On nonlinear Schrödinger equations

Jean Bourgain

Displaying similar documents to “On nonlinear Schrödinger equations”

Construction of blowup solutions for the nonlinear Schrödinger equation with critical nonlinearity

Jean Bourgain, W. Wang (1997)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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KAM techniques in PDE

Ricardo Pérez-Marco (2001-2002)

Séminaire Bourbaki

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Quasi Periodic Solutions of Nonlinear Random Schrödinger Equations

Wei-Min Wang (2004-2005)

Séminaire Équations aux dérivées partielles

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Quasi-periodic solutions of PDEs

Massimiliano Berti (2011-2012)

Séminaire Laurent Schwartz — EDP et applications

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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 $𝕋^{d}$ , $d \geq 2$ , and the $1$ - $d$ derivative wave equation. The proofs are based on both Nash-Moser implicit function theorems and KAM theory.

Lyapunov center theorem for some nonlinear PDE's : a simple proof

Dario Bambusi (2000)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Blow-up phenomena for critical nonlinear Schrödinger and Zakharov equations.

Merle, Frank (1998)

Documenta Mathematica

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Long time dynamics for the one dimensional non linear Schrödinger equation

Nicolas Burq, Laurent Thomann, Nikolay Tzvetkov (2013)

Annales de l’institut Fourier

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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

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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

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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 $s \geq - \frac{1}{4}$ (joint work with D. Tataru, [, ]) , and the solutions to KdV satisfy global a priori estimate in $H^{- 1}$ (joint work with T. Buckmaster []).