The focusing nonlinear Schrödinger equation serves as a universal model for the amplitude of a wave packet in a general one-dimensional weakly-nonlinear and strongly-dispersive setting that includes water waves and nonlinear optics as special cases. Rogue waves of infinite order are a novel family of solutions of the focusing nonlinear Schr¨odinger equation that emerge universally in a particular asymptotic regime involving a large-amplitude and near-field limit of a broad class of solutions of the same equation. In this talk, we will present several recent results on the emergence of these special solutions along with their interesting asymptotic and exact properties. Notably, these solutions exhibit anomalously slow temporaldecay and are connected to the third Painlev´e equation. Finally, we will extend the emergence of rogue waves of infinite order to the first several flows of the AKNS hierarchy — allowing for arbitrarily many simultaneous flows — and report on recent work regarding their space-time asymptotic behavior under a general flow from the hierarchy.[-]

We prove that the solution to the Benjamin-Ono equation on the line, with initial data given by minus a soliton, exhibits scattering in infinite time. Our approach relies on an explicit formula for solutions with rational initial data in L2 having only simple poles. This formula is expressed as a ratio of determinants involving contour integrals. Additionally, we develop some spectral properties of the Lax operator associated with the Benjamin-Ono equation. This work is in collaboration with Elliot Blackstone, Patrick Gérard, and Peter D. Miller[-]

The focusing Continuum Calogero–Moser (CCM) equation is a completely integrable PDE that describes a continuum limit of a particle gas interacting pairwise through an inverse square potential. This system is well-posed in the scaling-critical space L2 below the mass of the soliton, but above this threshold there are solutions that blow up in finite time. In this talk, we will discuss some new and existing results about solutions below the soliton mass threshold. This is based on joint work with Rowan Killip and Monica Visan.[-]

In this talk, I discuss the energy-critical half-wave maps equation (HWM). It has been known for quite some time that (HWM) is completely integrable with a Lax pair structure. However, the question about global-in-time existence of solutions has been completely open so far — even for smooth and sufficiently small initial data. I will present very recent results that prove global well-posedness for rational initial data (with no size restriction) along with a general soliton resolution result in the large-time limit. The proofs strongly exploit the Lax structure of (HWM) in combination with an explicit flow formula. This is joint work with Patrick Gérard (Paris-Saclay).[-]

Suspensions are ubiquitous in nature (sediments, clouds,biological fluids ... etc.) and in industry such as civil engineering (paints, polymers ... etc.) among many others. The rigorous derivation of fluid-kinetic models for suspensions has attracted a lot of attention in the last decade. This lecture aims at presenting a review of the main results that have been obtained.

The first session aims at introducing both the microscopic and the limiting equation and giving a formal derivation of the former one. The second session aims at presenting the main early results concerning the derivation of an effective model starting from the microscopic model in which particle positions and velocities are fixed or given. Such a system takes the following form for example
\begin{equation}\label{eq:Stokes}
\left \{
\begin{array}{rcl}
-\Delta u+\nabla p &=& f, \text{ on } \Omega\setminus \overline{\underset{i=1}{\overset{N}{\bigcup}} B(x_i,r)} \\
\text{div } u&=& 0, \text{ on } \Omega\setminus \overline{\underset{i=1}{\overset{N}{\bigcup}} B(x_i,r)} \\
u&=& V_i, \text{ on } \partial B(x_i,r)\\
u&=& 0, \text{ on } \partial \Omega
\end{array}
\right.
\end{equation}
where $\Omega$ a smooth open set of $\mathbb{R}^3$, $x_1, x_2, \cdots, x_N$ are the particles position, $r$ their radius and $V_i$ the given velocity of the $i$th particle. The aim is then to perform an asymptotic analysis when the number of particles $N$ becomes large while their radius $r$ becomes small, first results have been obtained in [1,2,3] where the limit equations depend on the scale of the holes and their typical distance; Stokes equation, Darcy equation or Stokes-Brinkman equation. After recalling the recent contributions, we will present a short argument giving insights about the derivation of the Brinkman term in a simple case.

The last session of this mini-course aims at presenting the results regarding the rigorous derivation of fluid-kinetic models when taking into account the fluid-particle interactions and particle dynamics. This means that we consider the Stokes equation [1] coupled to Newton laws where we neglect particles inertia (balance of force and torque) and the motion of the center of the particles $\dot{x}_i=V_i$.

The rigorous derivation of a fluid-kinetic model in this setting have been obtained in [6,5,7] in the case $\Omega=\mathbb{R}^3$ under some separation assumptions on the particles. The obtained equation is a Transport-Stokes equation
\begin{equation}\label{eq:TS}\tag{TS}
\left\{
\begin{array}{rcl}
- \Delta u + \nabla p &=& \rho g,\\
\text{div } u&=& 0, \\
\partial_t \rho +\text{div }( ( u + \gamma^{-1} V_{\mathrm{St}})\rho) &=& 0,
\end{array}
\right.
\end{equation}
where $\gamma = \lim Nr \in (0,\infty]$.

This result is related to the mean field limit of many particles interacting through a kernel and has been extensively studied for several different problems. We present the main ideas for such a derivation using the method of reflections and stability estimates through Wasserstein distance following the approach by M. Hauray [4]. We finish by emphasizing new results based on a mean-field argument for the derivation of models of suspensions.[-]