We describe recent results concerning the orbital and asymptotic stability of dark solitons and multi- solitons for the Landau-Lifshitz equation with an easy-plane anisotropy. This is joint work with André de Laire (University of Lille Nord de France), and by Yakine Bahri (Nice Sophia Antipolis University).

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

I will discuss a joint work with Bellazzini, Boussaid, Jeanjean about the existence and orbital stability of standing waves for NLS with a partial confinement in a supercritical regime. The main point is to show the existence of local minimizers of the constraint energy.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

We consider the defocusing nonlinear Schrödinger equations on a two-dimensional compact Riemannian manifold without boundary or a bounded domain in dimension two. In particular, we discuss the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

In form of a case study for the mKdV and the KdV2 equation we discuss a novel approach of representing frequencies of integrable PDEs which allows to extend them analytically to spaces of low regularity and to study their asymptotics. Applications include wellposedness results in spaces of low regularity as well as properties of the actions to frequencies map. This is joint work with Jan Molnar.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

The onset of a dispersive shock in solutions to the Kadomtsev-Petviashvili (KP) equations is studied numerically. First we study the shock formation in the dispersionless KP equation by using a map inspired by the characteristic coordinates for the one-dimensional Hopf equation. This allows to numerically identify the shock and to unfold the singularity. A conjecture for the KP solution near this critical point in the small dispersion limit is presented. It is shown that dispersive shocks for KPI solutions can have a second breaking where modulated lump solutions appear.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

The Green-Naghdi system is an asymptotic model for the water-waves system, describing the propagation of surface waves above a layer of ideal, homogeneous, incompressible and irrotational fluid, when the depth of the layer is assumed to be small with respect to wavelength of the flow. It can be seen as a perturbation of the standard quasilinear (dispersionless) Saint-Venant system, with additional nonlinear higher-order terms. Because of the latter, the well-posedness theory concerning the GN system is not satisfactory, in particular outside of the one-dimensional framework. We will discuss novel results, obtained with Samer Israwi, that emphasize the role of the irrotationality assumption.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

In this lecture, we investigate the behavior of the solutions to the nonlinear Schrodinger equation: (1) ( i@tu +
u = f(u); ujt=0 = u0 2 H1 rad(R2); where the nonlinearity f : C ! C is dened by (2) f(u) = p( p 4 juj) u with p > 1 and p(s) = es2 􀀀 pX􀀀1 k=0 s2k k!
Recall that the solutions of the Cauchy problem (1)-(2) formally satisfy the conservation laws: (3) M(u; t) = Z R2 ju(t; x)j2dx = M(u0) and (4) H(u; t) = Z R2  jru(t; x)j2 + Fp(u(t; x))  dx = H(u0) ; where Fp(u) = 1 4 p+1 􀀀p 4 juj

It is known (see [4], [6] and [2]) that global well-posedness for the Cauchy problem (1)-(2) holds in both subcritical and critical regimes in the functional space C(R;H1(R2)) L4(R;W1;4(R2)). Here the notion of criticity is related to the size of the initial Hamiltonian H(u0) with respect to 1. More precisely, the concerned Cauchy problem is said to be subcritical if H(u0) < 1, critical if H(u0) = 1 and supercritical if H(u0) > 1. Structures theorems originates in the elliptic framework in the studies by H. Brezis and J.- M. Coron in [3] and M. Struwe in [8]. The approach that we shall adopt in this article consists in comparing the evolution of oscillations and concentration eects displayed by sequences of solutions of the nonlinear Schrodinger equation (1)-(2) and solutions of the linear Schrodinger equation associated to the same sequence of Cauchy data. Our source of inspiration here is the pioneering works [1] and [7] whose aims were to describe the structure of bounded sequences of solutions to semilinear defocusing wave and Schrodinger equations, up to small remainder terms in Strichartz norms. The analysis we conducted in this work emphasizes that the nonlinear eect in this framework only stems from the 1-oscillating component of the sequence of the Cauchy data, using the terminology introduced in [5]. This phenomenon is strikingly dierent from those obtained for critical semi linear dispersive equations, such as for instance in [1, 7] where all the oscillating components induce the same nonlinear eect, up to a change of scale. To carry out our analysis, we have been led to develop a prole decomposition of bounded sequences of solutions to the linear Schrodinger equation both in the framework of Strichartz and Orlicz norms. The linear structure theorem we have obtained in this work highlights the distinguished role of the 1-oscillating component of the sequence of the Cauchy data. It turns out that there is a form of orthogonality between the Orlicz and the Strichartz norms for the evolution under the ow of the free Schrodinger equation of the unrelated component to the scale 1 of the Cauchy data (according to the vocabulary of [5]), while this is not the case for the 1-oscillating component.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

The incompressible Euler equation with free surface dictates the dynamics of the interface separating the air from a perfect incompressible fluid. This talk is about the controllability and the stabilization of this equation. The goal is to understand the generation and the absorption of water waves in a wave tank. These two problems are studied by two different methods: microlocal analysis for the controllability (this is a joint work with Pietro Baldi and Daniel Han-Kwan), and study of global quantities for the stabilization (multiplier method, Pohozaev identity, hamiltonian formulation, Luke’s variational principle, conservation laws…).

