Accueil > Les trimestres des LMB > Printemps 2015 > Colloques, conférences et ateliers > dispersive-pde > **Résumés**

# Résumés

**Adami, Riccardo** (Milano Bicocca) : Existence of ground states for the NLS on metric graphs

Résumé : The issue of minimizing the energy associated to the nonlinear Schrodinger equation on exotic domains has become nowadays very popular for its connection to Bose-Einstein condensation and related topics. if the spatial domain is branched, so that it can be reasonably tought of as a graph, then the mere issue of proving existence or non existence of a ground state is non trivial. we report on recent results on such topics. this is a joint research project with E. Serra and P. Tilli.

**Antonelli, Paolo** (L’Aquila) : Global existence results for some systems in quantum fluid dynamics

Résumé : In this talk we present some results regarding the Cauchy problem for quantum hydrodynamics systems. Such models arise in many physical contexts, such as the description of superfluidity phenomena, the dynamics of Bose-Einstein condensates or the modelling of semiconductor devices. I will discuss the existence of finite energy weak solutions by exploiting the analogy with a class of nonlinear Schrödinger equations through the Madelung transformations. The main advantage of this approach is that it does not need to define the velocity field in the vacuum region. I will conclude the exposition by discussing some ongoing and future research perspectives.

**Bellazzini, Jacopo** (Sassari) : Ground states for dipolar quantum gases in the unstable regime

Résumé : The Aim of the talk is to show the existence and stability/ instability properties of standing waves for a nonlinear Schrödinger equation arising in dipolar Bose-Einstein condensate in the unstable regime.

Two cases are presented : the first when the system is free, the second when gradually a trapping potential is add. In both cases this leads to the search of critical points of a constrained functional which is unbounded from below on the constraint. In the first case, by showing that the constrained functional has a so-called *mountain pass geometry*, we prove the existence of ground states and show that any ground state is orbitally unstable. In the second case we prove that, if the trapping potential is small, two different kind of standing waves appears : one corresponds to a local minima of the constrained energy functional and it consists in ground states, the other is again of *mountain pass type* but now correspond to excited states. We also prove that any ground state is a local minimizer. Despite the problem is mass supercritical and the functional unbounded from below the standing waves associated to the ground states turn to be orbitally stable.

Eventually we show that the addition of the trapping potential, however small, create a “gap” in the ground state energy level of the system. Joint work with Louis Jeanjean.

**Benzoni-Gavage, Sylvie** (Lyon I) : Modulated equations and dispersive shocks

Résumé : The zero dispersion limit in dispersive perturbations of hyperbolic PDEs is a challenging topic, which is well understood for the Korteweg-de Vries (KdV) equation only. For more general equations like the Euler—Korteweg (EK) system, a preliminary route consists in investigating modulation equations. These are indeed expected to govern, in an averaged sense, the propagation of oscillatory wave trains. In particular, dispersive shocks are oscillatory, unsteady patterns that can be associated with special solutions of modulated equations. The talk will report on our latest findings on the modulation equations for a class of Hamiltonian PDEs including both KdV and EK. This is work in progress with C. Mietka and M. Rodrigues.

**Chiron, David** (Nice) : Multiple branches of travelling waves in the 2D Gross-Pitaevskii model

Résumé : In this talk, we will be interested in travelling waves solutions in the two-dimensional Gross-Pitaevskii equation, which is a Nonlinar Schrödinger equation with the constraint of modulus one at infinity. This equation possesses a branch of travelling waves which has been numerically obtained by Jones and Roberts for speeds between 0 and the speed sound associated with acoustic waves. Some rigorous results justify the two extreme asymptotics : on the one hand for speeds close to 0, where we qualitatively observe vortices ; on the other hand for speeds close to the speed of sound, where the wave becomes a rarefaction wave close, after rescaling, to the Kadomtsev-Petviashvili I solitary wave. This branch may be obtained by minimizing the energy at fixed momentum. We shall present numerical results showing the existence of other branches of travelling waves, corresponding to the excited states. This is a joint work with Claire Scheid (Nice).

**De Bièvre, Stephan** (Lille 1) : Orbital stability via the energy-momentum method revisited

(Joint work with F. Genoud (University of Vienna) and S. Rota Nodari (Universite de Lille))

Résumé : In this talk, we (re)consider the energy-momentum method for proving the orbital stability of relative equilibria of Hamiltonian dynamical systems on Banach spaces. We will give a presentation which highlights the interplay that is at work in this method between (symplectic) geometry and (functional) analysis. We will in particular present an extension of the Vakitov-Kholokolov condition in the presence of higher dimensional symmetry groups which provides a variant on results by Grillakis-Shatah-Strauss (1990).

