A note on Berestycki-Cazenave's classical instability result for nonlinear Schr\"odinger equations

In this note we give an alternative, shorter proof of the classical result of Berestycki and Cazenave on the instability by blow-up for the standing waves of some nonlinear Schr\"odinger equations

High speed excited multi-solitons in nonlinear Schr\"odinger equations

We consider the nonlinear Schr\"odinger equation with a general nonlinearity. In dimension higher than 2, this equation admits travelling wave solutions with a fixed profile which is not the ground state. This kind of profiles are called excited states. In this paper, we construct solutions to NLS behaving like a sum of N excited states which spread up quickly as time grows (which we call multi-solitons). We also show that if the flow around one of these excited states is linearly unstable, then the multi-soliton is not unique, and is unstable

Chemical analysis of a single basic cell of porous anodic aluminium oxide templates

We prepared anodic aluminium oxide (AAO) templates with “honeycomb” geometry, i.e. hexagonally ordered circular pores. The structures were extensively studied and characterized by EPMA coupled with FEG-SEM and FEG-TEM coupled with EDX at meso and nanoscopic scales, in other words, at the scale of a single basic cell making up the highly ordered porous anodic film. The analyses allowed the identification of the chemical compounds present and the evaluation of their levels in the different parts of each cell. Of note was the absence of phosphates inside the “skeleton” and their high content in the “internal part”. Various models of porous anodic film growth are discussed on the basis of the results, contributing to a better understanding of AAO template preparation and selfnanostructuring phenomena

Minimal mass blow up solutions for a double power nonlinear Schr\"odinger equation

We consider a nonlinear Schr\"odinger equation with double power nonlinearity, where one power is focusing and mass critical and the other mass sub-critical. Classical variational arguments ensure that initial data with mass less than the mass of the ground state of the mass critical problem lead to global in time solutions. We are interested by the threshold dynamic and in particular by the existence of finite time blow up minimal solutions. For the mass critical problem, such an object exists thanks to the explicit conformal symmetry, and is in fact unique. For the focusing double power nonlinearity, we exhibit a new class of minimal blow up solutions with blow up rates deeply affected by the double power nonlinearity. The analysis adapts the recent approach developed by Rapha\"el and Szeftel for the construction of minimal blow up elements

Procréation médicalement assistée

National audienc

Embryon

National audienc

Standing waves in nonlinear Schrödinger equations

In the theory of nonlinear Schrödinger equations, it is expected that the solutions will either spread out because of the dispersive effect of the linear part of the equation or concentrate at one or several points because of nonlinear effects. In some remarkable cases, these behaviors counterbalance and special solutions that neither disperse nor focus appear, the so-called standing waves. For the physical applications as well as for the mathematical properties of the equation, a fundamental issue is the stability of waves with respect to perturbations. Our purpose in these notes is to present various methods developed to study the existence and stability of standing waves. We prove the existence of standing waves by using a variational approach. When stability holds, it is obtained by proving a coercivity property for a linearized operator. Another approach based on variational and compactness arguments is also presented. When instability holds, we show by a method combining a Virial identity and variational arguments that the standing waves are unstable by blow-up