Differential Equations

Fredholm groupoids

Event Date: Jan 22, 2020 in Differential Equations, Seminars

Resume : In many cases, the study of linear partial differential equations on a singular manifold can be related to that of a Lie groupoid whose action generates the (pseudo)differential operators of interest. Obtaining Fredholm conditions for these operators leads to the definition of Fredholm groupoids as recently introduced by Carvalho, Nistor and Qiao and also studied by Côme. I will introduce theses objects and give examples to illustrate the notion of Fredholm groupoids.  

Quantum Mean Field Asymptotics and Multiscale Analysis

Event Date: Jan 22, 2020 in Differential Equations, Seminars

Abstract: In a joint work with Z. Ammari, and F. Nier, we study how multiscale  analysis and quantum mean field asymptotics can be brought together. In  particular we study when a sequence of one-particle density matrices has  a limit with two components: one classical and one quantum. The introduction of “separating quantization for a family” provides a  simple criterion to check when those two types of limit are well  separated. We also give examples of explicit computations of such limits, and how to check that the separating assumption is satisfied. In this talk we will...

On the propagation of regularity for solutions of the Zakharov-Kuznetsov equation & Weakly coupled bound states of Pauli operators

Event Date: Nov 30, 1999 in Differential Equations, Seminars

Traveling waves for some nonlocal 1D Gross-Pitaevskii equations with nonzero conditions at infinity

Event Date: Dec 18, 2019 in Differential Equations, Seminars

Abstract:   We consider a nonlocal family of Gross-Pitaevskii equations with nonzero condition at infinity in dimension one. In this talk, we provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at infinity. Moreover, we show that the branch is orbitally stable. In this manner, this result generalizes known properties for the short-range interaction case given by a Dirac delta function. Our proof relies on the minimization of the energy at fixed momentum and a concentration-compactness...

Stability of nonlinear patterns in low dimensional Bose gases & “Long time existence for some Boussinesq type systems”

Event Date: Nov 27, 2019 in Differential Equations, Seminars

Title: Stability of nonlinear patterns in low dimensional Bose gases Abstract: In this talk I will present recent results, obtained in collaboration with prof. A. Corcho (UFRJ, Brazil), on the rigorous study of the orbital stability properties of the simplest nonlinear pattern in low dimensional Bose gases, the  black soliton solution. This is a solution of a  non-integrable defocusing Schrödinger model, represented by the  quintic Gross-Pitaevskii equation (5GP). Once the black soliton is characterized as a critical point of the associated Ginzburg-Landau energy of the 5GP, I will show some...

Strongly interacting kink-antikink pairs for scalar fields on a line

Event Date: Nov 20, 2019 in Differential Equations, Seminars

Abstract:   I will present a recent joint work with Michał Kowalczyk and Andrew Lawrie. A nonlinear wave equation with a double-well potential in 1+1 dimension admits stationary solutions called kinks and antikinks, which are minimal energy solutions connecting the two minima of the potential. We study solutions whose energy is equal to twice the energy of a kink, which is the threshold energy for a formation of a kink-antikink pair. We prove that, up to translations in space and time, there is exactly one kink-antikink pair having this threshold energy. I will explain the main...