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).

A Modern Introduction to the Mathematical Theory of Water Waves. By R. S. Johnson. Cambridge University Press, 1997. xiv+445 pp. Hardback ISBN 0 521 59172 4 £55.00; paperback 0 521 59832 X £19.95.

We observe that theN-state integrable chiral Potts model can be considered as a part of some new algebraic structure related to the six-vertex model. As a result, we obtain a functional equation which is supposed to determine all the eigenvalues of the chiral Potts model transfer matrix.

Chiral Potts model as a descendant of the six-vertex model

In this paper, we study on the conformable (2+1)-dimensional Ablowitz-KaupNewell-Segur equation in order to show the existence of complex combined dark-bright soliton solutions. To this purpose an effective method which is the sine-Gordon expansion method is used. The 2D and 3D surfaces under some suitable values of parameters are also plotted.

Complex solitons in the conformable (2+1)-dimensional Ablowitz-Kaup-Newell-Segur equation

A machine-generated list of 192 local solutions of the Heun equation is given. They are analogous to Kummer's 24 solutions of the Gauss hypergeometric equation, since the two equations are canonical Fuchsian differential equations on the Riemann sphere with four and three singular points, respectively. Tabulation is facilitated by the identification of the automorphism group of the equation with n singular points as the Coxeter group D n . Each of the 192 expressions is labeled by an element of D 4 . Of the 192, 24 are equivalent expressions for the local Heun function Hl, and it is shown that the resulting order-24 group of transformations of Hl is isomorphic to the symmetric group S 4 . The isomorphism encodes each transformation as a permutation of an abstract four-element set, not identical to the set of singular points.

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The 192 solutions of the Heun equation

A method is proposed for constructing finite-gap elliptic inx or/and int solutions of the Korteweg-de Vries equation. Dynamics of poles for two-gap elliptic solutions of the KdV equation are considered. Numerous examples of new elliptic solutions of the KdV equation are given.

Finite-gap elliptic solutions of the KdV equation

Real finite-gap regular solutions of the Kaup-Boussinesq equation

We find a new class of algebraic geometric solutions of Heun's equation with the accessory parameter belonging to a hyperelliptic curve Dependence of these solutions from the accessory parameter as well as their relation to Heun's polynomials is studied Methods of calculating the algebraic genus of the curve, and its branching points, are suggested Monodromy group is considered Numerous examples are given

Elliptic solitons and Heun's equation

Recently, considerable progress has been made in understanding the nature of the algebro-geometrical superposition principles for the solutions of nonlinear completely integrable evolution equations, and mainly for the equations related to hyperelliptic Riemann surfaces. Here we find such a superposition formula for particular real solutions of the KP and Boussinesq equations related to the nonhyperelliptic curve ω4 = (λ − E
1) (λ − E
2) (λ − E
3) (λ − E
4). It is shown that the associated Riemann theta function may be decomposed into a sum containing two terms, each term being the product of three one-dimensional theta functions. The space and time variables of the KP and Boussinesq equations enter into the arguments of these one-dimensional theta functions in a linear way.

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On the Riemann Theta Function of a Trigonal Curve and Solutions of the Boussinesq and KP Equations

The purpose of this paper is to determine a simple and efficient method of reduction of Riemann theta-functions of large genus to Riemann theta-functions of smaller genus.Figures: 1. Bibliography: 15 titles.

https://iopscience.iop.org/article/10.1070/SM1988v061n02ABEH003213/pdf

A. O. Smirnov

Papers

Finite-gap elliptic solutions of the KdV equation

Real finite-gap regular solutions of the Kaup-Boussinesq equation

Elliptic solitons and Heun's equation

On the Riemann Theta Function of a Trigonal Curve and Solutions of the Boussinesq and KP Equations

a Matrix Analogue of Appell's Theorem and Reductions of Multidimensional Riemann Theta-Functions