Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Multi angle  Mean field limits for Ginzburg-Landau vortices
Serfaty, Sylvia (Auteur de la Conférence) | CIRM (Editeur )

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

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Multi angle  Leapfrogging for the axisymmetric Gross-Pitaevskii equation
Smets, Didier (Auteur de la Conférence) | CIRM (Editeur )

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

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Multi angle  Soliton resolution for derivative NLS equation
Sulem, Catherine (Auteur de la Conférence) | CIRM (Editeur )

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

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Multi angle  Interactions of solitary waves for the nonlinear Schrödinger equations
Martel, Yvan (Auteur de la Conférence) | CIRM (Editeur )

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

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Multi angle  Solitons vs collapses
Kuznetsov, Evgenii (Auteur de la Conférence) | CIRM (Editeur )

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

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Multi angle  Almost sure scattering for the energy-critical Schrödinger equation in 4D with radial data
Visan, Monica (Auteur de la Conférence) | CIRM (Editeur )

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

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Multi angle  Geometric heat flows and caloric gauges
Tataru, Daniel (Auteur de la Conférence) | CIRM (Editeur )

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

Filtrer

Type
Domaine
Codes MSC

Z