states is that the energy tends to be localized in certain lattice sites forming breather-like excitations. Returning to statistical mechanics, such realizations, which would formally correspond to negative temperatures, are not possible (since the Hamil...

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An experimentally realizable scheme of periodic sign-changing modulation of the scattering length is proposed for Bose-Einstein condensates similar to dispersion-management schemes in fiber optics. Because of controlling the scattering length via the Feshbach resonance, the scheme is named Feshbach-resonance management. The modulational-instability analysis of the quasiuniform condensate driven by this scheme leads to an analog of the Kronig-Penney model. The ensuing stable localized structures are found. These include breathers, which oscillate between the Thomas-Fermi and Gaussian configuration, or may be similar to the 2-soliton state of the nonlinear Schrödinger equation, and a nearly static state ("odd soliton") with a nested dark soliton. An overall phase diagram for breathers is constructed, and full stability of the odd solitons is numerically established.