Table of contents

Sophie Kowalevski, 1850-1891

In 1889 Sophie Kowalevski published her famous paper `Sur le probleme de la rotation d'un corps solide autour d'un point fixe' (1889 Acta Math.12 177-232). This paper earned her a prize from the Paris Academy of Sciences and fame as an outstanding mathematician. It has now become an important part of her mathematical legacy. Another well known and frequently cited result of her research is the Cauchy-Kowalevski theorem proved in her Doctoral Dissertation in 1874. In both works she approached the solution of differential equations in terms of analytic or meromorphic functions, and in both cases solution led to specific restrictions on the differential equations in question. In the case of the Cauchy-Kowalevski theorem she found a sufficient condition for a very general ODE to have an analytic solution. In the problem of the rotation of a solid body around a fixed point she invented a new method, subsequently known as the Kowalevski-Painlevé method, and found a new case for the parameters of the equations when there is an extra integral of motion. This meant that this special case, since then referred to as the Kowalevski top, can be integrated in quadratures. In modern terminology the Kowalevski top was one of the first examples of an integrable system. The actual integration of the equations of motion was a hard but challenging task, and required enormous intuition. Even today, any `simplification' of her ingenious solution is a challenging problem. These two of her major mathematical achievements were at the frontiers of mathematical research at the end of the 19th century and they opened the way to what now exists as the theory of integrable systems.

A short biography of Sophie Kowalevski, with photographs and links to other sites, can be found in the MacTutor History of Mathematics Archive at http://www-history.mcs.st-and.ac.uk/history/.

In connection with the 150th anniversary of the birth of Sophie Kowalevski, one of us, Vadim Kuznetsov, together with Allan Fordy and Michael Semenov-Tian-Shansky, organized a Workshop on Mathematical Methods of Regular Dynamics which was held on 12-15 April 2000 in Tetley Hall, University of Leeds, UK. Details of this meeting can still be found on the web page: http://www.amsta.leeds.ac.uk/~vadim/work.html. It was well attended and not only paid tribute to Kowalevski's mathematical achievements but also served as a good opportunity for many researchers working in algebraic geometry, integrable systems and algebraic origins of integrability to meet and discuss the most recent results and to stimulate cooperation. Many areas were represented in 34 talks and 15 poster presentations. On behalf of organizers we want to thank the sponsors of the meeting: the London Mathematical Society, the Royal Society, the Russian Foundation of Basic Research and the Mathematical Physics Group of the Institute of Physics.

This Special Issue is a collection of research papers which were either reported during the meeting by their authors or submitted in reply to a call for papers announced in May 2000. It gives a good account of this vivid area of modern mathematical research. We hope you will enjoy reading it as much as we did. It should be mentioned that we have adopted the canonical spelling of `Sophie Kowalevski' but alternative forms such as `Sofia Kovalevskaya' and `Kowalewski' also exist in the literature.

The contents are split into four sections. The first section, Sophie Kowalevski's legacy consists mainly of papers directly related either to tops, discovering new integrable systems or to various methods used to identify, characterize or integrate an integrable system. The contributions by G Falqui, I V Komarov and D Markushevich discuss various new results for the Kowalevski top. B Gaffet establishes Liouville integrability of a new case in the equations describing spinning gas clouds. T Kozlova finds billiard systems with polynomial integrals of higher degree while O E Orel and P E Ryabov give a classification of Kirchhoff equations with an additional integral of fourth degree. K Nakagawa and H Yoshida provide a necessary condition for the integrability of homogeneous systems with two degrees of freedom. In the work of L A Piovan a construction of an integrable system in terms of algebro-geometric data is described. Integrable motion of an axisymmetric rigid body is the subject of T V Salnikova's work. The contribution of A Tsygvintsev deals with the problem of existence of polynomial first integrals.

