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## Section: Research Program

### Nonlinear Schrödinger equations

As well known, the (non)linear Schrödinger equation

with coupling constants $g\in ℝ,\lambda \in {ℝ}_{+}$ and real potential V (possibly depending also on time) models many phenomena of physics.

When in the equation (5) above one sets $\lambda =0,g\ne 0$, one obtains the nonlinear (focusing of defocusing) Schrödinger equation. It is used to model light propagation in optical fibers. In fact, it then takes the following form:

where $\beta$ and $\gamma$ are functions that characterize the physical properties of the fiber, $t$ is time and $z$ the position along the fiber. Several issues are of importance here. Two that will be investigated within the MEPHYSTO project are: the influence of a periodic modulation of the fiber parameters $\beta$ and $\gamma$ and the generation of so-called “rogue waves” (which are solutions of unusually high amplitude) in such systems.

If now $g\ne 0,\lambda \ne 0$ and $V$ is still random, but $|g|\ll \lambda$, a natural question is whether, and in which regime, one-dimensional Anderson localization perdures. Indeed, Anderson localization can be affected by the presence of the nonlinearity, which corresponds to an interaction between the electrons or atoms. Much numerical and some analytical work has been done on this issue (see for example [43] for a recent work at PhLAM, Laser physics department, Univ. Lille 1), but many questions remain, notably on the dependence of the result on the initial conditions, which, in a nonlinear system, may be very complex. The cold atoms team of PhLAM (Garreau-Szriftgiser) is currently setting up an experiment to analyze the effect of the interactions in a Bose-Einstein condensate on a closely related localization phenomenon called “dynamical localization”, in the kicked rotor, see below.