Isospectral deformations, the spectrum of Jacobi matrices, infinite continued fraction and difference operators. Application to dynamics on infinite dimensional systems

TL;DR

This paper explores the connections between isospectral deformations, Jacobi matrices, continued fractions, and difference operators, applying algebraic geometry to analyze integrable systems and their dynamics in infinite-dimensional settings.

Contribution

It introduces new algebraic geometric methods to study isospectral deformations and integrable systems, linking spectral theory with infinite continued fractions and difference operators.

Findings

Connections established between coadjoint orbits and integrable dynamics

Algebraic geometric techniques applied to infinite continued fractions and Jacobi matrices

Examples provided from mathematical physics demonstrating the theory

Abstract

This paper is devoted to the study of some connections between coadjoint orbits in infinite dimensional Lie algebras, isospectral deformations and linearization of dynamical systems. We explain how results from deformation theory, cohomology groups and algebraic geometry can be used to obtain insight into the dynamics of integrable systems. Another part will be dedicated to the study of infinite continued fraction, orthogonal polynomials, the isospectral deformation of periodic Jacobi matrices and general difference operators from an algebraic geometrical point of view. Some connections with Cauchy-Stieltjes transform of a suitable measure and Abelian integrals are given. Finally the notion of algebraically completely integrable systems is explained, techniques to solve such systems are presented and some interesting cases appear as coverings of such dynamical systems. These results are exemplified by several problems of dynamical systems of relevance in mathematical physics.

Equations

$$\frac{\partial u}{\partial t}-6u\frac{\partial u}{\partial x}+\frac{{\partial }^{3}u}{\partial x^3}=0,u(x,0)=u(x),x\in \mathbb{R}$$

$$\frac{dA}{dt}=[A,B]\equiv AB-BA,$$

$$A=-\frac{{\partial }^2}{\partial x^2}+u,B=4\frac{{\partial }^3}{\partial x^3}-3\left(u\frac{\partial }{\partial x}+\frac{\partial u}{\partial x}\right).$$

$$\frac{3}{4}\frac{{\partial }^{2}u}{\partial y^2}=\frac{\partial }{\partial x}\left(\frac{\partial u}{\partial t}-\frac{1}{4}\left(6u\frac{\partial u}{\partial x}+\frac{{\partial }^{3}u}{\partial x^3}\right)\right).$$

$$\dot{A}(t,h)=[A(t,h),B(t,h)]\text{ or }[B(t,h),A(t,h)],\left({}^{.}\equiv \frac{d}{dt}\right)$$

$$A(t,h)=\sum _{k=l}^m{A}_k(t)h^k,B(t,h)=\sum _{k=l}^m{B}_k(t)h^k,$$

$$A(t)=g(t)A(0)g(t{)}^{-1},$$

$$\dot{g}(t)=-A(t)g(t).$$

$$P(z,h)=\det (A-zI),$$

$$\mathcal{C}=\{(z,h):P(z,h)=0\},$$

$$\left\{H_1,H_2\right\}(a)=⟨⟨a,\left[{\nabla }_{{\mathcal{N}}^{\ast }}{H}_1,{\nabla }_{{\mathcal{N}}^{\ast }}{H}_2\right]⟩⟩,a\in {\mathcal{N}}^{\ast }.$$

$$\dot{a}=\left[a,p{r}_{\mathcal{K}}(\nabla H)\right],$$

$$\mathcal{L}=\sum _{-\infty }^m{A}_i{h}^i:A_i\in \mathcal{L},m\in \mathbb{Z}\text{mbox}free,$$

$$\left[\sum A_i{h}^i,\sum B_j{h}^j\right]=\sum _{i,j}\left[A_i,B_j\right]h^{i+j},$$

$${⟨\sum A_i{h}^i,\sum B_j{h}^j⟩}_k=\sum _{i+j=-k}⟨A_i,B_j⟩,$$

$$\mathcal{L}={\mathcal{L}}_{0,\infty }+{\mathcal{L}}_{-\infty ,-1}=\mathcal{K}+\mathcal{N},$$