In the Algebra and Analysis section contributions are assembled that deal with special functions and algebro-geometric methods applied to integrable systems. Nonlinear special functions, in particular the Painlevé transcendents, are treated in the papers by A W N Hone, A V Kitaev and M Mazzocco, whilst multivariable special functions appear in the contributions by A Yu Orlov and D M Scherbin, by F A Grunbaum and by T Sasaki and M Yoshida. These arise as well in connection with quantum many-body problems as is seen in the articles by H Braden and by S Kharchev and D Lebedev, whilst in the contributions by M Salerno et al and J C Eilbeck et al multi-variable Abelian functions play the central role. In the latter papers, the modern algebro-geometric constructions to integrable systems are exploited, whilst in the paper by T Sasaki and M Yoshida the geometry of moduli spaces of marked cubic surfaces is studied, giving rise to Appell-Lauricella systems of equations. Finally, numerical aspects are considered in the papers by E Celledoni and F A Grunbaum.

A number of contributions are situated on the crossroads between (differential) geometry and physics and therefore are placed in the section Geometry and Physics. The contribution by E D Belokolos deals with Kleinian functions and domain structures. The link between gauge theories and gravity on the one hand and integrable systems on the other hand is exhibited in the papers by D Bernard and N Regnault and by A Gorsky. The modern developments on systems of hydrodynamic type and dispersionless equations are represented by the contributions by O McCarthy and I Strachan and by E V Ferapontov. Lie algebraic aspects are represented in the papers by Ph Feinsilver and by L Casian and Y Kodama.

The last (not least) section is called Discrete Systems. Many of the modern developments in integrable systems and special functions deal with discrete systems governed by difference equations. The issue has a good representation of contributions in this direction: orthogonal polynomials and the discrete KP equation in the contribution by L Haine and P Iliev, discrete Painlevé equations in the contribution by A Ramani et al and in the form of Bäcklund transformations in the articles by R Hernández Heredero et al, A W N Hone et al and C Verhoeven and M Musette. Another type of discrete systems are `integrable' cellular automata which exhibit regular dynamics. These are represented in the contributions by K Hikami and R Inoue and by G R W Quispel et al.

We discuss the Poisson structure underlying the two-field Kowalevski gyrostat and the Kowalevski top. We start from their Lax structure and construct a suitable pencil of Poisson brackets which endows these systems with the structure of bi-Hamiltonian completely integrable systems. We study the Casimir functions of such pencils, and show how it is possible to frame the Kowalevski systems within the so-called Gel'fand-Zakharevich bi-Hamiltonian setting for integrable systems.

We consider an ordinary differential reduction of the gas-dynamical equations proposed by Ovsiannikov and Dyson, representing a tri-axial ellipsoidal gas cloud rotating as it expands into the vacuum. For a monatomic gas (γ = 5/3) without vorticity, the system has the Painlevé property and is integrable, at least in cases of rotation around a fixed axis. We present preliminary results concerning fully general states of rotation.

This paper constitutes a generalization to arbitrary states of rotation of an earlier work in which we showed Liouville integrability of the expanding and rotating gas cloud model of Ovsiannikov and of Dyson in cases of rotation around a fixed principal axis.

We present a review of some results on Kowalevski's top (KT) in classical and quantum mechanics. The following items are considered:

(i) the generalized KT (GKT) on the Lie algebras so(4), e(3) and so(3,1);

(ii) Kowalevski's gyrostat on these algebras;

(iii) action of the GKT;

(iv) quantum counterparts of the KT;

(v) semiclassical quantization of the GKT;

(vi) generalization of the KT by Chaplygin and Goryachev at l = 0 and its Lax representation.

Unsolved questions are also discussed.

The problem of the existence of polynomial-in-momenta first integrals for dynamical billiard systems is considered. Examples of billiards with irreducible integrals of third and fourth degree are constructed with the help of the integrable problems of Goryachev-Chaplygin and Kovalevsky from rigid body dynamics.

Kowalevski's curve of genus 2 is related to two other curves arising from the solution of the Kowalevski top by the method of spectral curves in the case when the angular momentum of the top is orthogonal to the gravity vector. One is the Bobenko-Reyman-Semenov-Tian-Shansky curve of genus 2, the other is the spectral curve of the Kuznetsov-Tsiganov Lax matrix, of genus 3. The relations between the curves are given by correspondences, that is, multivalued maps, inducing isogenies of the corresponding Jacobian or Prym varieties.

A necessary condition for the integrability of Hamiltonian systems with a two-dimensional homogeneous potential, due to Morales-Ruiz and Ramis, is extended for more general Hamiltonian systems of the form H = T(p) + V(q), with homogeneous functions T(p) and V(q).

