-deformed KP hierarchy and -deformed constrained KP hierarchy.
Quantum Euler-Poisson systems: Existence of stationary states
A one-dimensional quantum Euler-Poisson system for semiconductors for the electron density and the electrostatic potential in bounded intervals is considered. The existence and uniqueness of strong solutions with positive electron density is shown for quite general (possibly non-convex or non-monotone) pressure-density functions under a “subsonic” condition, i.e. assuming sufficiently small current densities. The proof is based on a reformulation of the dispersive third-order equation for the electron...
Quantum integrable 1D anyonic models: construction through braided Yang-Baxter equation.
Quasi-periodic solutions of Hamiltonian PDEs
We overview recent existence results and techniques about KAM theory for PDEs.
Quasi-periodic solutions of nonlinear random Schrödinger equations
Quasi-periodic solutions with Sobolev regularity of NLS on with a multiplicative potential
We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on , finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are then the solutions are . The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators...
Random data Cauchy problem for supercritical Schrödinger equations
Recent progress on the blow-up problem of Zakharov equations
Recent Results on the Cauchy Problem for Focusing and Defocusing Gross-Pitaevskii Hierarchies
In this paper, we review some of our recent results in the study of the dynamics of interacting Bose gases in the Gross-Pitaevskii (GP) limit. Our investigations focus on the well-posedness of the associated Cauchy problem for the infinite particle system described by the GP hierarchy.
Reductions of multicomponent mKdV equations on symmetric spaces of DIII-type.
Regularity bounds on Zakharov system evolutions.
Remarks on global existence and compactness for solutions in the critical nonlinear schrödinger equation in 2D
In the talk we shall present some recent results obtained with F. Merle about compactness of blow up solutions of the critical nonlinear Schrödinger equation for initial data in . They are based on and are complementary to some previous work of J. Bourgain about the concentration of the solution when it approaches to the blow up time.
Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain
We concentrate on the analysis of the critical mass blowing-up solutions for the cubic focusing Schrödinger equation with Dirichlet boundary conditions, posed on a plane domain. We bound from below the blow-up rate for bounded and unbounded domains. If the blow-up occurs on the boundary, the blow-up rate is proved to grow faster than , the expected one. Moreover, we state that blow-up cannot occur on the boundary, under certain geometric conditions on the domain.
Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain
In this paper we concentrate on the analysis of the critical mass blowing-up solutions for the cubic focusing Schrödinger equation with Dirichlet boundary conditions, posed on a plane domain. We bound the blow-up rate from below, for bounded and unbounded domains. If the blow-up occurs on the boundary, the blow-up rate is proved to grow faster than , the expected one. Moreover, we show that blow-up cannot occur on the boundary, under certain geometric conditions on the domain.
Remarques sur l'analyse semi-classique de l'équation de Schrödinger non linéaire
Robust control problems of vortex dynamics in superconducting films with Ginzburg-Landau complex systems.
Rôle des oscillations quadratiques dans des équations de Schrödinger non linéaire
Scattering amplitude for the Schrödinger equation with strong magnetic field
In this note, we study the scattering amplitude for the Schrödinger equation with constant magnetic field. We consider the case where the strengh of the magnetic field goes to infinity and we discuss the competition between the magnetic and the electrostatic effects.
Scattering for 1D cubic NLS and singular vortex dynamics
We study the stability of self-similar solutions of the binormal flow, which is a model for the dynamics of vortex filaments in fluids and super-fluids. These particular solutions form a family of evolving regular curves in that develop a singularity in finite time, indexed by a parameter . We consider curves that are small regular perturbations of for a fixed time . In particular, their curvature is not vanishing at infinity, so we are not in the context of known results of local existence...
Self-similar solutions and Besov spaces for semi-linear Schrödinger and wave equations
We prove that the initial value problem for the semi-linear Schrödinger and wave equations is well-posed in the Besov space , when the nonlinearity is of type , for . This allows us to obtain self-similar solutions, as well as to recover previously known results for the solutions under weaker smallness assumptions on the data.