Abstract:
The aim of this paper is to perform a numerical and analytical study of a rotating Bose Einstein condensate placed in a harmonic plus Gaussian trap, following the experiments of \cite{bssd}. The rotational frequency $\Omega$ has to stay below the trapping frequency of the harmonic potential and we find that the condensate has an annular shape containing a triangular vortex lattice. As $\Omega$ approaches $\omega$, the width of the condensate and the circulation inside the central hole get large. We are able to provide analytical estimates of the size of the condensate and the circulation both in the lowest Landau level limit and the Thomas-Fermi limit, providing an analysis that is consistent with experiment.

Abstract:
We consider a rotating Bose-Einstein condensate confined in combined harmonic and quartic traps, following recent experiments [V. Bretin, S. Stock, Y. Seurin and J. Dalibard, cond-mat/0307464]. We investigate numerically the behavior of the wave function which solves the three-dimensional Gross Pitaevskii equation. When the harmonic part of the potential is dominant, as the angular velocities $Omega$ increases, the vortex lattice evolves into a giant vortex. We also investigate a case not covered by the experiments or the previous numerical works: for strong quartic potentials, the giant vortex is obtained for lower $Omega$, before the lattice is formed. We analyze in detail the three dimensional structure of vortices.

Abstract:
For a Bose-Einstein condensate placed in a rotating trap and confined in the z axis, we set a framework of study for the Gross-Pitaevskii energy in the Thomas Fermi regime. We investigate an asymptotic development of the energy, the critical velocities of nucleation of vortices with respect to a small parameter $\ep$ and the location of vortices. The limit $\ep$ going to zero corresponds to the Thomas Fermi regime. The non-dimensionalized energy is similar to the Ginzburg-Landau energy for superconductors in the high-kappa high-field limit and our estimates rely on techniques developed for this latter problem. We also take the advantage of this similarity to develop a numerical algorithm for computing the Bose-Einstein vortices. Numerical results and energy diagrams are presented.

Abstract:
For a Bose-Einstein condensate placed in a rotating trap, we give a simplified expression of the Gross-Pitaevskii energy in the Thomas Fermi regime, which only depends on the number and shape of the vortex lines. Then we check numerically that when there is one vortex line, our simplified expression leads to solutions with a bent vortex for a range of rotationnal velocities and trap parameters which are consistent with the experiments.

Abstract:
We consider a rotating Bose-Einstein condensate in a harmonic trap and investigate numerically the behavior of the wave function which solves the Gross Pitaevskii equation. Following recent experiments [Rosenbuch et al, Phys. Rev. Lett., 89, 200403 (2002)], we study in detail the line of a single quantized vortex, which has a U or S shape. We find that a single vortex can lie only in the x-z or y-z plane. S type vortices exist for all values of the angular velocity Omega while U vortices exist for Omega sufficiently large. We compute the energy of the various configurations with several vortices and study the three-dimensional structure of vortices.

Abstract:
In this paper, we study the Ginzburg-Landau equations for a two dimensional domain which has small size. We prove that if the domain is small, then the solution has no zero, that is no vortex. More precisely, we show that the order parameter $\Psi$ is almost constant. Additionnally, we obtain that if the domain is a disc of small radius, then any non normal solution is symmetric and unique. Then, in the case of a slab, that is a one dimensional domain, we use the same method to derive that solutions are symmetric. The proofs use a priori estimates and the Poincar\'e inequality.

Abstract:
In this paper, we provide the different types of bifurcation diagrams for a superconducting cylinder placed in a magnetic field along the direction of the axis of the cylinder. The computation is based on the numerical solutions of the Ginzburg-Landau model by the finite element method. The response of the material depends on the values of the exterior field, the Ginzburg-Landau parameter and the size of the domain. The solution branches in the different regions of the bifurcation diagrams are analyzed and open mathematical problems are mentioned.

Abstract:
We classify the ground states and topological defects of a rotating two-component condensate when varying several parameters: the intracomponent coupling strengths, the intercomponent coupling strength and the particle numbers.No restriction is placed on the masses or trapping frequencies of the individual components. We present numerical phase diagrams which show the boundaries between the regions of coexistence, spatial separation and symmetry breaking. Defects such as triangular coreless vortex lattices, square coreless vortex lattices and giant skyrmions are classified. Various aspects of the phase diagrams are analytically justified thanks to a non-linear $\sigma$ model that describes the condensate in terms of the total density and a pseudo-spin representation.

Abstract:
We provide complete phase diagrams describing the ground state of a trapped spinor BEC under the combined effects of rotation and a Rashba spin-orbit coupling. The interplay between the different parameters (magnitude of rotation, strength of the spin-orbit coupling and interaction) leads to a rich ground state physics that we classify. We explain some features analytically in the Thomas-Fermi approximation, writing the problem in terms of the total density, total phase and spin. In particular, we analyze the giant skyrmion, and find that it is of degree 1 in the strong segregation case. In some regions of the phase diagrams, we relate the patterns to a ferromagnetic energy.

Abstract:
This paper deals with the study of the behaviour of the wave functions of a two-component Bose-Einstein condensate near the interface, in the case of strong segregation. This yields a system of two coupled ODE's for which we want to have estimates on the asymptotic behaviour, as the strength of the coupling tends to infinity. As in phase separation models, the leading order profile is a hyperbolic tangent. We construct an inner and outer solution that we match at a point using the fact that the Hamiltonian of the system is constant. Then, we use the properties of the associated linearized operator to perturb the constructed approximate solution into a genuine one for which we have an asymptotic expansion. Furthermore, we prove that the constructed heteroclinic solutions are linearly nondegenerate, in the natural sense, and that there is a spectral gap, independent of the large interaction parameter, between the zero eigenvalue (due to translations) at the bottom of the spectrum and the rest of the spectrum.