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).

Superstring theoryをめぐって(放談室)

Analogue gravity is a research programme which investigates analogues of
general relativistic gravitational fields within other physical systems,
typically but not exclusively condensed matter systems, with the aim of gaining
new insights into their corresponding problems Analogue models of (and for)
gravity have a long and distinguished history dating back to the earliest years
of general relativity In this review article we will discuss the history,
aims, results, and future prospects for the various analogue models We start
the discussion by presenting a particularly simple example of an analogue
model, before exploring the rich history and complex tapestry of models
discussed in the literature The last decade in particular has seen a
remarkable and sustained development of analogue gravity ideas, leading to some
hundreds of published articles, a workshop, two books, and this review article
Future prospects for the analogue gravity programme also look promising, both
on the experimental front (where technology is rapidly advancing) and on the
theoretical front (where variants of analogue models can be used as a
springboard for radical attacks on the problem of quantum gravity)

Analogue Gravity

Dispersion and nonlinearity play a fundamental role in wave motions in nature. The nonlinear shallow water equations that neglect dispersion altogether lead to breaking phenomena of the typical hyperbolic kind with the development of a vertical profile. In particular, the linear dispersive term in the Korteweg–de Vries equation prevents this from ever happening in its solution. In general, breaking can be prevented by including dispersive effects in the shallow water theory. The nonlinear theory provides some insight into the question of how nonlinearity affects dispersive wave motions. Another interesting feature is the instability and subsequent modulation of an initially uniform wave profile.

Solitons and the Inverse Scattering Transform

This paper develops a new approach in the analysis of the Camassa–Holm equation. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose solutions are obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global solutions, depending continuously on the initial data. Our solutions are conservative, in the sense that the total energy equals a constant, for almost every time.

/pdf/global-conservative-solutions-of-the-camassa-holm-equation-2gi6h3ifov.pdf

Global conservative solutions of the Camassa-Holm equation

Multipeakons and the Classical Moment Problem

We present an inverse scattering approach for computing n-peakon solutions of the Degasperis-Procesi equation (a modification of the Camassa-Holm (CH) shallow water equation) The associated non-self-adjoint spectral problem is shown to be amenable to analysis using the isospectral deformations induced from the n-peakon solution, and the inverse problem is solved by a method generalizing the continued fraction solution of the peakon sector of the CH equation

https://iopscience.iop.org/article/10.1088/0266-5611/19/6/001/pdf

Multi-peakon solutions of the Degasperis–Procesi equation

A closed form of the multi-peakon solutions of the Camassa-Holm equation is found using a theorem of Stieltjes on continued fractions. An explicit formula is obtained for the scattering shifts.

https://iopscience.iop.org/article/10.1088/0266-5611/15/1/001/pdf

Jacek Szmigielski

Papers

Multipeakons and the Classical Moment Problem

Multi-peakon solutions of the Degasperis–Procesi equation

Multi-peakons and a theorem of Stieltjes

Acoustic Scattering and the Extended Korteweg– de Vries Hierarchy

Multipeakons and a theorem of Stieltjes