Section: New Results

Schrödinger equations

In [18] , S. De Bièvre, G. Dujardin, and S. Rota-Noradi, in collaboration with physicists of the PhLAM laboratory in Lille, developed an analysis of the phenomenon of modulational instability in dispersion-kicked optical fibers. They proposed a genuine analysis of the phenomenon, together with estimates on physical properties such as the gain along the fibers, and they showed that their analysis actually fits both numerical and physical experiments.

In [20] , S. De Bièvre and G. Dujardin, in collaboration with physicists of the PhLAM laboratory in Lille, developed an analysis of the propagation along a periodically-modulated optic fiber of generalized Peregrine rogue waves. In particular, they provided a full analysis of the multiple compression points appearing in such waves.

In D. Bonheure and R. Nascimento [21] obtained new results on the existence and qualitative properties of waveguides for a mixed-diffusion NLS. They provided a full qualitative description of the waveguides when the fourth order dissipation is small.

S. De Bièvre, S. Rota Nodari, and F. Genoud (CEMPI visitor, September 2013) have explained the geometry underlying the so-called energy-momentum method for proving orbital stability in infinite dimensional Hamiltonian systems. Applications include the orbital stability of solitons of the NLS and Manakov equations. This work appeared as a chapter (120p) in the first volume of the CEMPI Lecture Notes in Mathematics, cf. [48] .

In [26] , Bonheure, S. Cingolani and M. Nys obtained new striking results on stationary solutions of the 3D NLS driven by an exterior magnetic field. They construct a new class of cylindrical solutions in the energy class which concentrate, in the semi-classical limit, on a circle of the plane through the equator. In contrast with the case of solutions localized around a single point, the concentration is driven by the electrical field as well as the magnetic field