**De Laire, André** (Lille 1) : The Landau-Lifshitz-Gilbert equation with rough initial data

résumé : In this talk we will consider the Landau-Lifshitz-Gilbert equation, a model describing the dynamics for the spin in ferromagnetic materials. We will discuss the Cauchy problem with initial conditions that are not in the standard Sobolev spaces, but are in the BMO (bounded mean oscillation) space. By using the stereographic projection, this problem is related to the Cauchy problem for a nonlinear Schrodinger equation with rough initial data.

**Felli, Veronica** (Milano Bicocca) : Unique continuation properties and essential self-adjointness for relativistic Schrödinger operators with singular potentials

Résumé : I will present some results obtained in collaboration with M. M. Fall about unique continuation properties for a class of relativistic fractional elliptic equations with singular potentials. I will also discuss the

problem of essential self-adjointness of the corresponding relativistic Schrödinger operator providing an explicit sufficient and necessary condition on the coefficient of the singular potential for essential self-adjointness.

**Genoud François** (Delft) : Stable solitons of the cubic-quintic NLS with a delta-function potential

Résumé : This talk is about the one-dimensional nonlinear Schrödinger equation with a combination of cubic focusing and quintic defocusing nonlinearities, and an attractive delta-function potential. All standing waves with a positive soliton profile can be determined explicitly in terms of elementary functions, and I will show by a bifurcation and spectral analysis that all these solutions are orbitally stable. A remarkable feature is a regime of bistability, where two stable solitons coexist. This is joint work with Boris Malomed and Rada Weishäupl.

**Georgiev, Vladimir** (Pisa) : On continuity of the solution map for the cubic 1d periodic NLW equation

Résumé : We study the uniform continuity of the solution map for the focussing cubic half-wave equation in periodic setting. We prove an ill-posedness result, in the sense that the corresponding solution map cannot be uniformly continuous in the Sobolev space of order $s \in (1/4,1/2)$. This a joint work with Nikolay Tzvetkov and Nicola Visciglia.

**Gérard, Patrick** (Paris-Sud) : Growth of Sobolev norms for the cubic Szegö equation

Résumé : The problem of finding and studying solutions of defocusing nonlinear Hamiltonian PDEs with large time unbounded Sobolev norms is still widely open. In the case of a simple model admitting an integrable structure, I will show how the study of quasiperiodic solutions allows to display this phenomenon for generic initial data. This is a jointwork with Sandrine Grellier.

**Klein, Christian** (Dijon) : Multidomain spectral method for Schrödinger equations

Résumé : A multidomain spectral method with compactified exterior domains combined with stable second and fourth order time integrators is presented for Schrödinger equations. The numerical approach allows high precision numerical studies of solutions on the whole real line. At examples for the linear and cubic nonlinear Schrödinger equation, this code is compared to transparent boundary conditions and perfectly matched layers approaches. The code can deal with asymptotically non vanishing solutions as the Peregrine breather being discussed as a model for rogue waves. It is shown that the Peregrine breather can be numerically propagated with essentially machine precision, and that localized perturbations of this solution can be studied.

**Martel, Yvan** (École polytechnique) : On multi-solitons for the quartic generalized Korteweg-de Vries equation

Résumé :

**Miot, Evelyne** (Paris-Sud) : Formation of Néel walls in a thin-film limit for the Landau-Lifshitz-Gilbert equation

Résumé : We consider thin ferromagnetic films described by a time dependent $S^2$-valued map called magnetization, which satisfies a Landau-Lifshitz-Gilbert equation.

We first study the Cauchy problem in the corresponding energy space. Then, we establish compactness properties of the magnetization in an asymptotical regime consistent with the formation of transition layers called Néel walls. This is joint work with Raphael Côte and Radu Ignat.

**Rota Nodari, Simona** (Lille 1) : On a nonlinear Schrödinger equation for nucleons

Résumé : In this talk we consider a model for a nucleon interacting with the $\sigma$ and $\omega$ mesons in the atomic nucleus. The model is relativistic, but we study it in the nuclear physics nonrelativistic limit where is described by a nonlinear Schrödinger-type equation with a mass which depends on the solution itself.

After discussing some previous results on the existence of positive solutions, I will prove the uniqueness and non-degeneracy of these ones. As an application, I will construct solutions to the relativistic $\sigma$ and $\omega$ model, which consists of one Dirac equation coupled to two Klein-Gordon equations. The talk is based on joint work with Mathieu Lewin.

**Sabin, Julien** (Paris-Sud) : Optimal trace ideals properties of the restriction operator and applications

Résumé : We study the spectral properties of the Fourier restriction operator to hypersurfaces of the euclidian space of dimension N, and extend the theorems of Stein-Tomas and Strichartz to systems of orthonormal functions. As an application, we deduce new Strichartz estimates for systems of orthonormal functions, showing the dispersive properties of infinite quantum systems. This is a joint work with Rupert Frank (Caltech).