There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

A comprehensive review of zonal flow phenomena in plasmas is presented. While the emphasis is on zonal flows in laboratory plasmas, planetary zonal flows are discussed as well. The review presents the status of theory, numerical simulation and experiments relevant to zonal flows. The emphasis is on developing an integrated understanding of the dynamics of drift wave–zonal flow turbulence by combining detailed studies of the generation of zonal flows by drift waves, the back-interaction of zonal flows on the drift waves, and the various feedback loops by which the system regulates and organizes itself. The implications of zonal flow phenomena for confinement in, and the phenomena of fusion devices are discussed. Special attention is given to the comparison of experiment with theory and to identifying directions for progress in future research.

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Zonal flows in plasma—a review

Preface Introduction 1. Electric properties of the dielectric fiber 2. Derivation of wave packet equation and introduction to soliton transmission systems 3. Inverse scattering transform and N-Soliton solutions 4. Perturbation methods 5. Effects on higher order terms 6. Reshaping of solitons and concept of guiding center (Average) solitons 7. Soliton transmission control 8. Interaction among solitons in same channel 9. Interaction among solitons in different channels 10. Femto second solitons 11. Soliton transmission experiments 12. Other applications of optical solitons 13. Spatial solitons, stability and chaos 14. Modulational instability 15. Dark solitons General references

Solitons in optical communications

We derive the nonlinear wave equation for an envelope of an electromagnetic wave in a monomode dielectric waveguide. Concrete examples are given for a single-mode optical fiber where the coefficients of resultant nonlinear Schrodinger equation with higher-order dispersion and dissipation (both linear and nonlinear) are given in terms of properties of the eigenfunction of the guided wave as well as of the material nonlinearity and dispersion. Using a newly-developed perturbation method, we show that the higher-order dispersions (linear and nonlinear) perserve the profile of a single soliton but to split up a bound N soliton ( N \geq 2 ) into individual solitons with different heights which propagate at different velocities. We also show that the higher-order nonlinear dissipation due to the induced Raman effect downshifts the carrier frequency of a single soliton in proportion to the distance of propagation and to the fourth power of the soliton amplitude.

Nonlinear pulse propagation in a monomode dielectric guide

We discuss the propagation of optical solitons in a monomode fiber as a model of long-distance-high-bit-rate transmission system. We give several new results which did not appear in our previous papers on this subject, such as (1) a derivation of the perturbed nonlinear Schrodinger equation from the Maxwell equation, (2) on the integrability of the perturbed nonlinear Schrodinger equation, (3) a discussion of the soliton as a stable fixed point of certain infinite-dimensional map generated by a transmission system with periodic excitations.

Optical solitons in a monomode fiber

Spectrum cascade in drift wave turbulence in a magnetized plasma as well as Rossby wave turbulence in an atmospheric pressure system are studied based on a three‐wave decay process derivable from the model equation applicable to both cases. The decay in the three‐way interaction occurs to smaller and larger values of ‖k‖. In a region of large wavenumbers this leads to the dual cascade; the energy spectrum cascades to smaller ‖k‖ and the enstrophy spectrum to larger ‖k‖, similar to the case of two‐dimensional Navier–Stokes turbulence. In a small wavenumber region a resonant three‐wave decay process dominates the cascade process, and an anisotropic spectrum develops. As a consequence of the cascade, zonal flows in the direction perpendicular to the direction of inhomogeneity appear which presents a potential implication for the particle confinement in a turbulent plasma.

Nonlinear behavior and turbulence spectra of drift waves and Rossby waves

By a proper design of the frequency-dependent gain characteristic of optical amplifiers, the parameters (amplitude η and velocity κ) of optical solitons in fibers can be made to approach a desired fixed point in κ − η space. The method is effective in controlling the random walk of solitons caused either by initial jitter and/or by amplifier noise (the Gordon–Haus effect) and in overcoming the bit-rate limitation that they provide.

Yuji Kodama

Papers

Solitons in optical communications

Nonlinear pulse propagation in a monomode dielectric guide

Optical solitons in a monomode fiber

Nonlinear behavior and turbulence spectra of drift waves and Rossby waves

Generation of asymptotically stable optical solitons and suppression of the Gordon-Haus effect.