A topical review of numerical and experimental studies of supercontinuum generation in photonic crystal fiber is presented over the full range of experimentally reported parameters, from the femtosecond to the continuous-wave regime. Results from numerical simulations are used to discuss the temporal and spectral characteristics of the supercontinuum, and to interpret the physics of the underlying spectral broadening processes. Particular attention is given to the case of supercontinuum generation seeded by femtosecond pulses in the anomalous group velocity dispersion regime of photonic crystal fiber, where the processes of soliton fission, stimulated Raman scattering, and dispersive wave generation are reviewed in detail. The corresponding intensity and phase stability properties of the supercontinuum spectra generated under different conditions are also discussed.

Supercontinuum generation in photonic crystal fiber

http://cds.cern.ch/record/262851/files/9405029.pdf

Supersymmetry and quantum mechanics

Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations

Symplectic structures, their Bäcklund transformations and hereditary symmetries

Attraction between the atoms of a Bose–Einstein condensate renders it unstable to collapse, although a condensate with a limited number of atoms1 can be stabilized2 by confinement in an atom trap. However, beyond this number the condensate collapses3,4,5. Condensates constrained to one-dimensional motion with attractive interactions are predicted to form stable solitons, in which the attractive forces exactly compensate for wave-packet dispersion1. Here we report the formation of bright solitons of 7Li atoms in a quasi-one-dimensional optical trap, by magnetically tuning the interactions in a stable Bose–Einstein condensate from repulsive to attractive. The solitons are set in motion by offsetting the optical potential, and are observed to propagate in the potential for many oscillatory cycles without spreading. We observe a soliton train, containing many solitons; repulsive interactions between neighbouring solitons are inferred from their motion.

Formation and propagation of matter-wave soliton trains

Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media

A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I

Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II

is encountered in various physical problems. Gross [1J and Pitaevskil' [2J, and later on Tsuzuki [3J, used this equation to describe the oscillations of a Bose gas at zero temperature. The problem of the evolution of the complex envelope of a monochromatic wave in a nonlinear medium [4,5J also leads to Eq. (1). Finally, interpreting t as a longitudinal coordinate, Eq. (1) can be treated as a two-dimensional variant of the well known \"parabolic\" equation[6J that describes stationary wave beams in a nonlinear medium.

Interaction between solitons in a stable medium

In this volume each of the contributors proposes his own test to recognize integrable PDEs. We believe that, independently from the basic definition of integrability, the test must satisfy some general requirements. Namely, it has to be 


effective (in other words, if an equation has passed through the test, then there are almost no doubts about its integrability);


sufficiently algorithmical, yet able to admit a proper realization in a symbolic computer language (like Reduce, Formac, Macsyma, MuMath, AMP, etc.);


applicable to a large class of PDEs.

Alexey Borisovich Shabat

Papers

Exact Theory of Two-dimensional Self-focusing and One-dimensional Self-modulation of Waves in Nonlinear Media

A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I

Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II

Interaction between solitons in a stable medium

The Symmetry Approach to Classification of Integrable Equations