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Sep 21, 2013
09/13

by
Laurent Thomann

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In this paper we consider the Schr\"odinger equation with power-like nonlinearity and confining potential or without potential. This equation is known to be well-posed with data in a Sobolev space $\H^{s}$ if $s$ is large enough and strongly ill-posed is $s$ is below some critical threshold $s_{c}$. Here we use the randomisation method of the inital conditions, introduced by N. Burq-N. Tzvetkov and we are able to show that the equation admits strong solutions for data in $\H^{s}$ for some...

Source: http://arxiv.org/abs/0901.4238v1

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49

Sep 23, 2013
09/13

by
Laurent Thomann

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In this paper we consider a Schrodinger equation on the circle with a quadratic nonlinearity. Thanks to an explicit computation of the first Picard iterate, we give a precision on the dynamic of the solution, whose existence was proved by C. E. Kenig, G. Ponce and L. Vega. We also show that the equation is well-posed in a space based on Lp norms in frequencies.

Source: http://arxiv.org/abs/0810.4691v1

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50

Sep 18, 2013
09/13

by
Laurent Thomann

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In this paper we consider supercritical nonlinear Schr\"odinger equations in an analytic Riemannian manifold $(M^d,g)$, where the metric $g$ is analytic. Using an analytic WKB method, we are able to construct an Ansatz for the semiclassical equation for times independent of the small parameter. These approximate solutions will help to show two different types of instabilities. The first is in the energy space, and the second is an immediate loss of regularity in higher Sobolev norms.

Source: http://arxiv.org/abs/0707.1785v1

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50

Sep 19, 2013
09/13

by
Laurent Thomann

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We prove the equivalence between the smoothing effect for a Schr\"odinger operator and the decay of the associate spectral projectors. We give two applications to the Schr\"odinger operator in dimension one.

Source: http://arxiv.org/abs/0910.1074v2

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112

Jul 20, 2013
07/13

by
Laurent Thomann

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Using variational methods, we construct approximate solutions for the Gross-Pitaevski equation which concentrate on circles in $\R^3$. These solutions will help to show that the $L^2$ flow is unstable for the usual topology and for the projective distance.

Source: http://arxiv.org/abs/math/0609807v3

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61

Jul 20, 2013
07/13

by
Laurent Thomann

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In this paper we are interested in constructing WKB approximations for the non linear cubic Schr\"odinger equation on a Riemannian surface which has a stable geodesic. These approximate solutions will lead to some instability properties of the equation.

Source: http://arxiv.org/abs/math/0609805v2

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3.0

Jun 28, 2018
06/18

by
Pierre Germain; Laurent Thomann

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We derive heuristically an integro-differential equation, as well as a shell model, governing the dynamics of the Lowest Landau Level equation in a high frequency regime.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1509.09080

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5.0

Jun 30, 2018
06/18

by
Tadahiro Oh; Laurent Thomann

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We consider the defocusing nonlinear wave equations (NLW) on the two-dimensional torus. In particular, we construct invariant Gibbs measures for the renormalized so-called Wick ordered NLW. We then prove weak universality of the Wick ordered NLW, showing that the Wick ordered NLW naturally appears as a suitable scaling limit of non-renormalized NLW with Gaussian random initial data.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1703.10452

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38

Sep 22, 2013
09/13

by
Emanuele Haus; Laurent Thomann

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We construct solutions to the quintic nonlinear Schr\"odinger equation on the circle with initial conditions supported on arbitrarily many different resonant clusters. This is a sequel of a work of Beno\^it Gr\'ebert and the second author.

Source: http://arxiv.org/abs/1210.7291v1

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Sep 22, 2013
09/13

by
Laurent Thomann; Nikolay Tzvetkov

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In this paper we construct a Gibbs measure for the derivative Schr\"odinger 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.

Source: http://arxiv.org/abs/1001.4269v3

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Sep 17, 2013
09/13

by
Benoît Grébert; Laurent Thomann

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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. P\"oschel and uses recent techniques of H. Eliasson and S.B. Kuksin. As an application we show that some 1D nonlinear Schr\"odinger equations with harmonic potential admits many quasi-periodic solutions. In a second application we prove the reducibility of the 1D Schr\"odinger equations with the harmonic potential and a quasi...

Source: http://arxiv.org/abs/1003.2793v2

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Jun 27, 2018
06/18

by
Frédéric Hérau; Laurent Thomann

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We prove a global existence result with initial data of low regularity, and prove the trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with small non linear term but with a possibly large exterior confining potential in dimension $d=2$ and $d=3$. The proof relies on a fixed point argument using sharp estimates (at short and long time scales) of the semi-group associated to the Fokker-Planck operator, which were obtained by the first author.

Topics: Analysis of PDEs, Mathematics

Source: http://arxiv.org/abs/1505.01698

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5.0

Jun 30, 2018
06/18

by
Zaher Hani; Laurent Thomann

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We consider the cubic nonlinear Schr\"odinger equation with harmonic trapping on $\mathbb{R}^D$ ($1\leq D\leq 5$). In the case when all but one directions are trapped (a.k.a "cigar-shaped" trap), following the approach of Hani-Pausader-Tzvetkov-Visciglia, we prove modified scattering and construct modified wave operators for small initial and final data respectively. The asymptotic behavior turns out to be a rather vigorous departure from linear scattering and is dictated by the...

