Laurent Thomann
University of Nantes
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Featured researches published by Laurent Thomann.
Communications in Mathematical Physics | 2011
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
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
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
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
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
Benoît Grébert; Eric Paturel; Laurent Thomann
L^{2}(\R^{d})
Archive for Rational Mechanics and Analysis | 2018
Patrick Gérard; Pierre Germain; Laurent Thomann
for any
Asymptotic Analysis | 2010
Laurent Thomann
d\geq 2
arXiv: Analysis of PDEs | 2018
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
Benoît Grébert; Tiphaine Jézéquel; Laurent Thomann
d=2