$$\mathcal{L}=\sum _{i\in \mathbb{Z}}{L}_i,\left[L_{i,}{L}_j\right]\subset L_{i+j},\left[L_0,L_0\right]=0,L_i^{\ast }=L_{-i}.$$

$$\left[\left(l_1,l_2\right),(l_1^{{}^{\prime }},l_2^{{}^{\prime }})\right]$$

$$⟨\left(l_1,l_2\right),(l_1^{{}^{\prime }},l_2^{{}^{\prime }})⟩$$

$$\mathcal{K}=\left\{(l,-l):l\in \mathcal{L}\right\},{\mathcal{K}}^{\perp }=\left\{(l,l):l\in \mathcal{L}\right\},$$

$$\mathcal{N}=\left\{(l_{-},l_{+}):l_{-}\in B^{-},l_{+}\in B^{+},p{r}_0(l_{-})=p{r}_0(l_{+})\right\},$$

$${\mathcal{N}}^{\perp }=\left\{(l_{-},l_{+}):l_{-}\in B^{-},l_{+}\in B^{+},p{r}_0(l_{+}+l_{-})=0\right\},$$

$$V_m=\left\{A=\sum _{i=1}^{m-1}{A}_i{h}^i+\alpha h^m,\alpha =diag({\alpha }_1,\cdots ,{\alpha }_n)\text{ fixed}\right\},$$

$$H={⟨f(A{h}^{-j}),h^k⟩}_1,$$

$$\dot{A}=\left[A,p{r}_{\mathcal{K}}(f^{\prime }(A{h}^{-j})h^{k-j})\right],$$

$$A=\sum _{i=0}^{m-1}{A}_i{h}^i+\alpha h^m,$$

$$\dot{A}=\left[A,a{d}_{\beta }a{d}_{\alpha }^{-1}{A}_{m-1}+\beta h\right],{\beta }_i=f^{\prime }\left({\alpha }_i\right),$$

$$V_{-j,k}=\sum _{-j\le i\le k}{L}_i\subseteq {\mathcal{L}\simeq \mathcal{K}}^{\perp },$$

$$\dot{l}=\left[l,(p{r}^{+}-\frac{1}{2}p{r}_0)\nabla H(l)\right],p{r}^{+}\text{mbox}projectiononto{B}^{+},$$

$${\int }_{s_1(0)}^{s_1(t)}{\omega }_k+{\int }_{s_2(0)}^{s_2(t)}{\omega }_k+\cdots +{\int }_{s_g(0)}^{s_g(t)}{\omega }_k=c_{k}t,0\le k\le g,$$

$$A=X+\alpha h,X\in so(n).$$

Full text

Isospectral deformations, the spectrum of Jacobi matrices,

infinite continued fraction and difference operators. Application to dynamics on infinite dimensional systems

A. Lesfari

Department of Mathematics

Faculty of Sciences

University of Chouaïb Doukkali

B.P. 20, 24000 El Jadida, Morocco.

E. mail : lesfariahmed@yahoo.fr.

Abstract. This paper is devoted to the study of some connections between coadjoint orbits in infinite dimensional Lie algebras, isospectral deformations and linearization of dynamical systems. We explain how results from deformation theory, cohomology groups and algebraic geometry can be used to obtain insight into the dynamics of integrable systems. Another part will be dedicated to the study of infinite continued fraction, orthogonal polynomials, the isospectral deformation of periodic Jacobi matrices and general difference operators from an algebraic geometrical point of view. Some connections with Cauchy-Stieltjes transform of a suitable measure and Abelian integrals are given. Finally the notion of algebraically completely integrable systems is explained, techniques to solve such systems are presented and some interesting cases appear as coverings of such dynamical systems. These results are exemplified by several problems of dynamical systems of relevance in mathematical physics.

MSC(2010): 37K10, 32C35, 14B12, 14H40, 35S05, 11A55, 37K20.

Key words: Integrable systems, Lax equations, cohomology group, spectral curves, deformation, Jacobian varieties, periodic Jacobi matrices, difference operators, continued fraction.

