Abstracts

Jean Dalibard (Collège de France, Paris) Ultracold Atomic Gases: A Tunable Laboratory for Soliton Physics

In this introductory talk, I will explain why cold atomic gases constitute a very flexible experimental platform for studying solitonic solutions of well-known nonlinear equations, such as the nonlinear Schrödinger equation or the Landau-Lifshitz equation. In a one-dimensional geometry, these gases allow us to observe well-known phenomena such as soliton collisions, as well as more surprising ones that exploit the very peculiar dispersion relation of solitonic objects.

Patrick Gérard (Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay) Solitons and  long time dynamics of some integrable equations

I will explain the role played by soliton solutions in the long time dynamics of some integrable equations enjoying special representation formulae. The most typical of these equations is the Benjamin-Ono equation, describing long, one way internal gravity waves in a two layer fluid with infinite depth. For this equation, I will  sketch how the representation formula leads to a proof of the soliton resolution conjecture.

Quentin Glorieux (Laboratoire Kastler Brossel, Sorbonne Université) Paraxial fluids of light: an experimental platform for nonlinear Schrödinger dynamics

Philippe Gravejat (CY Cergy Paris Université) Solitonic vortices for the Gross-Pitaevskii equation in a strip

The talk deals with the Gross-Pitaevskii equation in a two-dimensional strip following experiments in fermions and bosons, and numerical simulations showing evidence of solitons and solitonic vortices. A mathematical question is to construct solutions.
A first answer will be given by a minimization procedure under constraint, but only when the width of the strip is large enough. Next a perturbative approach close to the soliton will provide a rigorous construction of bifurcating two-dimensional stationary solutions with a fixed number of vortices.

This is joint work with André de Laire (University of Lille) and Didier Smets (Sorbonne University) on the one hand, and with Amandine Aftalion (CNRS and University Paris Saclay) and Étienne Sandier (Paris-East Créteil University) on the other hand.

Antonio Munoz Mateo (Universidad de la Laguna, Tenerife) Solitary waves in confined superfluids

The confinement of superfluids has demonstrated to be a useful tool to select the kind and stability of available nonlinear superfluid waves. In elongated traps, increasing values of the ratio between chemical potential and characteristic transverse trapping energy are associated with the existence of increasingly complex stationary waves made of transverse vortex patterns, while small ratios limit the spectrum to the lowest energy excitations: either a single transverse vortex line or just a planar soliton. Independently of their complexity, all these solitary waves share an exponential localization and an asymptotic phase difference along the axial direction; however, only those with the lowest energy, depending on the cross section, are dynamically stable. This family of nonlinear waves are referred to as Chladni solitons for their similarity to standing waves in metal plates. I will talk about their generation, linear excitations, particle-like character, and conditions for the formation of bound states in Bose-Einstein condensates of ultracold atomic gases with different geometries.

Luc Nguyen (Oxford University) Solitons and solitonic vortices as Moutain pass

We study critical points of the Ginzburg-Landau energy on 2D strips and 3D cylinders. In relation with recent experiments on fermionic and bosonic strips, we prove that there is a critical width of the strip under which the minimizer in some suitable space is the soliton while above it the minimizer is solitonic vortex. We manage to go further and characterize them as a mountain pass solutions. In 3D, we present generalizations and open questions.

Hélène Perrin (CNRS et Université Sorbonne Paris Nord) Dynamics of one-dimensional Bose gases: a model physical system for the nonlinear Schrödinger equation

We study the proliferation of solitons in a continuously shaken one-dimensional gas confined in a box trap. We model the dynamics with the non linear Schrödinger equation. We observe power laws in the momentum distribution, and a saturation in the energy transferred to the gas.

Giacomo Roati (CNR-INO and LENS, University of Florence) Vortex dynamics in strongly interacting Fermi superfluids

We investigate vortex matter in strongly interacting Fermi superfluids of ultracold atoms. By engineering vortex configurations on demand and by tracking vortex trajectories with high spatial resolution, we establish an ideal quantum laboratory for probing the fundamental mechanisms underlying vortex-driven instabilities and dissipation. Our approach opens the door to new insights into vortex-matter phenomena in strongly correlated superfluids.

Frédéric Rousset (Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay) Transverse stability and instability of solitary waves  

We will be interested in studying the stability of a one-dimensional solitary wave submitted to multi-dimensional perturbations (periodic or localized) in Hamiltonian  partial differential equations. We will present a general criterion for linear instability  and discuss it on various physical examples: Gross-Pitaevskii, water-waves, plasma models...

ROUND TABLE

Jérome Beugnon (Collège de France) Multi-solitons in two-component ultracold gases

Alberto Bramati (Laboratoire Kastler Brossel, Sorbonne Université) Quantum fluids of light: superfluidity, dark solitons and more

Quantum fluids of ligth are an ideal playground to study superfluidity, quantized vortices, dark solitons and more generally quantum phase transition in intrinsically out of equilibrium driven-dissipative systems.

I will briefly discuss the specificity of such systems and the potential of quantum fluids of light to investigate the superfluid-supersolid transition.