S. B. Medvedev

Physics and mathematics of dispersion-managed optical solitons

We review the main physical and mathematical properties of dispersion-managed (DM) optical solitons. Theory of DM solitons can be presented at two levels of accuracy: first, simple, but nevertheless, quantitative models based on ordinary differential equations governing evolution of the soliton width and phase parameter (the so-called chirp); and second, a comprehensive path-average theory that is capable of describing in detail both the fine structure of DM soliton form and its evolution along the fiber line. An analogy between DM soliton and a macroscopic nonlinear quantum oscillator model is also discussed.

Hamiltonian averaging and integrability in nonlinear systems with periodically varying dispersion

By applying Hamiltonian averaging and a quasi-identity-like transformation it is demonstrated that the averaged dynamics of high-frequency nonlinear waves in systems with periodically varying dispersion can be described in a particular limit by the integrable nonlinear Schrödinger equation.

Interaction of two fractions in a degenerate bose gas at finite temperatures

Free expansion of Bose–Einstein condensates of rubidium atoms at finite temperatures has been analyzed experimentally and theoretically. It has been shown that the interaction between condensed and noncondensed atoms is manifested most clearly by a decrease in the density of atoms in the center of the expanding cloud as compared to the theoretical prediction for a pure condensate.

Evolution of a steady state of the two-dimensional Gross-Pitaevskii equation

Expansion of a steady state of the Gross-Pitaevskii equation after switching off the external field has been investigated. It has been shown that the evolution of the aspect ratio of the localized solution is described by the one-dimensional oscillator equation with renormalized time. The renormalization is determined by the evolution of the width or the second moment of the solution. It has been found that the aspect ratio is monotonically inverted in infinite time in the case of the linear Schrödinger equation and does not reach the inverse value in the nonlinear case.

The theory of optical communication lines with a short-scale dispersion management

We investigate, theoretically and numerically, properties of dispersion-managed (DM) solitons in fiber lines with the dispersion compensation period L much shorter than the amplification distance Za. We present the path-averaged theory of DM transmission lines with a short-scale management in the case of asymmetric maps. Applying a quasi-identical transformation, we demonstrate that the path-averaged dynamics in such systems can be described by an integrable model in some limits.

Inverse problem for stationary state of a Bose-Einstein condensate

The Bose-Einstein condensate is generally described by the three-dimensional equations. However, the two-dimensional case is of great interest too. First, these results can be directly used to study the planar condensates [1]. Second, it serves as a test version for numerical calculations and as a quasi-two-dimensional approximation for the complete three-dimensional problem. An additional advantage of the two-dimensional case is that with the help of variational approach one can obtain analytical results for the stationary state and for the expansion of the condensate [2].

Quasilinear theory for the nonlinear Schrödinger equation with periodic coefficients

The nonlinear Schrödinger equation with periodic coefficients is analyzed under the condition of large variation in the local dispersion. The solution after n periods is represented as the sum of the solution to the linear part of the nonlinear Schrödinger equation and the nonlinear first-period correction multiplied by the number of periods n. An algorithm for calculating the quasilinear solution with arbitrary initial conditions is proposed. The nonlinear correction to the solution for a sequence of Gaussian pulses is obtained in the explicit form.