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Dive into the research topics where Laurent Thomann is active.

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Featured researches published by Laurent Thomann.


Communications in Mathematical Physics | 2011

KAM for the Quantum Harmonic Oscillator

Benoît Grébert; Laurent Thomann

In this paper we prove an abstract KAM theorem for infinite dimensional Hamiltonians systems. This result extends previous works of S.B. Kuksin and J. Pöschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an application we show that some 1D nonlinear Schrödinger equations with harmonic potential admits many quasi-periodic solutions. In a second application we prove the reducibility of the 1D Schrödinger equations with the harmonic potential and a quasi periodic in time potential.


Nonlinearity | 2010

Gibbs measure for the periodic derivative nonlinear Schrödinger equation

Laurent Thomann; Nikolay Tzvetkov

In this paper we construct a Gibbs measure for the derivative Schrodinger equation on the circle. The construction uses some renormalisations of Gaussian series and Wiener chaos estimates, ideas which have already been used by the second author in a work on the Benjamin-Ono equation.


Annales De L Institut Henri Poincare-analyse Non Lineaire | 2012

Resonant dynamics for the quintic nonlinear Schrödinger equation

Benoît Grébert; Laurent Thomann

Abstract We consider the quintic nonlinear Schrodinger equation (NLS) on the circle i ∂ t u + ∂ x 2 u = ± ν | u | 4 u , ν ≪ 1 , x ∈ S 1 , t ∈ R . We prove that there exist solutions corresponding to an initial datum built on four Fourier modes which form a resonant set (see Definition 1.1), which have a nontrivial dynamic that involves periodic energy exchanges between the modes initially excited. It is noticeable that this nonlinear phenomenon does not depend on the choice of the resonant set. The dynamical result is obtained by calculating a resonant normal form up to order 10 of the Hamiltonian of the quintic NLS and then by isolating an effective term of order 6. Notice that this phenomenon cannot occur in the cubic NLS case for which the amplitudes of the Fourier modes are almost actions, i.e. they are almost constant.


Analysis & PDE | 2015

On the continuous resonant equation for NLS: II. Statistical study

Pierre Germain; Zaher Hani; Laurent Thomann

We consider the continuous resonant (CR) system of the 2D cubic nonlinear Schrodinger (NLS) equation. This system arises in numerous instances as an effective equation for the long-time dynamics of NLS in confined regimes (e.g. on a compact domain or with a trapping potential). The system was derived and studied from a deterministic viewpoint in several earlier works, which uncovered many of its striking properties. This manuscript is devoted to a probabilistic study of this system. Most notably, we construct global solutions in negative Sobolev spaces, which leave Gibbs and white noise measures invariant. Invariance of white noise measure seems particularly interesting in view of the absence of similar results for NLS.


Analysis & PDE | 2014

Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator

Aurélien Poiret; Didier Robert; Laurent Thomann

Thanks to an approach inspired from Burq-Lebeau \cite{bule}, we prove stochastic versions of Strichartz estimates for Schrodinger with harmonic potential. As a consequence, we show that the nonlinear Schrodinger equation with quadratic potential and any polynomial non-linearity is almost surely locally well-posed in


Nonlinearity | 2013

Beating effects in cubic Schrödinger systems and growth of Sobolev norms

Benoît Grébert; Eric Paturel; Laurent Thomann

L^{2}(\R^{d})


Archive for Rational Mechanics and Analysis | 2018

On the Cubic Lowest Landau Level Equation

Patrick Gérard; Pierre Germain; Laurent Thomann

for any


Asymptotic Analysis | 2010

A remark on the Schrödinger smoothing effect

Laurent Thomann

d\geq 2


arXiv: Analysis of PDEs | 2018

A pedestrian approach to the invariant Gibbs measures for the 2-d defocusing nonlinear Schrödinger equations

Tadahiro Oh; Laurent Thomann

. Then, we show that we can combine this result with the high-low frequency decomposition method of Bourgain to prove a.s. global well-posedness results for the cubic equation: when


Journal of Nonlinear Science | 2015

Stability of Large Periodic Solutions of Klein–Gordon Near a Homoclinic Orbit

Benoît Grébert; Tiphaine Jézéquel; Laurent Thomann

d=2

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Nikolay Tzvetkov

Institut Universitaire de France

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Nicolas Burq

University of Paris-Sud

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Pierre Germain

Courant Institute of Mathematical Sciences

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Zaher Hani

Georgia Institute of Technology

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Tadahiro Oh

University of Edinburgh

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Tiphaine Jézéquel

École normale supérieure de Cachan

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