Topics: Mathematics, Analysis of PDEs, Mathematical Physics

Source: http://arxiv.org/abs/1408.6213

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109

Sep 23, 2013
09/13

by
Benoît Grebert; Laurent Thomann

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We consider the quintic nonlinear Schr\"odinger equation (NLS) on the circle. We prove that there exist solutions corresponding to an initial datum built on four Fourier modes which form a resonant set, which have a non trivial dynamic that involves periodic energy exchanges between the modes initially excited. It is noticeable that this nonlinear phenomena does not depend on the choice of the resonant set. The dynamical result is obtained by calculating a resonant normal form up to order...

Source: http://arxiv.org/abs/1111.6827v1

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Jun 26, 2018
06/18

by
Benoît Grébert; Eric Paturel; Laurent Thomann

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We consider the cubic nonlinear Schr{\"o}dinger equation on the spatial domain $\mathbb{R}\times \mathbb{T}^d$, and we perturb it with a convolution potential. Using recent techniques of Hani-Pausader-Tzvetkov-Visciglia, we prove a modified scattering result and construct modified wave operators, under generic assumptions on the potential. In particular, this enables us to prove that the Sobolev norms of small solutions of this nonresonant cubic NLS are asymptotically constant.

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1502.07699

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52

Sep 18, 2013
09/13

by
Benoît Grébert; Tiphaine Jézéquel; Laurent Thomann

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We consider the Klein-Gordon equation on a Riemannian surface which is globally well-posed in the energy space. This equation has an homoclinic orbit to the origin, and in this paper we study the dynamics close to it. Using a strategy from Groves-Schneider, we show that there are many solutions which stay close to this homocline during all times. We point out that the solutions we construct are not small.

Source: http://arxiv.org/abs/1211.4155v2

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Jun 26, 2018
06/18

by
Pierre Germain; Zaher Hani; Laurent Thomann

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We study the continuous resonant (CR) equation which was derived by Faou-Germain-Hani as the large-box limit of the cubic nonlinear Schr\"odinger equation in the small nonlinearity (or small data) regime. We first show that the system arises in another natural way, as it also corresponds to the resonant cubic Hermite-Schr\"odinger equation (NLS with harmonic trapping). We then establish that the basis of special Hermite functions is well suited to its analysis, and uncover more of the...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1501.03760

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4.0

Jun 30, 2018
06/18

by
Nicolas Burq; Laurent Thomann; Nikolay Tzvetkov

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We show, by the means of several examples, how we can use Gibbs measures to construct global solutions to dispersive equations at low regularity. The construction relies on the Prokhorov compactness theorem combined with the Skorokhod convergence theorem. To begin with, we consider the non linear Schr\"odinger equation (NLS) on the tri-dimensional sphere. Then we focus on the Benjamin-Ono equation and on the derivative nonlinear Schr\"odinger equation on the circle. Next, we construct...

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1412.7499

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5.0

Jun 30, 2018
06/18

by
Rafik Imekraz; Didier Robert; Laurent Thomann

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We study integrability and continuity properties of random series of Hermite functions. We get optimal results which are analogues to classical results concerning Fourier series, like the Paley-Zygmund or the Salem-Zygmund theorems. We also consider the case of series of radial Hermite functions, which are not so well-behaved. In this context, we prove some L^p bounds of radial Hermite functions, which are optimal when p is large.

Topics: Mathematics, Spectral Theory, Analysis of PDEs, Classical Analysis and ODEs

Source: http://arxiv.org/abs/1403.4913

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19

Jun 28, 2018
06/18

by
Tadahiro Oh; Geordie Richards; Laurent Thomann

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We consider the defocusing generalized KdV equations on the circle. In particular, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure. In handling a nonlinearity of an arbitrary high degree, we make use of the Hermite polynomials and the white noise functional.

Topics: Analysis of PDEs, Mathematics, Probability

Source: http://arxiv.org/abs/1509.06873

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58

Sep 20, 2013
09/13

by
Nicolas Burq; Laurent Thomann; Nikolay Tzvetkov

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In this article, we first present the construction of Gibbs measures associated to nonlinear Schr\"odinger 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 $L^2$...

Source: http://arxiv.org/abs/1002.4054v1

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75

Sep 22, 2013
09/13

by
Nicolas Burq; Laurent Thomann; Nikolay Tzvetkov

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We prove the existence of infinite energy global solutions of the cubic wave equation in dimension greater than 3. The data is a typical element on the support of suitable probability measures.

Source: http://arxiv.org/abs/1210.2086v1

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19

Jun 26, 2018
06/18

by
Pierre Germain; Zaher Hani; Laurent Thomann

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We consider the continuous resonant (CR) system of the 2D cubic nonlinear Schr{\"o}dinger (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....

Topics: Mathematics, Analysis of PDEs

Source: http://arxiv.org/abs/1502.05643

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55

Sep 22, 2013
09/13

by
Pierre Albin; Hans Christianson; Jeremy L. Marzuola; Laurent Thomann

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We consider the nonlinear Schr\"odinger equation on a compact manifold near an elliptic periodic geodesic. Using a geometric optics construction, we construct quasimodes to a nonlinear stationary problem which are highly localized near the periodic geodesic. We show the nonlinear Schr\"odinger evolution of such a quasimode remains localized near the geodesic, at least for short times.

Source: http://arxiv.org/abs/1103.3253v1