Displaying similar documents to “On nonlinear Schrödinger equations”

Quasi-periodic solutions of PDEs

Massimiliano Berti (2011-2012)

Séminaire Laurent Schwartz — EDP et applications

Similarity:

The aim of this talk is to present some recent existence results about quasi-periodic solutions for PDEs like nonlinear wave and Schrödinger equations in 𝕋 d , d 2 , and the 1 - d derivative wave equation. The proofs are based on both Nash-Moser implicit function theorems and KAM theory.

Long time dynamics for the one dimensional non linear Schrödinger equation

Nicolas Burq, Laurent Thomann, Nikolay Tzvetkov (2013)

Annales de l’institut Fourier

Similarity:

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 L 2 critical...

Nonlinear Schrödinger equation on four-dimensional compact manifolds

Patrick Gérard, Vittoria Pierfelice (2010)

Bulletin de la Société Mathématique de France

Similarity:

We prove two new results about the Cauchy problem in the energy space for nonlinear Schrödinger equations on four-dimensional compact manifolds. The first one concerns global well-posedness for Hartree-type nonlinearities and includes approximations of cubic NLS on the sphere as a particular case. The second one provides, in the case of zonal data on the sphere, local well-posedness for quadratic nonlinearities as well as a necessary and sufficient condition of global well-posedness...

Bounds for KdV and the 1-d cubic NLS equation in rough function spaces

Herbert Koch (2011-2012)

Séminaire Laurent Schwartz — EDP et applications

Similarity:

We consider the cubic Nonlinear Schrödinger Equation (NLS) and the Korteweg-de Vries equation in one space dimension. We prove that the solutions of NLS satisfy a-priori local in time H s bounds in terms of the H s size of the initial data for s - 1 4 (joint work with D. Tataru, [, ]) , and the solutions to KdV satisfy global a priori estimate in H - 1 (joint work with T. Buckmaster []).