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

In the 1960’s G. B. Whitham suggested a non-local version of the KdV equation as a model for water waves. Unlike the KdV equation it is not integrable, but it has certain other advantages. In particular, it has the same dispersion relation as the full water wave problem and it allows for wave breaking. The equation has a family of periodic, travelling wave solutions for any given wavelength. Whitham conjectured that this family contains a highest wave which has a cusp at the crest. I will outline a proof of this conjecture using global bifurcation theory and precise information about an integral operator which appears in the equation. Joint work with M. Ehrnström.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

We consider the existence of periodic traveling waves in a bidirectional Whitham equation, combining the full two-way dispersion relation from the incompressible Euler equations with a canonical quadratic shallow water nonlinearity. Of particular interest is the existence of a highest, cusped, traveling wave solution, which we obtain as a limiting case at the end of the main bifurcation branch of $2pi$-periodic traveling wave solutions. Unlike the unidirectional Whitham equation, containing only one branch of the full Euler dispersion relation, where such a highest wave behaves like $|x|^{1/2}$ near its peak, the cusped waves obtained here behave like $|xlog|x||$ at their peak and are smooth away from their highest points. This is joint work with Mathew A. Johnson and Kyle M. Claassen at University of Kansas.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

It is conjectured that bounded solutions of the focusing energy-critical wave equation decouple asymptotically as a sum of a radiation term and a finite number of solitons . In this talk, I will review recent works on the subject, including the proof of a weak form of this conjecture (joint work with Hao Jia, Carlos Kenig and Frank Merle)

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

We consider the linear wave equation and the linear Schr dingier equation outside a compact, strictly convex obstacle in R^d with smooth boundary. In dimension d = 3 we show that for both equations, the linear flow satises the (corresponding) dispersive estimates as in R^3. For d>3, if the obstacle is a ball, we show that there exists at least one point (the Poisson spot) where the dispersive estimates fail. This is joint work with Gilles Lebeau.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

In this talk we will present some results on the dynamics of vortex filaments according to a model introduced by Klein, Majda and Damodaran, focusing on the issue of collisions. This is a joint work with Valeria Banica and Erwan Faou.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

The aim of this talk is to present some qualitative properties of a coupled Maxwell-Schrödinger system. First, I will describe conditions for the existence of minimizers with prescribed charge in terms of a coupling constant e. Secondly, I will study the existence of ground states for the stationary problem, the uniqueness of ground states for small e and finish with the orbital stability for the quadratic nonlinearity. This is a joint work with Tatsuya Watanabe.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

The Hartree equation can be derived from the N-body Heisenberg equation by the mean-field limit assuming that the particle number N tends to infinity. The first rigorous result in this direction is due to Spohn (1980) (see also [Bardos-Golse-Mauser, Meth. Applic. Anal. 7:275-294, (2000)] for more details), and is based on analyzing the Dyson series representing the solution of the BBGKY hierarchy in the case of bounded interaction potentials.This talk will (1) provide an explicit convergence rate for the Spohn method, and (2) interpolate the resulting convergence rate with the vanishing h bound obtained in [Golse-Mouhot-Paul, Commun. Math. Phys. 343:165-205 (2016)] by a quantum variant of optimal transportation modulo O(h) terms. The final result is a bound for a Monge-Kantorovich-type distance between the Husimi transforms of the Hartree solution and of the first marginal of the N-body Heisenberg solution which is independent of h and vanishes as N tends to infinity. (Work in collaboration with T. Paul and M. Pulvirenti).

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

We will discuss special regularity properties of solutions to the IVP associated to the k-generalized KdV equations. We show that for data u0 2 H3=4+(R) whose restriction belongs to Hk((b;1)) for some k 2 Z+ and b 2 R, the restriction of the corresponding solution u(
; t) belongs to Hk((;1)) for any  2 R and any t 2 (0; T). Thus, this type of regularity propagates with innite speed to its left as time evolves. This kind of regularity can be extended to a general class of nonlinear dispersive equations. Recently, we proved that the solution ow of the k-generalized KdV equation does not preserve other kind of regularities exhibited by the initial data u0.

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)

In order to study the transverse (in) stability of a line soliton, we consider the 2-d Zakharov-Rubenchik/Benney-Roskes system with initial data localized by a line soliton. The new terms in perturbed system lead to some diculties, for example, the lack of mass conservation. In this talk, I will present our recent work on this problem. This is a joint work with Norbert Mauser and Jean-Claude Saut. 1

• Thematic program: Classical and Quantum Transport (2016/2017)
• Event: Workshop "Recent progress on the qualitative properties of nonlinear dispersive equations and systems" (2016)