1 Introduction

The discovery towards the end of the 19th century by Poincaré [51] that most nonlinear dynamical systems are not completely integrable marked the end of a long and fruitful interaction between Hamiltonian mechanics and algebraic geometry and the interest in this subject decreased for more than half a century. In fact many algebraic geometrical results such that elliptic and hyperelliptic curves, Abelian integrals, Riemann surfaces, etc., have their origin in problems of mechanics. Fortunately the discovery, 50 years ago, by Gardner, Greene, Kruskal and Miura [15] that the Korteweg-de Vries (KdV) equation [30] :

[TABLE]

could be integrated by spectral methods has generated an enormous number of new ideas in the area of Hamiltonian completely integrable dynamical systems. The resolution of this problem has led to unexpected connections between mechanics, spectral theory, Lie groups, algebraic geometry and even differential geometry, which have provided new insights into the old mechanical problems of last centuries and many new ones as well. Lax [35] showed that this equation is equivalent to the so-called Lax equation :

[TABLE]

where $A$ (Sturm-Liouville) and $B$ are the differential operators in $x$ :

[TABLE]

Lax equation means that, under the time evolution of the system, the linear operator $A(t)$ remains similar to $A(0)$. So the spectrum of $A$ is conserved, i.e. it undergoes an isospectral deformation. The eigenvalues of $A$, viewed as functionals, represent the integrals (constants of the motion) of the KdV equation. Thereafter, Mc Kean-van Moerbeke [44], Dubrovin-Novikov [12] solved the periodic problem for the KdV equation (for $x\in S^{1}$) in terms of a linear motion on a real torus. This torus is the real part of the Jacobi variety of a hyperelliptic curve with branch points defined by the simple periodic and anti-periodic spectrum of $A$. Also the motion is a straight line in the variables of the well known Abel-Jacobi map. A parallel theory related to Jacobi matrices had its origin in the periodic Toda problem [57] (discretized version of the KdV equation). Krichever [33] generalized these ideas to differential operators of any order, inspired by special examples of Zaharov-Shabat [61], among which is the important Kadomtsev-Petviashvili (KP) equation [24] :

[TABLE]

Also this theory was generalized to difference operators of any order by van Moerbeke and Mumford [60]. A one-to-one correspondence was established between curves of a certain type and classes of isospectral difference operators. They worked out a systematic method which provides an algebraic map from the invariants manifolds defined by the intersection of the constants of the motion to the Jacobi variety of an algebraic curve associated to Lax equation. Subsequently, these ideas led Mumford [47] to show the absence of isospectral (here means that the spectrum is given for all Floquet multipliers) deformations for Laplace-like two-dimensional periodic difference operators by relating the Picard variety with the class of such isospectral operators and by showing that for a generic class of such operators, the Picard variety is trivial. Therefore isospectral flows appear only in the case of one-dimensional operators. I shall not discuss here Mumford’s result on the absence of isospectral deformations for two-dimensional difference operators. However given a dynamical Hamiltonian system, it remains often hard to fit it into any of those general frameworks. But luckily, most of the problems possess the much richer structure of the so-called algebraic complete integrability (concept introduced et systematized by Adler and van Moerbeke). A dynamical system is algebraic completely integrable in the sense of Adler-van Moerbeke [5, 7, 40] if it can be linearized on a complex algebraic torus (Abelian variety).

Currently, the problem of finding and integrating nonlinear dynamical Hamiltonian systems, has attracted a considerable amount of attention in recent decades. Beside the fact that many such systems have been on the subject of powerful and beautiful theories of mathematics, another motivation for its study is : the concepts of integrability which are applying to an increasing number of physical systems, biological phenomena, population dynamics, chemical rate equations, to mention only a few. However, it seems still hopeless to describe or even to recognize with any facility, those nonlinear systems which are integrable, though they are exceptional. It is well known that the classical approach to solving nonlinear integrable dynamical systems was dominated by the question whether such systems can be solved by quadratures, i.e., by a finite number of algebraic operations including the inverting of functions. The method was based on solving the Hamilton-Jacobi equation by separation of variables, after an appropriate change of coordinates; for every problem finding this transformation required a great deal of ingenuity. The solutions of these problems can be expressed in terms of theta functions related to Riemann surfaces. It must be emphasized that the classical approach to proving that a system is integrable by quadratures (in terms of hyperelliptic integrals) was something very unsystematic and required a great deal of luck and ingenuity; Jacobi himself was very much aware of this difficulty and in his famous [22].

