Solitary waves in nonlocal NLS with dispersion averaged saturated nonlinearities

A nonlinear Schr\"odinger equation (NLS) with dispersion averaged nonlinearity of saturated type is considered. Such a nonlocal NLS is of integro-differential type and it arises naturally in modeling fiber-optics communication systems with periodically varying dispersion profile (dispersion management). The associated constrained variational principle is shown to posses a ground state solution by constructing a convergent minimizing sequence through the application of a method similar to the classical concentration compactness principle of Lions. One of the obstacles in applying this variational approach is that a saturated nonlocal nonlinearity does not satisfy uniformly the so-called strict sub-additivity condition. This is overcome by applying a special version of Ekeland's variational principle.Comment: 24 page

Dispersion managed solitons in the presence of saturated nonlinearity

The averaged dispersion managed nonlinear Schrödinger equation with saturated nonlinearity is considered. It is shown that under rather general assumptions on the saturated nonlinearity, the ground state solution corresponding to the dispersion managed soliton can be found for both zero residual dispersion and positive residual dispersion. The same applies to di_raction management solitons, which are a discrete version describing certain waveguide arrays

On some nonlinear and nonlocal effective equations in kinetic theory and nonlinear optics

This thesis deals with some nonlinear and nonlocal effective equations arising in kinetic theory and nonlinear optics. First, it is shown that the homogeneous non-cutoff Boltzmann equation for Maxwellian molecules enjoys strong smoothing properties: In the case of power-law type particle interactions, we prove the Gevrey smoothing conjecture. For Debye-Yukawa type interactions, an analogous smoothing effect is shown. In both cases, the smoothing is exactly what one would expect from an analogy to certain heat equations of the form $\partial_t u = f(-\Delta)u$, with a suitable function $f$, which grows at infinity, depending on the interaction potential. The results presented work in arbitrary dimensions, including also the one-dimensional Kac-Boltzmann equation. In the second part we study the entropy decay of certain solutions of the Kac master equation, a probabilistic model of a gas of interacting particles. It is shown that for initial conditions corresponding to $N$ particles in a thermal equilibrium and $M\leq N$ particles out of equilibrium, the entropy relative to the thermal state decays exponentially to a fraction of the initial relative entropy, with a rate that is essentially independent of the number of particles. Finally, we investigate the existence of dispersion management solitons. Using variational techniques, we prove that there is a threshold for the existence of minimisers of a nonlocal variational problem, even with saturating nonlinearities, related to the dispersion management equation