We study the topology of energy surfaces and obtain the bifurcation set for the integrable problem of the motion of a rigid body in a fluid. We also describe all bifurcations of Liouville tori and calculate the Fomenko-Zieschang invariant. To do this we use methods of studying integrable Hamiltonian systems, for which the Lax representation and separation of variables are not known. During our study we reveal some new topological effects. In particular, we observe the bifurcation of two tori into four, which had not been observed in mechanical systems previously.

We use the algebro-geometric data given by a genus-2 Jacobian, a curve and a line bundle on the Jacobian, and the action of a group of translates on the theta sections of this line bundle, to reconstruct an integrable system: the geodesic motion on {SO}(4), metric II (so termed after Adler and van Moerbeke).

We consider the problem of the existence of additional analytical first integrals in some Hamiltonian systems, which are close to integrable, namely, the motion of a rigid body is close to a dynamically symmetric one.

The problem of motion of a rigid body in an ideal liquid (the Kirchhoff problem) and the similar problem of rotation of a rigid body with a fixed point in an axisymmetric force field with a quadratic potential are investigated. The existence of hyperbolic periodic and asymptotic trajectories is shown. It is proved that perturbed trajectories are crossed but do not coincide. This is the reason for the absence of an additional analytical first integral in the perturbed problem.

The problem of perturbed motion of a dynamically symmetric rigid body along an absolutely smooth horizontal plane is considered. Non-integrability of this problem is proved by the method of splitting asymptotic surfaces.

We consider systems of ordinary differential equations with a quadratic homogeneous right-hand side. We give a new simple proof of an earlier result, which gives the necessary conditions for the existence of polynomial first integrals. The necessary conditions for the existence of a polynomial symmetry field are given. It is proved that an arbitrary homogeneous first integral of a given degree is a linear combination of a fixed set of polynomials.

Suppose we have a natural Hamiltonian H of n particles on the line, centre-of-mass momentum P and a further independent quantity Q, cubic in the momenta. If these are each S_n invariant and mutually Poisson commute we have the Calogero-Moser system with potential V = (1/6)∑_i≠j℘(q_i-q_j) + const.

The main purpose of this paper is to test the performance of Lie group integrators as applied to a semi-discrete version of the KdV equation.

We set out a method for giving explicit algebraic coordinates on varieties of elliptic solitons, which consists of finding the spectral curve by elimination and solving the Jacobi inversion problem by the use of Kleinian functions and their identities. As an example we solve a five-particle elliptic Calogero-Moser system, whose spectral curve turns out to be completely reducible over three equianharmonic elliptic curves.

For a finite reflection group G there is a rich theory developed by Dunkl, Heckman and Opdam leading to the notion of a commuting set of Bessel differential operators. These systems play an important role in the study of Calogero-Moser systems and other problems of physical interest. When G acts on the line one recovers the usual Bessel function with a well known power series expansion at the origin. We obtain some such expansions in the case of G = A₂ acting in the plane and we use these to produce plots of some of these functions.

Similarity reductions of the Hirota-Satsuma system and another gauge-related system yield non-autonomous Hamiltonian systems with quartic potentials. We present classes of special solutions and Bäcklund transformations which are interpreted in terms of the action of an affine Weyl group on the space of parameters. Some other quartic oscillators related to coupled Painlevé-type equations are briefly considered. We show how separation of variables also has an application in this context.

The integral representations for the eigenfunctions of N particle quantum open and periodic Toda chains are constructed within the framework of the quantum inverse scattering method. Both periodic and open N-particle solutions have essentially the same structure, being written as a generalized Fourier transform over the eigenfunctions of the N-1 particle open Toda chain with the kernels satisfying the Baxter equations of second and first order, respectively. In the latter case this leads to recurrent relations which result in a representation of Mellin-Burnes-type solutions of an open chain. As a byproduct, we obtain the Gindikin-Karpelevich formula for the Harish-Chandra function in the case of the {GL}(N,) group.