This paper is organized as follows : Section 1 is an introduction to the subject. Section 2 concerns nonlinear integrable dynamical systems which can be written as Lax equations with a spectral parameter. Such equations have no a priori Hamiltonian content. However, through the Adler-Kostant-Symes construction, we can produce Hamiltonian dynamical systems on coadjoint orbits in the dual space to a Lie algebra whose equations of motion take the Lax form. We outline an algebraic-geometric interpretation of the flows of these systems, which are shown to describe linear motion on a complex torus. The relationship between spectral theory and these systems is a fundamental aspect of the modern theory of nonlinear integrable dynamical systems (see for example [7,39, 45]). We present a Lie algebra theoretical schema leading to integrable systems, based on the Kostant-Kirillov coadjoint action. Many problems on Kostant-Kirillov coadjoint orbits in subalgebras of infinite dimensional Lie algebras (Kac-Moody Lie algebras) yield large classes of extended Lax pairs. A general statement leading to such situations is given by the Adler-Kostant-Symes theorem [1, 31, 55] and the van Moerbeke-Mumford linearization method provides an algebraic map from the complex invariant manifolds of these systems to the Jacobi variety (or some subabelian variety of it) of the spectral curve. The complex flows generated by the constants of the motion are straight line motions on these varieties. We present also the Griffith’s linearization method based on the observation that the tangent space to any deformation lies in a suitable cohomology group and that on algebraic curves, higher cohomology can always be eliminated using duality theory. We explain how results from deformation theory and algebraic geometry can be used to obtain insight into the dynamics of integrable systems. These conditions are cohomological and the Lax equations turn out to have a natural cohomological interpretation. These results are exemplified by several problems of dynamical integrable systems : Euler-Arnold equations for the geodesic flow on the special orthogonal group (the rotation group), Jacobi geodesic flow on the ellipsoid, Neumann motion of a point on the sphere, Lagrange top, periodic infinite band matrix, $n$-dimensional rigid body and Toda lattice. The periodic Toda lattice consists of isospectral deformations of periodic Jacobi matrices. The first flow describes a periodic chain of particles interacting with an exponential potential. The flow is conjugated to a motion of an auxiliary spectrum. Jacobi’s map transforms this motion into a linear flow on the Jacobi variety of the hyperelliptic curve attached to the matrix. The system is periodic or quasiperiodic. Section 3 is devoted to the study of infinite continued fraction, orthogonal polynomials, the isospectral deformation of periodic Jacobi matrices and general difference operators from an algebraic geometrical point of view. Complex Jacobi matrices play an important role in the study of asymptotics and zero distribution of formal orthogonal polynomials. Similarly, some connections with Cauchy-Stieltjes transform of a suitable measure and Abelian integrals are given. In Section 4 the notion of algebraically completely integrable Hamiltonian systems is explained and techniques to solve such systems are presented. Some important problems will be studied namely the periodic $5$-particle Kac-van Moerbeke lattice and the generalized periodic Toda systems. Also some interesting cases of integrable systems for example the Ramani-Dorizzi-Grammaticos (RDG) series of integrable potentials, a generalized Hénon-Heiles Hamiltonian, which will be studied in this section, appear as coverings of algebraic completely integrable systems. The manifolds invariant by the complex flows are coverings of Abelian varieties and these systems are called algebraic completely integrable in the generalized sense. The later are completely integrable in the sense of Arnold-Liouville. Also we shall see how some algebraic completely integrable systems can be constructed from known algebraic completely integrable in the generalized sense.

