## [ Abstracts ]

On some constrained minimization problems (Mihai Maris)
Abstract (TBA)
Finite time extinction by nonlinear damping for the Schrödinger equation (Clément Gallo)
We consider the Schrödinger equation on a compact manifold, in the presence of a nonlinear damping term, which is homogeneous and sublinear. For initial data in the energy space, we construct a weak solution, defined for all positive time, which is shown to be unique. In the one-dimensional case, we show that it becomes zero in finite time. In the two and three-dimensional cases, we prove the same result under the assumption of extra regularity on the initial datum.
Existence, stability, concentration properties and dynamics of solitons in Nonlinear Schrödinger Equation in presence of a strong nonlinearity (Marco Ghimenti)
I present some result about the dynamics of solitons (stable solitary waves) for the Nonlinear Schroedinger equation in presence of an external potential. We'll focus on the role on the nonlinearity in formation, stability and concentration properties of the soliton. I will discuss both the case of confining potential and bounded potential. The second result is obtained by a suitable notion of barycenter that seem to be useful to study the dynamics of solitons in more general frameworks.
Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations (Louis Jeanjean)
The aim of this talk is to study the existence and the stability of standing waves with prescribed $$L^2$$-norm for a class of Schrödinger-Poisson-Slater equations in $$\mathbb R^{3}$$ $$\label{evolution1} i\psi_{t}+ \Delta \psi - (|x|^{-1}*|\psi|^{2}) \psi+|\psi|^{p-2}\psi=0 %\ \ \text{ in } \mathbb R^{3},$$ when $$p \in (\frac{10}{3},6)$$. The standing waves are found as critical points of the associated energy functional $F(u) = \frac{1}{2}\int_{\mathbb R^3}|\nabla u|^2 dx + \frac{1}{4}\int_{\mathbb R^3}\int_{\mathbb R^3} \frac{|u(x)|^2 |u(y)|^2}{|x-y|}dxdy - \frac{1}{p} \int_{\mathbb R^3}|u|^p dx$ on the constraints given by $S(c) = \{u\in H^{1}(\mathbb R^{3}): \|u\|_{2}^2=c\}$ where $$c>0$$ is given. For the value of $$p \in (\frac{10}{3},6)$$ considered the functional $$F$$ is unbounded from below on $$S(c)$$ and the existence of critical points is obtained by a mountain pass theorem on $$S(c)$$. In order to show the compactness of the Palais-Smale sequences, we prove the monotonicity of the mountain pass energy levels $$\gamma(c)$$ as well as a localization lemma for a specific sequence. Our main result is that standing waves with prescribed $$L^2$$-norm exist provided that $$c>0$$ is sufficiently small. We shall also see that when $$c>0$$ is not small a non-existence result is expected. The solutions obtained are shown to be strongly unstable. Finally we draw a comparison between the Schrödinger-Poisson-Slater equation and the classical nonlinear Schrödinger equation.

## [ Registration ]

No registration is required.

## [ Place ]

The workshop will be held in the Mathematics Institute of Toulouse, seminar room MIP on the second floor of building 1R3.

## [ Organization ]

The workshop is organized by Stefan Le Coz, with the support of the ANR project Esonse and the IMT.

## [ Bonus ]

Some pictures of Toulouse (images from Flickr, click on the image for original link).