We consider deformations of 2×2 and 3×3 matrix linear ODEs with rational coefficients with respect to singular points of Fuchsian type which do not satisfy the well known system of Schlesinger equations (or its natural generalization). Some general statements concerning the reducibility of such deformations for 2×2 ODEs are proved. An explicit example of the general non-Schlesinger deformation of 2×2-matrix ODEs of the Fuchsian type with four singular points is constructed and application of such deformations to the construction of special solutions of the corresponding Schlesinger systems is discussed. Some examples of isomonodromic and non-isomonodromic deformations of 3×3 matrix ODEs are considered. The latter arise as the compatibility conditions with linear ODEs with non-single-valued coefficients.

Using the -problem and dual -problem, we derive bilinear relations which allow us to construct integrable hierarchies in different parametrizations, their Darboux-Bäcklund transformations and to analyse constraints for them in a very simple way. Scalar KP, BKP and CKP hierarchies are considered as examples.

In this paper, we classify all values of the parameters α, β, γ and δ of the Painlevé VI equation such that there are rational solutions. We give a formula for them up to the birational canonical transformations and the symmetries of the Painlevé VI equation.

We present the fermionic representation for the q-deformed hypergeometric functions related to Schur polynomials. We show that these multivariate hypergeometric functions are tau-functions of the KP hierarchy, and at the same time they are the ratios of Toda lattice tau-functions, considered by Takasaki, evaluated at certain values of higher Toda lattice times. The variables of the hypergeometric functions are related to the higher times of those hierarchies via a Miwa change of variables. The discrete Toda lattice variable shifts parameters of hypergeometric functions. Hypergeometric functions of type _pΦ_s can also be viewed as a group 2-cocycle for the ΨDO on the circle (the group times are higher times of TL hierarchy and the arguments of a hypergeometric function). We obtain the determinant representation and the integral representation of a special type of KP tau-functions, these results generalize some of the results of Milne concerning multivariate hypergeometric functions. We write down a system of partial differential equations for these tau-functions (string equations).

Model potentials for quantum dots with smooth boundaries, realistic in the whole range of energies, are introduced, starting from the integrable motion of a particle on a sphere under the action of an external quadratic field. We show that in the case of rotational invariant potentials, the associated 2D Schrödinger equation has exact zero-energy eigenfunctions, in terms of which the whole discrete spectrum can be characterized.

The relation between the uniformizing equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces and an Appell-Lauricella hypergeometric system in nine variables is clarified.

The domain structure of a scalar field or a vector potential and a solenoidal field on a plane is shown to have a symmetry of the Kleinian group. This allows us to build a classification of domain structures for superconductors and magnetics by means of the Kleinian groups and explain their main properties. A number of examples of the domain branching on a plane boundary of superconductors and magnetics are described. Generalizations of the theory allow us to take in account the more general types of field, the symmetry of the crystal lattice, the three-dimensionality of space and more general functionals of the free energy.

Using a Lax pair based on the affine SL(2,) Kac-Moody algebra, we solve two-dimensional reduced vacuum Einstein's equations. We obtain explicit determinant formulae for metric coefficients with no quadrature left. This Lax connection also allows a new approach to the Poisson algebra of two-dimensional reduced gravity. In particular, we show that it leads to pure c-number r-matrices and modified Yang-Baxter equations. We explain how one can construct classical observables within this framework.

This paper concerns the topology of isospectral real manifolds of certain Jacobi elements associated with real split semisimple Lie algebras. The manifolds are related to the compactified level sets of the generalized (non-periodic) Toda lattice equations defined on the semisimple Lie algebras. We then give a cellular decomposition and the associated chain complex of the manifold by introducing coloured Dynkin diagrams which parametrize the cells in the decomposition. We also discuss the Morse chain complex of the manifold.

The approach of Berezin to the quantization of so(n,2) via generalized coherent states is considered in detail. A family of n commuting observables is found in which the basis for an associated Fock-type representation space is expressed. An interesting feature is that computations can be done by explicit matrix calculations in a particular basis. The basic technical tool is the Leibniz function, the inner product of coherent states.

Some general properties of compatible Poisson brackets of hydrodynamic type are discussed, in particular: (a) an invariant differential-geometric criterion of the compatibility based on the Nijenhuis tensor which is slightly different from those existing in the literature; (b) the Lax pair with a spectral parameter governing compatible Poisson brackets in the diagonalizable case; (c) the connection of this problem with the class of surfaces in Euclidean space which possess non-trivial deformations preserving the Weingarten operator.

