Stability of attractive Bose-Einstein condensates in a periodic potential.

Using a standing light wave potential, a stable quasi-one-dimensional attractive dilute-gas Bose-Einstein condensate can be realized. In a mean-field approximation, this phenomenon is modeled by the cubic nonlinear Schrödinger equation with attractive nonlinearity and an elliptic function potential of which a standing light wave is a special case. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schrödinger equation nor in the integrable nonlinear Schrödinger equation. Their stability is examined using analytic and numerical methods. Trivial-phase solutions are experimentally stable provided they have nodes and their density is localized in the troughs of the potential. Stable time-periodic solutions are also examined.

Mathematical frontiers in optical solitons.

Solitons are localized concentrations of field energy, resulting from a balance of dispersive and nonlinear effects. They are ubiquitous in the natural sciences. In recent years optical solitons have arisen in new and exciting contexts that differ in many ways from the original context of coherent propagation in a uniform medium. We review recent developments in incoherent spatial solitons and in gap solitons in periodic structures.

Bose-Einstein condensates in standing waves: the cubic nonlinear Schrödinger equation with a periodic potential.

We present a new family of stationary solutions to the cubic nonlinear Schrödinger equation with an elliptic function potential. In the limit of a sinusoidal potential our solutions model a quasi-one-dimensional dilute gas Bose-Einstein condensate trapped in a standing light wave. Provided that the ratio of the height of the variations of the condensate to its dc offset is small enough, both trivial phase and nontrivial phase solutions are shown to be stable. Recent developments allow for experimental investigation of these predictions.

Stability of repulsive Bose-Einstein condensates in a periodic potential.

The cubic nonlinear Schrödinger equation with repulsive nonlinearity and an elliptic function potential models a quasi-one-dimensional repulsive dilute gas Bose-Einstein condensate trapped in a standing light wave. New families of stationary solutions are presented. Some of these solutions have neither an analog in the linear Schrödinger equation nor in the integrable nonlinear Schrödinger equation. Their stability is examined using analytical and numerical methods. All trivial-phase stable solutions are deformations of the ground state of the linear Schrödinger equation. Our results show that a large number of condensed atoms is sufficient to form a stable, periodic condensate. Physically, this implies stability of states near the Thomas-Fermi limit.

Observation of the breakup of a prechirped N-soliton in an optical fiber.

We present what is believed to be the first experimental evidence showing the breakup of a chirped N-soliton pulse into an ordered train of fundamental solitons, as predicted by theory. We also present numerical experiments that confirm this phenomenon. Implications for optical communications systems that use chirped pulses are discussed.

Modulational stability of plane waves in nonreturn-to-zero communications systems with dispersion management.

Dispersion-managed optical transmission lines, with dispersion periodically switched between the normal and anomalous regimes, offer significantly better performance than transmission lines with constant dispersion by reducing the dispersion penalty and spectral broadening owing to self-phase modulation. We analyze the evolution of plane waves in a dispersion-managed transmission line, using Floquet theory, and show them to be modulationally stable, provided that the average dispersion is zero or negative (normal dispersion) and that the switching is fast enough, and to be unstable when anomalous dispersion dominates. These results indicate that the transition regions between 1's and 0's are primarily responsible for pulse deformations.