Titres et résumés

Patrick Gérard (Université Paris-Saclay)

Title: Integrable equations of Benjamin-Ono type.

Abstract: In this mini-course, I will study two integrable nonlinear dispersive equations on the line : the Benjamin-Ono equation and the more recently introduced Calogero-Moser derivative nonlinear Schrödinger equation.
The following topics will be discussed.
1. Local wellposedness in Sobolev spaces, Lax pair, explicit formulae.
2. Conservation laws and global wellposedness.
3. Traveling waves and multi-solitons.
4. Zero-dispersion limit.
 
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Valeria Banica (Sorbonne Université)

Title: TBA.

Abstract: TBA.
 
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Yannick Bonthonneau (Université Sorbonne Paris Nord)

Title: TBA.

Abstract: TBA.
 
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Nicolas Camps (Nantes Université)

Title: TBA.

Abstract: TBA.
 
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Alberto Enciso (Instituto de Ciencias Matemáticas, Madrid)

Title: TBA.

Abstract: TBA.
 
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Gregory Faye (CNRS, Université Paul Sabatier, Toulouse)

Title: The local limit theorem for complex valued sequences: the parabolic case.

Abstract: We give a complete expansion, at any accuracy order, for the iterated convolution of a complex valued integrable sequence in one space dimension. The remainders are estimated sharply with generalized Gaussian bounds. The result applies in probability theory for random walks as well as in numerical analysis for studying the large time behavior of numerical schemes. This is joint work with Jean-François Coulombel.  
 
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Pierre Germain (Imperial College, Londres)

Title: TBA.

Abstract: TBA.
 
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Bernard Helffer (Nantes Université)

Title: On Courant and Pleijel theorems for sub-Riemannian Laplacians.

Abstract: We are interested in the number of nodal domains of eigenfunctions of sub-Laplacians on sub-Riemannian manifolds. Specifically, we investigate the validity of Pleijel's theorem, which states that, as soon as the dimension is strictly larger  than 1, the number of nodal domains of an eigenfunction corresponding to the k-th eigenvalue is strictly (and uniformly, in a certain sense) smaller than k for large k.
In the first part of the talk we reduce this question from the case of general sub-Riemannian manifolds to that of nilpotent groups.
In the second part, we analyze in detail the case where the nilpotent group is a Heisenberg group times a Euclidean space. Along the way we improve known bounds on the optimal constants in the Faber--Krahn and isoperimetric inequalities on these groups.
This is a joint work with Rupert Frank (LMU München)
 
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Mikaela Iacobelli (ETH Zurich)

Title: Stability and singular limits in plasma physics.

Abstract: In this talk, we will present two kinetic models that are used to describe the evolution of charged particles in plasmas: the Vlasov-Poisson system and the Vlasov-Poisson system with massless electrons. These systems model respectively the evolution of electrons, and ions in a plasma. We will discuss the well-posedness of these systems, the stability of solutions, and their behavior under singular limits. Finally, we will introduce a new class of Wasserstein-type distances specifically designed to tackle stability questions for kinetic equations.
 
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Kevin Le Balc’h (INRIA, Sorbonne Université)

Title: TBA.
Abstract: TBA.
 
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Yvan Martel (Université Versailles Saint-Quentin)
Title: Asymptotic stability of small solitary waves for the one-dimensional cubic-quintic Schrödinger equation with an internal mode.

Abstract: I will present a result concerning the asymptotic stability of small solitary waves for the one-dimensional Schrödinger equation with a cubic-quintic, focusing-focusing nonlinearity. For this model, the linearized operator around a small solitary wave has an internal mode whose contribution to the dynamics is handled by checking a nonlinear condition related to the Fermi golden rule.
Reference: Yvan Martel, Asymptotic stability of small standing solitary waves of the one-dimensional cubic-quintic Schrödinger equation, arXiv:2312.11016
 
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Ewelina Zatorska (University of Warwick)

Title: TBA.

Abstract: TBA.
 
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