Alexander S. Kovalev

Gap and out-gap solitons in modulated systems of finite length: exact solutions in the slowly varying envelope limit

We discuss nonlinear excitations in finite-size one-dimensional modulated systems. Considering a binary modulated discrete nonlinear Schrodinger chain of large but finite length with periodic boundary conditions, we obtain exact elliptic-function solutions corresponding to stationary excitations in the slowly varying envelope limit. From these solutions, we analyze how the transformation between (localized) gap and (delocalized) out-gap solitons manifests itself in a system of finite length. The analogue of a localized gap soliton appears through a bifurcation at a critical point, so that gap soliton analogues exist only for chains longer than a critical value, which scales inversely proportional to the modulation depth. The total norm of these gap–out-gap states is found to be a monotonic function of the frequency, always inside a nonlinear gap with edges defined by the main nonlinear modes which approach the linear spectrum gap boundaries in the small-amplitude limit. The transformation from a gap to an out-gap state is associated with a particular frequency, close to the lower boundary of the linear gap; at this point the elliptic functions become trigonometric, corresponding to a finite-size analogue of an algebraic soliton. We compare the scenario with earlier results obtained numerically for purely discrete chains with few degrees of freedom.

Soliton propagation through a disordered system: statistics of the transmission delay.

We have studied the soliton propagation through a segment containing random pointlike scatterers. In the limit of small concentration of scatterers when the mean distance between the scatterers is larger than the soliton width, a method has been developed for obtaining the statistical characteristics of the soliton transmission through the segment. The method is applicable for any classical particle traversing through a disordered segment with the given velocity transformation after each act of scattering. In the case of weak scattering and relatively short disordered segment the transmission time delay of a fast soliton is mostly determined by the shifts of the soliton center after each act of scattering. For sufficiently long segments the main contribution to the delay is due to the shifts of the amplitude and velocity of a fast soliton after each scatterer. Corresponding crossover lengths for both cases of light and heavy solitons have been obtained. We have also calculated the exact probability density function of the soliton transmission time delay for a sufficiently long segment. In the case of weak identical scatterers the latter is a universal function which depends on a sole parameter--the mean number of scatterers in a segment.

On the theory of autosoliton propagation in optical fibers guided by in-line nonlinear devices

A theoretical model is developed to describe the propagation of ultra-short optical pulses in fiber transmission systems in the quasi-linear regime, with periodically inserted in-line lumped nonlinear optical devices. Stable autosoliton solutions are obtained for a particular application of the general theory.