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The cubic Szegö equation has been introduced as a toy model for totally non dispersive evolution equations. It turned out that it is a complete integrable Hamiltonian system for which we built a non linear Fourier transform giving an explicit expression of the solutions.
This explicit formula allows to study the dynamics of the solutions. We will explain different aspects of it: almost-periodicity of the solutions in the energy space, uniform analyticity for a large set of initial data, turbulence phenomenon for a dense set of smooth initial data in large Sobolev spaces.
From joint works with Patrick Gérard.[-]

The cubic Szegö equation has been introduced as a toy model for totally non dispersive evolution equations. It turned out that it is a complete integrable Hamiltonian system for which we built a non linear Fourier transform giving an explicit expression of the solutions.
This explicit formula allows to study the dynamics of the solutions. We will explain different aspects of it: almost-periodicity of the solutions in the energy space, uniform ...[+]

35B40 ; 35B15 ; 35Q55 ; 37K15 ; 47B35

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To illustrate specifically quantum behaviours, the talk will consider three typical problems for non-linear kinetic models evolving through pair collisions at temperatures not far from absolute zero. Based on those examples, a number of differences between quantum and classical Boltzmann theory is discussed in more general term.

82D50 ; 76Y05 ; 82D30 ; 35Q60 ; 35Q55

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We will discuss the convergence (in the semiclassical limit) of a solution to the Hartree-Fock equation towards an operator, whose Wigner transform is a solution to the Vlasov equation. We will consider both cases of positive and zero temperature. The results we will present are part of a project in collaboration with N. Benedikter, M. Porta and B. Schlein.

82C22 ; 82C10 ; 35Q40 ; 35Q55

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We consider the nonlinear Schrödinger equation in the partially periodic setting $\mathbb{R}^d\times \mathbb{T}$. We present some recent results obtained in collaboration with N. Tzvetkov concerning the Cauchy theory and the long-time behavior of the solutions.

nonlinear Schrödinger equation - Cauchy theory - scattering

35Q55 ; 35B40 ; 35P25

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This talk is devoted to the study of the following inverse boundary value problem: given a Riemannian manifold with boundary determine the magnetic potential in a dynamical Schrödinger equation in a magnetic field from the observations made at the boundary.

inverse problem - Schrödinger equation - magnetic field

35R30 ; 35Q55 ; 35R35 ; 35Q60

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Anyons are by definition particles with quantum statistics different from those of bosons and fermions. They can occur only in low dimensions, 2D being the most relevant case for this talk. They have hitherto remained hypothetical, but there is good theoretical evidence that certain quasi-particles occuring in quantum Hall physics should behave as anyons.

I shall consider the case of tracer particles immersed in a so-called Laughlin liquid. I will argue that, under certain circumstances, these become anyons. This is made manifest by the emergence of a particular effective Hamiltonian for their motion. The latter is notoriously hard to solve even in simple cases, and well-controled simplifications are highly desirable. I will discuss a possible mean-field approximation, leading to a one-particle energy functional with self-consistent magnetic field.[-]

Anyons are by definition particles with quantum statistics different from those of bosons and fermions. They can occur only in low dimensions, 2D being the most relevant case for this talk. They have hitherto remained hypothetical, but there is good theoretical evidence that certain quasi-particles occuring in quantum Hall physics should behave as anyons.

I shall consider the case of tracer particles immersed in a so-called Laughlin liquid. I ...[+]

82B10 ; 81S05 ; 35P15 ; 35Q40 ; 35Q55 ; 81V70

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We will review in this talk some mathematical results concerning stochastic models used by physicist to describe BEC in the presence of fluctuations (that may arise from inhomogeneities in the confinement parameters), or BEC at finite temperature. The results describe the effect of those fluctuations on the structures - e.g. vortices - which are present in the deterministic model, or the convergence to equilibrium in the models at finite temperature. We will also describe the numerical methods which have been developed for those models in the framework of the ANR project Becasim. These are joint works with Reika Fukuizumi, Arnaud Debussche, and Romain Poncet.[-]

