
We study thinfilm limits of the full threedimensional GinzburgLandau model for a superconductor in an applied magnetic field oriented obliquely to the film surface. We obtain Γconvergence results in several regimes, determined by the asymptotic ratio between the magnitude of the parallel applied magnetic field and the thickness of the film. Depending on the regime, we show that there may be a decrease in the density of Cooper pairs. We also show that in the case of variable thickness of the film, its geometry will affect the effective applied magnetic field, thus influencing the position of vortices.
I will describe results on singularity (non)formation and stability, in the energycritical 2D setting, for some nonlinear Schroedingertype systems of geometric originthe Schroedinger map and LandauLifshitz equationwhich model dynamics of ferromagnets and liquid crystals.
In this work, we establish an appropriate 2D Strichartz type estimate for the linear wave equation set on a bounded domain with either Dirichlet or Neumann type boundary conditions. The proof follows BurqLebeauPlanchon work in 3D and solely based on spectral projection estimates due to Sogge. Our Strichart estimate enables us to solve the nonlinear problem with exponential nonlinearity. We define a trichotomy for the cauchy problem, prove the wellposedness on two sides of the trichotomy and a sort of instability on the the last side.
This is a joint work with R. Jrad.
Near absolute zero, a gas of quantum particles can condense into an unusual state of matter, called BoseEinstein condensation, that behaves like a giant quantum particle. It's only recently that we've been able to make the rigorous connection between the physics of the microscopic dynamics and the mathematics of the macroscopic model, the cubic nonlinear Schrodinger equation (NLS). I'll discuss joint work with Benjamin Schlein and Gigliola Staffilani on twodimensional cases of BoseEinstein condensationand the periodic case is especially interesting, because of techniques from analytic number theory and applications to quantum computing. As time permits, I'll also mention work in progress on computational quantum manybody systems and phase transitions for the invariant measures of the NLS.
I will present recent necessary and sufficient conditions for the existence of bifurcation points along ground state and excited state branches in nonlinear Schroedinger equations. The possible types of bifurcations and their effect on the dynamical stability of the bound states will also be discussed.
This is joint work with D. Pelinovsky (McMaster), P. Kevrekidis (U. Mass.) and V. Natarajan (U. of Illinois).
Consider the nonlinear magnetic Schrödinger equation for u : R^{3} ×R → C,

A class of semilinear elliptic equations, which describes timeindependent solutions to a degenerate parabolic equation modeling population dynamics, is studied. Under suitable assumptions all solutions are compactly supported, moreover, multiplicity of solutions is shown by the methods of variations.
Excited states of BoseEinstein condensates are considered in the ThomasFermi limit, in the framework of the GrossPitaevskii equation with repulsive interatomic interactions in a harmonic potential. The relative dynamics of dark solitons (density dips on the localized condensates) with respect to the harmonic potential and to each other is approximated using the averaged Lagrangian method and the LyapunovSchmidt reductions. This permits a complete characterization of the equilibrium positions of the dark solitons as a function of the chemical potential parameters. It also yields an analytical handle on the oscillation frequencies of dark solitons around such equilibria. The asymptotic predictions are generalized for an arbitrary number of dark solitons and are corroborated by numerical computations for two and threesoliton configurations.
A system of hydrodynamic type is a system of quasilinear firstorder PDEs; the quasilinear nature, remarkably and beautifully, allows us to study such systems using finitedimensional differentialgeometric methods. To say such a system is Hamiltonian is to say that it is composed of some Poisson bracket and some Hamiltonian function. The motivating question is: given a system of hydrodynamic type, how can we determine whether or not it is Hamiltonian? We certainly can't test all Poisson brackets and all Hamiltonian functions! I'll present a recent answer to this question for systems of hydrodynamic type with three equations.
Discrete solitons of the discrete nonlinear Schrödinger (dNLS) equation become compactly supported in the anticontinuum limit of the zero coupling between lattice sites. Eigenvalues of the linearization of the dNLS equation at the discrete soliton determine its spectral and linearized stability. All unstable eigenvalues of the discrete solitons near the anticontinuum limit were characterized earlier for this model. Here we analyze the resolvent operator and prove that it is uniformly bounded in the neighborhood of the continuous spectrum if the discrete soliton is simply connected in the anticontinuum limit. This result rules out existence of internal modes (neutrally stable eigenvalues of the discrete spectrum) of such discrete solitons near the anticontinuum limit.
We study the spectral properties of matrix Hamiltonians generated by linearizing nonlinear Schrödinger equations about soliton solutions. Using a hybrid analyticnumerical proof, we show that there are no embedded eigenvalues for the 3dimensional cubic nonlinearity, and other nonlinearities. Though we focus on the 3d cubic problem, the goal of this work is to present a new, robust, algorithm for verifying the spectral properties needed for stability analysis. We also present several cases for which our approach is inconclusive and speculate on ways to extend the method.
This is joint work with J. L. Marzuola (Columbia University).
In this talk we consider the "condensateintimevaryingbox" problem in one space dimension,

This is joint work with Karine Beauchard (Cachan) and Horst Lange (Cologne).
We consider the dynamic stability of pinned multivortex solutions to GinzburgLandau equations with external potential in R^{2}. For a sufficiently small external potential with widely spaced nondegenerate critical points, there exists a perturbed multivortex (pinned) solution whose vortex centers are near critical points of the potential. We show that multivortex solutions which are concentrated near local maxima of the potential are orbitally stable w.r.t. gradient and Hamiltonian dynamics.
Linear second order elliptic equation describing heat or mass diffusion and convection on a given velocity field is considered in R^{3}. The corresponding operator L may not satisfy the Fredholm property. In this case, solvability conditions for the equation L u = f are not known. In this work, we derive solvability conditions in H^{2}(R^{3}) for the non selfadjoint problem by relating it to a selfadjoint Schrödinger type operator, for which solvability conditions are obtained in our previous work.