Porquerolles

Program

The program of the workshop will consist in two 3 hours minicourses given by

J.-C.Saut (University Paris 11) and
J.Yngvason (Vienna University),

as well as 50 mn lectures given by the following speakers:

   R. Anton (Université Paris 6)
      V. Banica (Université d'Evry)
N. Berloff (University of Cambridge)
      L. Bronsard (Mac Master University)
   R. Carles (Université de Montpellier)
   D. Chiron (Université de Nice)
   I. Danaila (Université Paris 6)
   C. Gallo (McMaster University)
Ph. Gravejat (Université Paris 9)
   R. Jerrard (University of Toronto)
   K. Nakanishi (Kyoto University)
   Y. Pomeau (the Univeristy of Arizona)
S. Rica (Ecole Normale Supérieure, Paris)
   I.M. Sigal (University of Toronto)
   D. Smets (Université Paris 6)
   D. Spirn (University of Minnesota)

Titles and Abstracts

R.Anton (Université Paris 6)
Title : Global existence for Gross-Pitaevskii equation on three dimensional exterior domains.
Abstract : We prove global existence in the energy space for the Gross-Pitaevskii equation on exterior domains of dimension three. We use a Strichartz estimate adapted to the domain. This estimate follows from a semi-classical dispersive estimate combined with a smoothing effect.

N. Berloff (University of Cambridge)
Title : Spontaneous rotating vortex lattices in a pumped decaying condensate.
Abstract : Injection and decay of particles in an inhomogeneous quantum condensate can significantly change its behaviour. We model trapped, pumped, decaying condensates by a complex Gross-Pitaevskii equation and analyse the density and currents in the steady state. With homogeneous pumping, rotationally symmetric solutions are unstable. Stability may be restored by a finite pumping spot. However if the pumping spot is larger than the Thomas-Fermi cloud radius, then rotationally symmetric solutions are replaced by solutions with spontaneous arrays of vortices. These vortex arrays arise without any rotation of the trap, spontaneously breaking rotational symmetry. carles WKB analysis for the Gross-Pitaevskii equation with non-trivial boundary conditions at infinity.

V. Banica (Université d'Evry)
Title : Scattering for a Gross-Pitaevskii type equation.
Abstract : We prove scattering around the constant solution for a 1-d Gross-Pitaevskii equation with time dependent coefficients. This equation appears, via Hasimoto's transform, when considering selfsimilar solutions of the binormal flow equation. The binormal flow of curves is known to be connected by the local induction approximation with filamentary vortex dynamics in 3-d incompressible inviscid fluids. This is a joint work with Luis Vega.

L.Bronsard (Mc Master University)
Title : Global minimizers for anisotropic superconductors

Abstract : We study periodic minimizers of the Anisotropic Ginzburg-Landau and Lawrence-Doniach models for anisotropic superconductors, in various limiting regimes. We are particularly interested in determining the direction of the internal magnetic field (and vortex lattice) as a function of the applied external magnetic strength and its orientation with respect to the axes of anisotropy. We identify the corresponding lower critical fields, and compare the Lawrence-Doniach and anisotropic Ginzburg-Landau minimizers in the periodic setting. This talk represents joint work with S. Alama and E. Sandier.

R. Carles (Université de Montpellier)
Title : WKB analysis for the Gross-Pitaevskii equation with non-trivial boundary conditions at infinity.
Abstract : We consider the semi-classical limit for the Gross–Pitaevskii equation. In order to consider non-trivial boundary conditions at infinity, we work in Zhidkov spaces rather than in Sobolev spaces. For the usual cubic nonlinearity, we obtain a point-wise description of the wave function as the Planck constant goes to zero, so long as no singularity appears in the limit system. We also discuss the link with Madelung transform.

D.Chiron (Université de Nice)
Title : The KdV/KP-I limit of the Nonlinear Schrodinger Equation.
Abstract : We justify rigorously the convergence of the amplitude of solutions of Nonlinear-Schr\"odinger type Equations with non zero limit at infinity to an asymptotic regime governed by the Korteweg-de Vries equation in dimension 1 and the Kadomtsev-Petviashvili I equation in dimensions 2 and more. This is joint work with Frederic Rousset.

I.Danaïla (Université Paris 6)
Title : Vortex configurations in Bose-Einstein condensates : a 3D numerical approach.
Abstract : Vortex states of a rotating Bose-Einstein condensate are computed by imaginary time propagation of the three-dimensional Gross-Pitaevskii equation. To accurately resolve sharp gradients in presence of vortices we use sixth-order compact finite differences schemes. Three-dimensional numerical results correspond to experiments performed in LKB. A rich variety of vortex configurations (single-line vortex, Abrikosov lattice, giant vortex) is found using different trapping potentials. Configurations with arrays of condensates in optical lattices are also presented.