We will review in this talk some mathematical results concerning stochastic models used by physicist to describe BEC in the presence of fluctuations (that may arise from inhomogeneities in the confinement parameters), or BEC at finite temperature. The results describe the effect of those fluctuations on the structures - e.g. vortices - which are present in the deterministic model, or the convergence to equilibrium in the models at finite ...[+]

35Q55 ; 60H15 ; 65M06

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The leapfrogging is the name given to a regime of interaction between vortex rings with the same axis of symmetry in incompressible fluids. We will explain where it comes from and indicate a rigorous derivation in the case of the axisymmetric Gross-Pitaevskii equation.

35Q55 ; 35Q56 ; 76B47

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In spite of enormous success of the theory of integrable systems, at least three important problems are not resolved yet or are resolved only partly. They are the following:
1. The IST in the case of arbitrary bounded initial data.
2. The statistical description of the systems integrable by the IST. Albeit, the development of the theory of integrable turbulence.
3. Integrability of the deep water equations.
These three problems will be discussed in the talk.[-]

In spite of enormous success of the theory of integrable systems, at least three important problems are not resolved yet or are resolved only partly. They are the following:
1. The IST in the case of arbitrary bounded initial data.
2. The statistical description of the systems integrable by the IST. Albeit, the development of the theory of integrable turbulence.
3. Integrability of the deep water equations.
These three problems will be discussed ...[+]

37K10 ; 35C07 ; 35C08 ; 35Q53 ; 35Q55 ; 76B15 ; 76Fxx

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We consider the Derivative Nonlinear Schrödinger equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities). We prove global wellposedness and give a full description of the long-time behavior of the solutions in the form of a finite sum of localized solitons and a dispersive component. Our analysis provides explicit formulae for the multi-soliton component as well as the correction dispersive term. We use the inverse scattering approach and the nonlinear steepest descent method of Deift and Zhou (1993) revisited by the $\bar{\partial}$-analysis of Dieng-McLaughlin (2008) and complemented by the recent work of Borghese-Jenkins-McLaughlin (2016) on soliton resolution for the focusing nonlinear Schrödinger equation. This is a joint work with R. Jenkins, J. Liu and P. Perry.[-]

We consider the Derivative Nonlinear Schrödinger equation for general initial conditions in weighted Sobolev spaces that can support bright solitons (but exclude spectral singularities). We prove global wellposedness and give a full description of the long-time behavior of the solutions in the form of a finite sum of localized solitons and a dispersive component. Our analysis provides explicit formulae for the multi-soliton component as well as ...[+]

35Q55 ; 37K15 ; 37K40 ; 35P25 ; 35A01

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Documents 35Q55 20 résultats

Various aspects of the dynamics of the cubic Szegö solutions - Grellier, Sandrine (Auteur de la Conférence) | H Nouveau

On low temperature kinetic theory; spin diffusion, anyons, Bose Einstein condensates - Arkeryd, Leif (Auteur de la Conférence) | Nouveau

From the Hartree-Fock dynamics to the Vlasov equation - Saffirio, Chiara (Auteur de la Conférence) | Nouveau

Scattering for NLS in $\mathbb{R}^d\times \mathbb{T}$ - Visciglia, Nicola (Auteur de la Conférence) | H Nouveau

Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field - Bellassoued, Mourad (Auteur de la Conférence) | H Nouveau

Emergent anyons in quantum Hall physics - Rougerie, Nicolas (Auteur de la Conférence) | H Nouveau

Inhomogeneities and temperature effects in Bose-Einstein condensates - de Bouard, Anne (Auteur de la Conférence) | H Nouveau

Leapfrogging for the axisymmetric Gross-Pitaevskii equation - Smets, Didier (Auteur de la Conférence) | H Nouveau

Unresolved problems in the theory of integrable systems - Zakharov, Vladimir (Auteur de la Conférence) | H Nouveau

Soliton resolution for derivative NLS equation - Sulem, Catherine (Auteur de la Conférence) | H Nouveau