We consider small perturbations of the Zakharov-Shabat nonlinear Schrr odinger equation on 0; ] with vanishing or periodic boundary conditions; we prove a Nekhoroshev type result for solutions starting in the neighbourhood (in the H 1 topology) of the majority of small amplitude nite dimensional invariant tori of the linearized system. More precisely we will prove that along the considered solu...

The one-dimensional Schrodinger equation is considered when the potential and its rst moment are absolutely integrable. The transmission coe cient vanishes at zero energy in the generic case, and it never vanishes in the exceptional case. It is shown that any nontrivial exceptional potential can always be fragmented into two generic potentials. Furthermore, any nontrivial potential, generic or...

A fundamental postulate of quantum mechanics states that a measurement of a quantum system causes the collapse of the wave function. This is in contrast to the smooth, continuous, deterministic time evolution of the wave function in between measurements, governed by the Schrodinger equation. The latter is similar to classical physics; the former is unique to quantum mechanics. (Note that many ...

For a typical NRZ pulse propagating in existing normally dispersive bers, we provide a uniform description of the optical transmission. There are three distinct regimes of the governing nonlinear Schrr odinger equation: the fully nonlinear dispersive regime (ND), which covers a small region of the pulse; and two limiting asymptotic regimes-nonlinear, weakly dispersive (ZD) (for the bulk of the ...

Quasi-periodic solutions with Liouvillean frequency of forced nonlinear Schr\"odinger equation are constructed. This is based on an infinite dimensional KAM theory for frequency.

Optical turbulence, which develops in the transverse directions when a laser beam propagates through a medium, is considered in the framework of the two-dimensional Schrodinger equation with a weak cubic nonlinearity. This classical model of optical turbulence was studied by many authors, but stationary (Kolmogorov) spectra that transfer transverse energy of the waves from large spatial scales...

The eld equations describing the propagation of electromagnetic waves in a nonlinear dielectric medium whose polarization responds locally to the electric eld as an anharmonic oscillator with potential V (P) have smooth solutions global in space and time for arbitrary smooth initial data as soon as V has bounded derivatives of order less than or equal to three. This is true in spite of the fact...

We study and compare the decoherent histories approach, the environment-induced decoherence and the localization properties of the solutions to the stochastic Schrr odinger equation in quantum jump simulation and quantum state diiusion approaches, for a quantum two-level system model. We show, in particular, that there is a close connection between the decoherent histories and the quantum jump ...

The one-dimensional Schrr odinger equation is considered when the potential and its rst moment are absolutely integrable. The transmission coeecient vanishes at zero energy in the generic case, and it never vanishes in the exceptional case. It is shown that any nontrivial exceptional potential can always be fragmented into two generic potentials. Furthermore, any nontrivial potential, generic o...

Schrr odinger processes are deened as mixtures of Brownian bridges which preserve the Markov property. In nite dimensions, they can be characterized as h-transforms in the sense of Doob for some space-time harmonic function h of Brownian motion, and also as solutions to a large deviation problem introduced by Schrr odinger which involves minimization of relative entropy with given marginals. As...