Almost global existence of small solutions to quadratic nonlinear Schördinger equations in three space dimensions.
Almost sure global well-posedness for the radial nonlinear Schrödinger equation on the unit ball II: the 3d case
We extend the convergence method introduced in our works [8–10] for almost sure global well-posedness of Gibbs measure evolutions of the nonlinear Schrödinger (NLS) and nonlinear wave (NLW) equations on the unit ball in to the case of the three dimensional NLS. This is the first probabilistic global well-posedness result for NLS with supercritical data on the unit ball in . The initial data is taken as a Gaussian random process lying in the support of the Gibbs measure associated to the equation,...
Almost sure well-posedness for the periodic 3D quintic nonlinear Schrödinger equation below the energy space
We also prove a long time existence result; more precisely we prove that for fixed there exists a set , such that any data , evolves up to time into a solution with , . In particular we find a nontrivial set of data which gives rise to long time solutions below the critical space , that is in the supercritical scaling regime.
An algorithm for quantum propagation through electron level crossings.
An application of semi-classical analysis to the asymptotic study of the supercooling field of a superconducting material
An inverse problem for a nonlinear Schrödinger equation.
Anisotropic inverse problems and Carleman estimates
This note reports on recent results on the anisotropic Calderón problem obtained in a joint work with Carlos E. Kenig, Mikko Salo and Gunther Uhlmann [8]. The approach is based on the construction of complex geometrical optics solutions to the Schrödinger equation involving phases introduced in the work [12] of Kenig, Sjöstrand and Uhlmann in the isotropic setting. We characterize those manifolds where the construction is possible, and give applications to uniqueness for the corresponding anisotropic...
Arbitrary growth and oscillation versus finite time blow up in nonlinear Schrödinger equations
Aspects of Long-time Behaviour of Solutions of Non-linear Hamiltonian Evolution Equations.
Asymptotic Analysis of a Schrödinger-Poisson System with Quantum Wells and Macroscopic Nonlinearities in Dimension 1
2000 Mathematics Subject Classification: 35Q02, 35Q05, 35Q10, 35B40.We consider the stationary one dimensional Schrödinger-Poisson system on a bounded interval with a background potential describing a quantum well. Using a partition function which forces the particles to remain in the quantum well, the limit h®0 in the nonlinear system leads to a uniquely solved nonlinear problem with concentrated particle density. It allows to conclude about the convergence of the solution.
Asymptotic behavior for a quadratic nonlinear Schrödinger equation.
Asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation
Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential
Asymptotics of solutions to Schrödinger equations with singular magnetic and electric potentials is investigated. By using a Almgren type monotonicity formula, separation of variables, and an iterative Brezis–Kato type procedure, we describe the exact behavior near the singularity of solutions to linear and semilinear (critical and subcritical) elliptic equations with an inverse square electric potential and a singular magnetic potential with a homogeneity of order −1.
Asymptotic behaviour for Schrödinger equations with a quadratic nonlinearity in one-space dimension.
Asymptotic expansion of small analytic solutions to the quadratic nonlinear Schrödinger equations in two-dimensional spaces.
Asymptotic stability of solitary waves for nonlinear Schrödinger equations
Attractors of a Schrödinger-Poisson system