Les équations d’Euler, des ondes et de Korteweg-de Vries comme limites asymptotiques de l’équation de Gross-Pitaevskii
Limitations on the control of Schrödinger equations
We give the definitions of exact and approximate controllability for linear and nonlinear Schrödinger equations, review fundamental criteria for controllability and revisit a classical “No-go” result for evolution equations due to Ball, Marsden and Slemrod. In Section 2 we prove corresponding results on non-controllability for the linear Schrödinger equation and distributed additive control, and we show that the Hartree equation of quantum chemistry with bilinear control is not controllable...
Local existence theory for the generalized Schrödinger equation
Local ill-posedness of the 1D Zakharov system.
Local minimizers with vortex filaments for a Gross-Pitaevsky functional
This paper gives a rigorous derivation of a functional proposed by Aftalion and Rivière [Phys. Rev. A64 (2001) 043611] to characterize the energy of vortex filaments in a rotationally forced Bose-Einstein condensate. This functional is derived as a Γ-limit of scaled versions of the Gross-Pitaevsky functional for the wave function of such a condensate. In most situations, the vortex filament energy functional is either unbounded below or has only trivial minimizers, but we establish the existence...
Local properties of stationary solutions of some nonlinear singular Schrödinger equations.
We study the local behaviour of solutions of the following type of equation,-Δu - V(x)u + g(u) = 0 when V is singular at some points and g is a non-decreasing function. Emphasis is put on the case when V(x) = c|x|-2 and g has a power-like growth.
Local well posedness of nonlinear Schrödinger equations
Local well-posedness for a higher order nonlinear Schrödinger equation in Sobolev spaces of negative indices.
Local Well-Posedness for Higher Order Nonlinear Dispersive Systems.
Long time dynamics for the one dimensional non linear Schrödinger equation
In this article, we first present the construction of Gibbs measures associated to nonlinear Schrödinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures). Finally, we prove that the Gibbs measures are indeed invariant by the flow of the equation. As a byproduct of our analysis, we give a global well-posedness and scattering result for the critical and super-critical...
Long-time asymptotics for the nonlinear heat equation with a fractional Laplacian in a ball
The nonlinear heat equation with a fractional Laplacian , is considered in a unit ball . Homogeneous boundary conditions and small initial conditions are examined. For 3/2 + ε₁ ≤ α ≤ 2, where ε₁ > 0 is small, the global-in-time mild solution from the space with κ < α - 1/2 is constructed in the form of an eigenfunction expansion series. The uniqueness is proved for 0 < κ < α - 1/2, and the higher-order long-time asymptotics is calculated.
Lyapunov control of a quantum particle in a decaying potential
Magnetic bottles for the Neumann problem : curvature effects in the case of dimension 3 (general case)
Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint
This paper is devoted to an analysis of vortex-nucleation for a Ginzburg-Landau functional with discontinuous constraint. This functional has been proposed as a model for vortex-pinning, and usually accounts for the energy resulting from the interface of two superconductors. The critical applied magnetic field for vortex nucleation is estimated in the London singular limit, and as a by-product, results concerning vortex-pinning and boundary conditions on the interface are obtained.
Mathematical models for laser-plasma interaction
We address here mathematical models related to the Laser-Plasma Interaction. After a simplified introduction to the physical background concerning the modelling of the laser propagation and its interaction with a plasma, we recall some classical results about the geometrical optics in plasmas. Then we deal with the well known paraxial approximation of the solution of the Maxwell equation; we state a coupling model between the plasma hydrodynamics and the laser propagation. Lastly, we consider the...
Mathematical models for laser-plasma interaction
We address here mathematical models related to the Laser-Plasma Interaction. After a simplified introduction to the physical background concerning the modelling of the laser propagation and its interaction with a plasma, we recall some classical results about the geometrical optics in plasmas. Then we deal with the well known paraxial approximation of the solution of the Maxwell equation; we state a coupling model between the plasma hydrodynamics and the laser propagation. Lastly, we consider the...
Maximizers for the Strichartz and the Sobolev-Strichartz inequalities for the Schrödinger equation.
Mean-field evolution of fermionic systems
We study the dynamics of interacting fermionic systems, in the mean-field regime. We consider initial states which are close to quasi-free states and prove that, under suitable assumptions on the inital data and on the many-body interaction, the quantum evolution of the system is approximated by a time-dependent quasi-free state. In particular we prove that the evolution of the reduced one-particle density matrix converges, as the number of particles goes to infinity, to the solution of the time-dependent...
Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation
We discuss the Hartree equation arising in the mean-field limit of large systems of bosons and explain its importance within the class of nonlinear Schrödinger equations. Of special interest to us is the Hartree equation with focusing nonlinearity (attractive two-body interactions). Rigorous results for the Hartree equation are presented concerning: 1) its derivation from the quantum theory of large systems of bosons, 2) existence and stability of Hartree solitons, and 3) its point-particle (Newtonian)...
Mechanical analogy for the wave-particle: Helix on a vortex filament.