We solve the time-dependent Gross-Pitaevskii equation by a variational ansatz to calculate the excitation spectrum of a Bose-Einstein condensate in a trap. The trial wave function is a Gaussian which allows an essentially analytical treatment of the problem. Our results reproduce numerical calculations over the whole range from small to large particle numbers, and agree exactly with the Stringari results in the strong interaction limit. Excellent agreement is obtained with the recent JILA experiment and predictions for the negative scattering length case are also made.

We obtain analytic solutions to the Gross-Pitaevskii equation with negative scattering length in highly asymmetric traps. We find that in these traps the Bose-Einstein condensates behave like quasiparticles and do not expand when the trapping in one direction is eliminated. The results can be applicable to the control of the motion of Bose-Einstein condensates.

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Using Lie group theory and canonical transformations we construct explicit solutions of nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities. We present the general theory, use it to show that localized nonlinearities can support bound states with an arbitrary number solitons and discuss other applications of interest to the field of nonlinear matter waves. Introduction.-Solitons are self-localized nonlinear waves which are sustained by an equilibrium between dispersion and nonlinearity and appear in a great variety of physical contexts [1]. In particular, these nonlinear structures have been generated recently in ultracold atomic bosonic gases cooled down below the Bose-Einstein transition temperature [2,3,4]. In those systems the effective nonlinear interactions are a result of the elastic two-body collisions between the condensed atoms.These interactions can be controlled by the so-called Feschbach resonance (FR) management [5], which has been used to generate bright solitons [3,6], induce collapse [7], etc. Recently, the control in time of the condensate scattering length has been the basis for many theoretical proposals to obtain different types of nonlinear structures such as periodic waves [8], shock waves [9], stabilized solitons [10], etc.Interactions can also be made spatially dependent by acting on the spatial dependence of either the magnetic field or the laser intensity (in the case of optical control of FR [11]) acting on the Feschbach resonances. This possibility has motivated in the last years a strong theoretical interest on nonlinear phenomena in Bose-Eintein condensates (BECs) with spatially inhomogeneous interactions. Several phenomena have been studied in quasi-one dimensional scenarios such as the emission of solitons [12] and the dynamics of solitons when the space modulation of the nonlinearity is a random [14], linear [15], periodic [16], or localized function [17]. The existence and stability of solutions has been studied in Ref. [18].In this paper we construct general classes of nonlinearity modulations and external potentials for which explicit solutions can be constructed. To do so we will use Lie group theory and canonical tranformations connecting problems with inhomogeneous nonlinearities with simpler ones having an homogeneous nonlinearity. We will show that localized nonlinearities can support bound

We study an example of exact parametric resonance in a extended system ruled by nonlinear partial differential equations of nonlinear Schrödinger type.It is also conjectured how related models not exactly solvable should behave in the same way. The results have applicability in recent experiments in Bose-Einstein condensation and to classical problems in Nonlinear Optics.

Using similarity transformations we construct explicit nontrivial solutions of nonlinear Schrödinger equations with potentials and nonlinearities depending both on time and on the spatial coordinates. We present the general theory and use it to calculate explicitly nontrivial solutions such as periodic (breathers), resonant, or quasiperiodically oscillating solitons. Some implications to the field of matter waves are also discussed.

We demonstrate, through numerical simulations, the controllable emission of matter-wave bursts from a Bose-Einstein condensate in a shallow optical dipole trap. The process is triggered by spatial variations of the scattering length along the trapping axis. In our approach, the outcoupling mechanisms are atom-atom interactions and thus, the trap remains unaltered. Once emitted, the matter wave forms a robust soliton. We calculate analytically the parameters for the experimental implementation of these matter-wave bursts of solitons.

We find a new type of optical vector soliton that originates from trapping of a dipole mode by the soliton-induced waveguides. These solitons, which appear as a consequence of the vector nature of the two-component system, are more stable than the previously found optical vortex solitons and represent a new type of extremely robust nonlinear vector structure.