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).

This book describes the remarkable connections that exist between the classical differential geometry of surfaces and modern soliton theory. The authors also explore the extensive body of literature from the nineteenth and early twentieth centuries by such eminent geometers as Bianchi, Darboux, Backlund, and Eisenhart on transformations of privileged classes of surfaces which leave key geometric properties unchanged. Prominent amongst these are Backlund-Darboux transformations with their remarkable associated nonlinear superposition principles and importance in soliton theory. It is with these transformations and the links they afford between the classical differential geometry of surfaces and the nonlinear equations of soliton theory that the present text is concerned. In this geometric context, solitonic equations arise out of the Gaus-Mainardi-Codazzi equations for various types of surfaces that admit invariance under Backlund-Darboux transformations. This text is appropriate for use at a higher undergraduate or graduate level for applied mathematicians or mathematical physics.

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Bäcklund and Darboux transformations : geometry and modern applications in soliton theory

A classification of discrete integrable systems on quad-graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the three-dimensional consistency. This property yields, among other features, the existence of the discrete zero curvature with a spectral parameter. For all integrable systems of the obtained exhaustive list, the so called three-leg forms are found. This establishes Lagrangian and symplectic structures for these systems, and the connection to discrete systems of the Toda type on arbitrary graphs. Generalizations of these ideas to the three-dimensional integrable systems and to the quantum context are also discussed.

Classification of integrable equations on quad-graphs. The consistency approach

The main purpos e of this chapter is to present a direct and systematic way of finding exact solutions and Backlund transformations of a certain class of nonlinear evolution equations. The nonlinear evolution equations are transformed, by changing the dependent variable(s), into bilinear differential equations of the following special form 
 
$$ F\left( {\frac{\partial }{{\partial t}} - \frac{\partial }{{\partial {t^1}}},\frac{\partial }{{\partial x}} - \frac{\partial }{{\partial {x^1}}}} \right)f(t,x)f({t^1},{x^1}){|_{t = {t^1},x = {x^1}}} = 0 $$ 
 
, which we solve exactly using a kind of perturbational approach.

Direct Methods in Soliton Theory (非線形現象の取扱いとその物理的課題に関する研究会報告)

By means of a special variable separation approach, a common formula with some arbitrary functions has been obtained for some suitable physical quantities of various (2+1)-dimensional models such as the Davey-Stewartson (DS) model, the Nizhnik-Novikov-Veselov (NNV) system, asymmetric NNV equation, asymmetric DS equation, dispersive long wave equation, Broer-Kaup-Kupershmidt system, long wave-short wave interaction model, Maccari system, and a general (N+M)-component Ablowitz-Kaup-Newell-Segur (AKNS) system. Selecting the arbitrary functions appropriately, one may obtain abundant stable localized interesting excitations such as the multidromions, lumps, ring soliton solutions, breathers, instantons, etc. It is shown that some types of lower dimensional chaotic patterns such as the chaotic-chaotic patterns, periodic-chaotic patterns, chaotic line soliton patterns, chaotic dromion patterns, fractal lump patterns, and fractal dromion patterns may be found in higher dimensional soliton systems. The interactions between the traveling ring type soliton solutions are completely elastic. The traveling ring solitons pass through each other and preserve their shapes, velocities, and phases. Some types of localized weak solutions, peakons, are also discussed. Especially, the interactions between two peakons are not completely elastic. After the interactions, the traveling peakons also pass through each other and preserve their velocities and phases, however, they completely exchange their shapes.

Localized excitations in (2+1)-dimensional systems.

Some new integrable nonlinear evolution equations in 2 + 1 dimensions

Induced Surfaces and Their Integrable Dynamics

(1+1)-dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems

It is shown that the discrete Darboux system, descriptive of conjugate lattices in Euclidean spaces, admits constraints on the (adjoint) eigenfunctions which may be interpreted as discrete orthogonality conditions on the lattices. Thus, it turns out that the elementary quadrilaterals of orthogonal lattices are cyclic. Orthogonal lattices on lines, planes and spheres are discussed and the underlying integrable systems in one, two and three dimensions are derived explicitly. A discrete analogue of Bianchi's Ribaucour transformation is set down and particular orthogonal lattices are given. As a by–product, discrete Dini surfaces are obtained.

Boris Konopelchenko

Papers

Some new integrable nonlinear evolution equations in 2 + 1 dimensions

Induced Surfaces and Their Integrable Dynamics

(1+1)-dimensional integrable systems as symmetry constraints of (2+1)-dimensional systems

Three–dimensional integrable lattices in Euclidean spaces: conjugacy and orthogonality

On (2+1)-dimensional nonlinear systems of Loewner-type