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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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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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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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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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quasi-periodic wave equation - laplacian - hamiltonian system - symplectic form - time dependance
37K55 ; 35Q55
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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 solutions of the semi-discrete Schrödinger equation (where time is continuous and spacial variable is discrete), $\partial_tu = i(\Delta_du + V u)$, where $\Delta_d$ is the standard discrete Laplacian on $\mathbb{Z}^n$ and $u : [0, 1] \times \mathbb{Z}^d \to \mathbb{C}$. Uncertainty principle states that a non-trivial solution of the free equation (without potential) cannot be sharply localized at two distinct times. We discuss different extensions of this result to equations with bounded potentials. The continuous case was studied in a series of articles by L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega.
The talk is mainly based on joint work with Ph. Jaming, Yu. Lyubarskii, and K.-M. Perfekt.
We consider solutions of the semi-discrete Schrödinger equation (where time is continuous and spacial variable is discrete), $\partial_tu = i(\Delta_du + V u)$, where $\Delta_d$ is the standard discrete Laplacian on $\mathbb{Z}^n$ and $u : [0, 1] \times \mathbb{Z}^d \to \mathbb{C}$. Uncertainty principle states that a non-trivial solution of the free equation (without potential) cannot be sharply localized at two distinct times. We discuss ...
35Q55 ; 35A02
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Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation, etc. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. We will present a new result on the derivation of a mean-field limit equation for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzburg-Landau) equation.
Ginzburg-Landau type equations are models for superconductivity, superfluidity, Bose-Einstein condensation, etc. A crucial feature is the presence of quantized vortices, which are topological zeroes of the complex-valued solutions. We will present a new result on the derivation of a mean-field limit equation for the dynamics of many vortices starting from the parabolic Ginzburg-Landau equation or the Gross-Pitaevskii (=Schrodinger Ginzb...
35Q55 ; 35Q56 ; 82D55
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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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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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I will present two cases of strong interactions between solitary waves for the nonlinear Schrödinger equations (NLS). In the mass sub- and super-critical cases, a work by Tien Vinh Nguyen proves the existence of multi-solitary waves with logarithmic distance in time, extending a classical result of the integrable case (1D cubic NLS equation). In the mass-critical case, a work by Yvan Martel and Pierre Raphaël gives a new class of blow up multi-solitary waves blowing up in infinite time with logarithmic rate.
These special behaviours are due to strong interactions between the waves, in contrast with most previous works on multi-solitary waves of (NLS) where interactions do not affect the general behaviour of each solitary wave.
I will present two cases of strong interactions between solitary waves for the nonlinear Schrödinger equations (NLS). In the mass sub- and super-critical cases, a work by Tien Vinh Nguyen proves the existence of multi-solitary waves with logarithmic distance in time, extending a classical result of the integrable case (1D cubic NLS equation). In the mass-critical case, a work by Yvan Martel and Pierre Raphaël gives a new class of blow up ...
35Q55 ; 76B25 ; 35Q51 ; 35C08
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This talk is devoted to solitons and wave collapses which can be considered as two alternative scenarios pertaining to the evolution of nonlinear wave systems describing by a certain class of dispersive PDEs (see, for instance, review [1]). For the former case, it suffices that the Hamiltonian be bounded from below (or above), and then the soliton realizing its minimum (or maximum) is Lyapunov stable. The extremum is approached via the radiation of small-amplitude waves, a process absent in systems with finitely many degrees of freedom. The framework of the nonlinear Schrodinger equation, the ZK equation and the three-wave system is used to show how the boundedness of the Hamiltonian H, and hence the stability of the soliton minimizing H can be proved rigorously using the integral estimate method based on the Sobolev embedding theorems. Wave systems with the Hamiltonians unbounded from below must evolve to a collapse, which can be considered as the fall of a particle in an unbounded potential. The radiation of small-amplitude waves promotes collapse in this case.
This work was supported by the Russian Science Foundation (project no. 14-22-00174).
This talk is devoted to solitons and wave collapses which can be considered as two alternative scenarios pertaining to the evolution of nonlinear wave systems describing by a certain class of dispersive PDEs (see, for instance, review [1]). For the former case, it suffices that the Hamiltonian be bounded from below (or above), and then the soliton realizing its minimum (or maximum) is Lyapunov stable. The extremum is approached via the radiation ...
35Q53 ; 35Q55 ; 37K10 ; 37N10 ; 76B15
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Inspired by a recent result of Dodson-Luhrmann-Mendelson, who proved almost sure scattering for the energy-critical wave equation with radial data in four dimensions, we establish the analogous result for the Schrödinger equation.
This is joint work with R. Killip and J. Murphy.
35Q55 ; 35L05 ; 35R60
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Choosing favourable gauges is a crucial step in the study of nonlinear geometric dispersive equations. A very successful tool, that has emerged originally in work of Tao on wave maps, is the use of caloric gauges, defined via the corresponding geometric heat flows. The aim of this talk is to describe two such flows and their associated gauges, namely the harmonic heat flow and the Yang-Mills heat flow.
70S15 ; 35Q53 ; 35Q55
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It is possible to model dissipation effects subjected by a particle by interactions between the particle and its environment. This seminal idea dates back to Caldeira-Leggett in the ’80ies. The specific case of a particle interacting with vibrational degrees of freedom has been thoroughsly investigated by S. De Bièvre and his collaborators. We will go back to these issues in the framework of kinetic equations, and also consider quantum versions of the problem based on couplings with the Schrödinger equation. We are particularly interested in stability issues. We will describe ; through rigorous statements and numerical experiments, analogies and differences with the case of a single classical particle and with the standard coupling with the Poisson equation.
It is possible to model dissipation effects subjected by a particle by interactions between the particle and its environment. This seminal idea dates back to Caldeira-Leggett in the ’80ies. The specific case of a particle interacting with vibrational degrees of freedom has been thoroughsly investigated by S. De Bièvre and his collaborators. We will go back to these issues in the framework of kinetic equations, and also consider quantum versions ...
35Q40 ; 35Q51 ; 35Q55
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Usual numerical methods become inefficient when they are applied to highly oscillatory evolution problems (order reduction or complete loss of accuracy). The numerical parameters must indeed be adapted to the high frequencies that come into play to correctly capture the desired information, and this induces a prohibitive computational cost. Furthermore, the numerical resolution of averaged models, even at high orders, is not sufficient to capture low frequencies and transition regimes. We present (very briefly) two strategies allowing to remove this obstacle for a large class of evolution problems : a 2-scale method and a micro/macro method. Two different frameworks will be considered : constant frequency, and variable - possibly vanishing - frequency. The result of these approaches is the construction of numerical schemes whose order of accuracy no longer depends on the frequency of oscillation, one then speaks of uniform accuracy (UA) for these schemes. Finally, a new technique for systematizing these two methods will be presented. Its purpose is to reduce the number of inputs that the user must provide to apply the method in practice. In other words, only the values of the field defining the evolution equation (and not its derivatives) are used.These methods have been successfully applied to solve a number of evolution models: non-linear Schrödinger and Klein-Gordon equations, Vlasov-Poisson kinetic equation with strong magnetic field, quantum transport in graphene.
Usual numerical methods become inefficient when they are applied to highly oscillatory evolution problems (order reduction or complete loss of accuracy). The numerical parameters must indeed be adapted to the high frequencies that come into play to correctly capture the desired information, and this induces a prohibitive computational cost. Furthermore, the numerical resolution of averaged models, even at high orders, is not sufficient to ...
65L05 ; 35Q55 ; 37L05
