We calculated, within the Gross-Pitaevskii formalism, the critical number of atoms for Bose-Einstein condensates with two-body attractive interactions in cylindrical traps with different frequency ratios. In particular, by using the trap geometries considered by Roberts et al. ͓Phys. Rev. Lett. 86, 4211 ͑2001͔͒, we show that the theoretical maximum critical numbers are given approximately by N c ϭ0.55(l 0 /͉a͉). Our results also show that, by exchanging the frequencies z and , the geometry with Ͻ z favors the condensation of larger number of particles. We also simulate the time evolution of the condensate when changing the ground state from aϭ0 to aϽ0 using a 200 ms ramp. A conjecture on higher-order nonlinear effects is also added in our analysis with an experimental proposal to determine its signal and strength.

In the framework of the Gross-Pitaevskii mean field approach it is shown that the supersonic flow of a Bose-Einstein condensate can support a new type of pattern-an oblique dark soliton. The corresponding exact solution of the Gross-Pitaevskii equation is obtained. It is demonstrated by numerical simulations that oblique solitons can be generated by an obstacle inserted into the flow.PACS numbers: 03.75.KkIntroduction. It is known that the nonlinear and dispersive properties of a Bose-Einstein condensate (BEC) can lead to the formation of various nonlinear structures (see, e.g., [1]). Until recently, most research has been focused on experimentally observed vortices and bright and dark solitons. Furthermore, the formation of dispersive shock waves in BECs with repulsive interactions between atoms was considered theoretically in [2,3] and studied experimentally in rotating [4] and non-rotating [5] condensate, where it was shown that dispersive shocks are generated as a result of the evolution of large disturbances in the BEC. However, another important type of nonlinear structure, namely a spatial dark soliton, can also be realized in a BEC. The first experimental evidence of their generation has recently appeared [6]. In fact, the existence of oblique spatial solitons in a BEC has a natural physical basis if the Cherenkov generation of dispersive sound waves by a small obstacle in the supersonic flow of a BEC is considered and the effect of increasing the size of the obstacle (i.e. the amplitude of the waves) is determined. Evidently, along with dispersion, nonlinear effects become equally important at finite distances from the obstacle, so that the Cherenkov cone breaks-up into a spatial structure consisting of one or several dark solitons. Such a structure represents the dispersive analog of the well-known steady spatial shock generated in the supersonic flow of a viscous compressible fluid past an obstacle. In this sense, it is the spatial counterpart of the one-dimensional expanding dispersive shock [2]-[5] generated in the evolution of large disturbances in a BEC. In the simplest case, the nonlinear wave structure would consist of a single spatial dark soliton given by a steady solution of the equations governing the BEC flow. Motivated by this physical consideration and the results of experiments [6], in this Letter we shall develop the theory of spatial dark solitons in the framework of the Gross-Pitaevskii (GP) mean field approach.

The theory of optical dispersive shocks generated in the propagation of light beams through photorefractive media is developed. A full one-dimensional analytical theory based on the Whitham modulation approach is given for the simplest case of a sharp steplike initial discontinuity in a beam with one-dimensional striplike geometry. This approach is confirmed by numerical simulations, which are extended also to beams with cylindrical symmetry. The theory explains recent experiments where such dispersive shock waves have been observed.

We consider formation of dissipationless shock waves in Bose-Einstein condensates with repulsive interaction between atoms. It is shown that for big enough initial inhomogeneity of density, interplay of nonlinear and dispersion effects leads to wave breaking phenomenon followed by generation of a train of dark solitons. Analytical theory is confirmed by numerical simulations. Experiments on free expansion of Bose-Einstein condensate (BEC) have shown [1] that evolution of large and smooth distributions of BEC is described very well by hydrodynamic approximation [2] where dispersion and dissipation effects are neglected. At the same time, it is well known from classical compressible gas dynamics (see, e.g., [3]) that typical initial distributions of density and velocity can lead to wave breaking phenomenon when formal solution of hydrodynamical equations becomes multivalued. It means that near the wave breaking point one cannot neglect dispersion and/or dissipation effects which prevent formation of a multivalued region of a solution. If the dissipation effects dominate the dispersion ones, then the multivalued region is replaced by the classical shock, i.e., narrow layer with strong dissipation within, which separates smooth regions with different values of density, fluid velocity and other physical parameters. This situation was studied in classical gas dynamics and found many practical applications. If, however, the dispersion effects dominate dissipation ones, then the region of strong oscillations is generated in the vicinity of the wave breaking point [4,5]. Observation of dark solitons in BEC [6][7][8] shows that the main role in dynamics of BEC is played by dispersion and nonlinear effects taken into account by the standard Gross-Pitaevskii (GP) equation [9], and dissipation effects are relatively small and can be considered as perturbation. Hence, there are initial distributions of BEC which can lead to formation of dissipationless shock waves. Here we shall consider such a possibility.The starting point of our consideration is the fact that the sound velocity in BEC is proportional to the square root from its density (see, e.g., [9] and references therein). Thus, if we create inhomogeneous BEC with high density hump (with density ϳ 1 ) in the center of lower density distribution (with density ϳ 0 ), and after that release this central part of BEC, then the high density hump will tend to expand with velocity ϳ ͱ 1 greater than the sound velocity ϳ ͱ 0 of propagation of disturbance in lower density BEC. As a result, wave breaking and formation of dissipationless shock wave can occur in this case. Note that initial distributions of this type were realized in experiment [10] on measurement of sound velocity in BEC and in the recent experiment [11]. In [10] the hump's density 1 was too small to generate shocks (see below). In experiment [11] generation of shock oscillations was apparently observed.The theory of dissipationless shock waves in media described by a one-dimensional (1D) nonlinear Schrödinger (NLS...

Recent experimental and theoretical advances in the creation and description of bright matter wave solitons are reviewed. Several aspects are taken into account, including the physics of soliton train formation as the nonlinear Fresnel diffraction, soliton-soliton interactions, and propagation in the presence of inhomogeneities. The generation of stable bright solitons by means of Feshbach resonance techniques is also discussed.