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).

Difference Schemes for Solving the Generalized Nonlinear Schrödinger Equation

Basic properties of cold Bose atoms in optical lattices are reviewed. The main principles of correct self-consistent description of arbitrary systems with Bose-Einstein condensate are formulated. Theoretical methods for describing regular periodic lattices are presented. A special attention is paid to the discussion of Bose-atom properties in the frame of the boson Hubbard model. Optical lattices with arbitrary strong disorder, induced by random potentials, are treated. Possible applications of cold atoms in optical lattices are discussed, with an emphasis of their usefulness for quantum information processing and quantum computing. An important feature of the present review article, distinguishing it from other review works, is that theoretical fundamentals here are not just mentioned in brief, but are thoroughly explained. This makes it easy for the reader to follow the principal points without the immediate necessity of resorting to numerous publications in the field.

Cold Bosons in Optical Lattices

The nonlinear propagation of femtosecond pulses in photonic-crystal fibers is investigated theoretically without the use of the slowly varying envelope approximation. Low-intensity supercontinuum generation caused by fission of higher-order solitons into red-shifted fundamental solitons and blue-shifted nonsolitonic radiation is studied in a large range of fiber and pulse parameters. It is shown that phase matching of degenerate four-wave mixing can be achieved in an extremely broad frequency range from the IR to the UV. Spontaneous generation of new frequency components and parametric amplification by four-wave mixing as well as its possible overlap with soliton fission are studied in detail.

Supercontinuum generation, four-wave mixing, and fission of higher-order solitons in photonic-crystal fibers

Local discontinuous Galerkin methods for nonlinear Schrödinger equations

We survey some of the applications of soliton theory in laser physics. We briefly treat the theory of optical self-focussing and optical filamentation in neutral dielectrics and plasmas where the governing equation is the non-linear Schrodinger equation or one of its generalisations. We establish the connection of this theory with 1-dimensional Langmuir turbulence in plasmas. We treat the theory of optical self-induced transparency (SIT) at greater length and develop the Maxwell-Bloch (MB), reduced Maxwell-Bloch (RMB), SIT and sine-Gordon (s-G) equations to describe it. An optical three-wave interaction is related to the s-G equation; and reference is made to recent work on solitons in stimulated Raman scattering. The RMB equations are solved by a Zakharov-Shabat-AKNS inverse scattering scheme. The inhomogeneously broadened RMB equations have the unusual feature that ln a is not a constant of the motion. However, the sharp line RMB equations have two infinite sets of conserved densities, and the system constitutes one more example of a completely integrable infinite dimensional Hamiltonian system. In terms of scattering data the Hamiltonian of the RMB equations separates into soliton, breather and `background', that is `radiation', contributions. The sine-Gordon equation and its separable Hamiltonian are found as a special case of the RMB equations and its Hamiltonian. Averaged Lagrangian techniques are independently used to relate the c-number MB, RMB and SIT equations and to analyse the slowly varying phase and envelope approximations by which the SIT equations are derived from the MB or RMB equations. The connection of this Lagrangian theory with the Hamiltonian theory is not established. The Hamiltonian formalism in terms of the scattering data is used to quantise the RMB and s-G equations. The RMB, like the s-G, has the discrete energy level spectrum associated with a quantised breather; but only in the s-G limit is the quantised system easy to interpret. The quantised s-G is used to model a `coarse grained' operator theory of strictly resonant sharp line optical pulse propagation. The validity of such a description is examined. The physics of the c-number RMB equations is also discussed and particularly the relation of the c-number breather solutions to the 2π-pulse solutions of the SIT equations. The c-number RMB breather solutions provide a more general theory of SIT valid (within the 2-level atom model) at all electromagnetic field intensities and restricted only by the low density condition which permits the neglect of back scattering. Finally we look at four problems in resonant non-linear optics from a physical point of view. These are degenerate SIT and singular perturbation theory for it; the collision of oppositely directed resonant optical pulses; the theory of super-radiance; and the theory of optical self-focussing in resonant SIT.

http://iopscience.iop.org/1402-4896/20/3-4/011/pdf/1402-4896_20_3-4_011.pdf

P M Jack

Papers

Solitons in Laser Physics