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 $g=0,\lambda \ne 0$, $V$ is a random potential, and ${\varphi }_0$ is deterministic, this is the standard random Schrödinger equation describing for example the motion of an electron in a random medium. The main issue in this setting is the determination of the regime of Anderson localization, a property characterized by the boundedness in time of the second moment $\int x^2{\left|\varphi (t,x)\right|}^{2}dx$ of the solution. If this second moment remains bounded in time, the solution is said to be localized. Whereas it is known that the solution is localized in one dimension for all (suitable) initial data, both localized and delocalized solutions exist in dimension 3 and it remains a major open problem today to prove this, cf. [41].

If now $g\ne 0,\lambda \ne 0$ and $V$ is still random, but $\left|g\right|\ll \lambda$, 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.