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.

We study the dynamics of small vortex clusters with few (2-4) co-rotating vortices in Bose-Einstein condensates by means of experiments, numerical computations, and theoretical analysis. All of these approaches corroborate the counter-intuitive presence of a dynamical instability of symmetric vortex configurations. The instability arises as a pitchfork bifurcation at sufficiently large values of the angular momentum that induces the emergence and stabilization of asymmetric rotating vortex configurations. The latter are quantified in the theoretical model and observed in the experiments. The dynamics is explored both for the integrable two-vortex system, where a reduction of the phase space of the system provides valuable insight, as well as for the non-integrable three-(or more) vortex case, which additionally admits the possibility of chaotic trajectories.

No abstract

A quantized vortex dipole is the simplest vortex molecule, comprising two countercirculating vortex lines in a superfluid. Although vortex dipoles are endemic in two-dimensional superfluids, the precise details of their dynamics have remained largely unexplored. We present here several striking observations of vortex dipoles in dilute-gas Bose-Einstein condensates, and develop a vortex-particle model that generates vortex line trajectories that are in good agreement with the experimental data. Interestingly, these diverse trajectories exhibit essentially identical quasiperiodic behavior, in which the vortex lines undergo stable epicyclic orbits.