We discuss dualities in the integrable dynamics behind the exact solution to the N = 2 SUSY Yang-Mills theory. It is shown that T duality in the string theory is related to the separation of variables procedure in the dynamical system. We argue that there are analogues of S duality as well as three-dimensional mirror symmetry in the many-body systems of Hitchin type governing the low-energy effective actions.

We construct a Gaudin-type lattice model as the Wess-Zumino-Witten model on elliptic curves at the critical level. Bethe eigenvectors are obtained by the bosonization technique.

The property of two metrics on one manifold having the same geodesics is equivalent to a special kind of integrability of the geodesic flows of these metrics (both in the classical and in the quantum sense). This gives us nontrivial restrictions on the topology of the manifold, allows us to construct new examples of such pairs of metrics on the sphere and to give a local description of geodesically equivalent metrics near the points where the eigenvalues of one metric with respect to the other bifurcate.

Multicomponent KdV systems are defined in terms of a set of structure constants and, as shown by Svinolupov, if these define a Jordan algebra the corresponding equations may be said to be integrable, at least in the sense of having higher-order symmetries, recursion operators and hierarchies of conservation laws. In this paper the dispersionless limits of these Jordan KdV equations are studied. Recursion laws for conserved densities are given under the assumption that the algebra possesses a unity element. Sufficient conditions are given for the unitized counterpart of a diagonalizable non-unital system to be diagonalizable. Hamiltonian structure is discussed within the context of D_N Jordan algebras and ^N scattering problems.

Krall's polynomials are orthogonal polynomials that are also eigenfunctions of a differential operator. We exhibit an analogue of Krall's polynomials within the context of rank-one commutative rings of difference operators. The corresponding spectral curves are unicursal curves with equations v² = u^{2R + 1}(u + 1)^{2S + 1}, R = 0,1,2,..., S = 0,1,2,.... Our analogues of Krall's polynomials are rational functions, which satisfy an orthogonality relation on the circle. The proof of the orthogonality relations combines the discrete Kadomtsev-Petviashvili bilinear identities, the cuspidal character of the singularities of the spectral curves, together with an extra symmetry of the problem.

In this paper we study one aspect of the continuous symmetries of the Toda equation. Namely, we establish a correspondence between Bäcklund transformations and continuous symmetries of the Toda equation. A symmetry transformation acting on a solution of the Toda equation can be seen as a superposition of Bäcklund transformations. Conversely, a Bäcklund transformation can be written, at least formally, as a composition of infinitely many higher symmetry transformations. This result reinforces the opinion that the presence of sufficiently many continuous symmetries for discrete equations is an indication of their integrability.

We study an integrable cellular automaton which is called the box-ball system (BBS). The BBS can be derived directly from the integrable differential-difference equation by either ultradiscretization or crystallization. We clarify the integrable structure and the hidden symmetry of the BBS.

Elementary, one- and two-point, Bäcklund transformations are constructed for the generic case of the sl(2) Gaudin magnet. The spectrality property is used to construct these explicitly given, Poisson integrable maps which are time discretizations of the continuous flows with any Hamiltonian from the spectral curve of the 2×2 Lax matrix.

We study piecewise-linear integrable systems. The associated piecewise-linear solitons, piecewise-linear integrable maps and piecewise-linear Lax representation are discussed.

We study the q-Painlevé V equation which can be obtained from the degeneration of the q-P_VI (in the form of the asymmetric q-P_III) equation and present its geometrical description. Based on the bilinear formulation we obtain the equations for the multi-dimensional τ-functions of q-P_V (in the form of nonautonomous Hirota-Miwa systems) which lives in the weight lattice of the A₄ affine Weyl group. This geometrical approach furnishes in a straightforward way the Miuras and the Schlesingers of q-P_V.

We extend the N-soliton solutions of the Kaup-Kupershmidt equation on a nonzero background decreasing as (x + 1/a)^-2. These new solutions describe the interaction of N solitary waves with a static bell-shaped wave. We give the conditions so that the Bäcklund transformation relating those solutions and the N-solitons of the Sawada-Kotera equation will be satisfied.