C. Gallo (McMaster university).
Title : Eigenvalues of a nonlinear ground state in the Thomas-Fermi approximation.

Ph. Gravejat (Université Paris 9).
Title : Transonic limit of the travelling waves for the two-dimensional Gross-Pitaevskii equation.
Abstract : We provide a rigorous mathematical derivation of the convergence in the long-wave transonic limit of the minimizing travelling waves for the two-dimensional Gross-Pitaevskii equation towards ground states for the Kadomtsev-Petviashvili equation.

R. Jerrard (University of Toronto).
Title : Semilinear wave equations and timelike minimal surfaces in the Minkowski space.

K. Nakanishi (Kyoto University).
Title : Scattering for the 3D Gross-Pitaevskii equation.
Abstract : We study large time asymptotic behavior for G-P in three spatial dimensions.
We prove that if the initial energy is small and localized then the solution disperses to the constant equilibrium according to the linearized equation. This also implies that every plane wave for the defocusing NLS is asymptotically stable.
The dispersive nature is much harder to analyze than the solutions around zero, because of (1) singular amplification of the zero frequency, and (2) strong interaction between parallel waves. We introduce quadratic transforms of the solution to remove the singularity (1), and a bilinear Fourier multiplier estimate to deal with the resonant interactions (2).
This is joint work with Stephen Gustafson and Tai-Peng Tsai.

Y. Pomeau (University of Arizona).
Title : Degenerate ground state in BEC and in superconductors.
Abstract : In general the ground state of quantum systems is not degenerate, although it would be interesting for many applications to have such a degeneracy, to store information for instance in a superposition of two ground state wave functions. I show that such a degenerate ground state may exist in conveniently shaped Bose Einstein condensates, as well in superconductors. .

S.Rica (Ecole Normale Supérieure, Paris).
Title : Coherent molecules and nanocrystal of impurities in Bose-Einstein condensates.
Abstract : I will discuss a work in progress with David Roberts at LANL, on the dynamics of a Bose-Einstein condensate with the presence of a number of impurities. I will present first the case of a single impurity, which in equations means a Gross-Pitaevskii equation (the condensate) coupled with a Schrödinger equation (the impurity). Then I will generalize to N different interacting impurities. Finally, I will present a phase diagram and show that an interesting situation will arise whenever the impurities form bound states, i.e. coherent molecules or coherent nanocrystals which can exhibit supersolid behavior.

J.-C.Saut (Université Paris 11).
Title : Recent progress and open problems on the Gross-Pitaevskii equation.

I.M.Sigal (University of Toronto).
Title : Dynamics of solitons in Gross-Pitaevskii equation.
Abstract : It is a common understanding in Physics that dynamics at a given scale originate from dynamics on a finer scale. In this talk I will demonstrate how this works for the case of dynamics of Bose gas in the mean-field regime. In this regime solutions of the many-body Schroedinger equation can be approximated by (products of) solutions of the Hartree or nonlinear Schroedinger (or Gross-Pitaevskii) equations with external potentials. In turn low energy solutions of the latter equations can be reduced to dynamics of rigidly moving well localized structures - solitons (ground states). I will review some recent results on and state open problems in this subject.

D. Smets (Université Paris 6).
Title : On the linear wave regime of the Gross-Pitaevskii equation.
Abstract : We study a long wave-length asymptotics for the Gross-Pitaevskii equation corresponding to perturbation of a constant state of modulus one. We exhibit lower bounds on the first occurence of possible zeros (vortices) and compare the solutions with the corresponding solutions to the linear wave equation or variants. This is a joint work with F. Bethuel and R. Danchin.

D.Spirn (University of Minnesota) .
Title : Dynamics of vortices in a damped Gross-Pitaevskii equation.
Abstract : We consider the dynamics of a Gross-Pitaevskii type equation of mixed type with both parabolic and Schrodinger terms. Under the limit of a large coupling constant, vortices condense down to points and satisfy a first order ODE. This is joint work with M. Kurzke, C. Melcher, and R. Moser.

J. Yngvason (Vienna University).
Title : The Gross-Pitaevskii equation and the quantum many-body problem.
Abstract : 1. Derivation of the (stationary) 3D Gross-Pitaevskii equation from the 3D quantum many-body problem.
2. Dimensional reduction, i.e., derivation of a low dimensional Gross-Pitaevskii equation from the 3D many body problem.
3. The Gross-Pitaevskii equation in rapidly rotating anharmonic traps.