The concept of algebraic complete integrability is quite effective in small dimensions and has the advantage to lead to global results, unlike the existing criteria for real analytic integrability, which, at this stage are perturbation results. In fact, the overwhelming majority of dynamical systems, Hamiltonian or not, are non-integrable and possess regimes of chaotic behavior in phase space. I shall not discus here the weaker notion of analytic integrability; the perturbation techniques developed in that context are of a totally different nature. In recent years, other important results have been obtained following studies on the KP and KdV hierarchies. The use of tau functions related to infinite dimensional Grassmannians, Fay identities, vertex operators and the Hirota’s bilinear formalism led to obtaining important results concerning these algebras of infinite order differential operators. In addition many problems related to algebraic geometry, combinatorics, probabilities and quantum gauge theory,…, have been solved explicitly by methods inspired by techniques from the study of dynamical integrable systems. An account of these results will appear elsewhere.

2 Coadjoint orbits in Kac-Moody Lie algebras, isospectral

deformations and linearization

A Lax equation (with parameter $h$) is given by a differential equation of the form

[TABLE]

where

[TABLE]

are matrices or operators and depend on a parameter $h$ (spectral parameter) whose coefficients $A_{k}$ and $B_{k}$ are matrices in Lie algebras. The pair $(A,B)$ is called Lax pair. The bracket $[,]$ is the usual Lie bracket of matrices. The equation (2.1) establishes a link between the Lie group theoretical and the algebraic geometric approaches to complete integrability of dynamical systems. The solution to (2.1) has the form

[TABLE]

where $g(t)$ is a matrix defined as

[TABLE]

We form the polynomial

[TABLE]

where $z$ is another variable and $I$ the $n\times n$ identity matrix. We define the curve (spectral curve) $\mathcal{C},$ to be the normalization of the complete algebraic curve whose affine equation is $P(z,h)=0$.

Theorem 1

A flow of the type (2.1) preserves the spectrum of $A$ for every $h\in\mathbb{C}$ and its characteristic polynomial $P(z,h)\equiv\det\left(A-zI\right)$. The latter defines an algebraic curve

[TABLE]

for almost all $(z,h)\in\mathcal{C}$, which is time independent, i.e., its coefficients $tr\left(A^{n}\right)$ are integrals of the motion (equivalently, the matrix $A$ undergoes an isospectral deformation).

The matrix $A-zI$, has a one-dimensional null-space, defining a holomorphic line bundle on the curve $\mathcal{C}$. Whenever the entries of the $A_{k}$ are moving in time, the curve $\mathcal{C}$ doest not move, inducing a motion on the set of line bundles. The set of holomorphic line bundles on an algebraic curve forms a group for the operation of tensoring $\otimes$ and the full set with a given topological type is parametrized by the points of a $g$-dimensional complex algebraic torus, where $g$ is the genus of the curve. This torus that we note, $Jac(\mathcal{C})$, is the Jacobian or Picard variety of the curve. When $\mathcal{C}$ is an elliptic curve, $Jac(\mathcal{C})$ is isomorphic to $\mathcal{C}$. Since the flow (2.1) induces deformations of line bundles, their topological type remains unchanged and therefore it induces a motion on the Jacobian variety; under some checkable condition on $A$ and $B$, du to Griffiths [18] (see further for details). In addition, some flows on Kostant-Kirillov coadjoint orbits in subalgebras of infinite dimensional Lie algebras (Kac-Moody Lie algebras) yield large classes of extended Lax pairs. A general statement leading to such situations is given by the Adler-Kostant-Symes theorem :

Theorem 2

Let $\mathcal{L}$ be a Lie algebra paired with itself via a non-degenerate, ad-invariant bilinear form $\langle,\rangle$, $\mathcal{L}$ having a vector space decomposition $\mathcal{L}=\mathcal{K}+\mathcal{N}$ with $\mathcal{K}$ and $\mathcal{N}$ Lie subalgebras. Then, with respect to $\langle,\rangle$, we have the splitting $\mathcal{L}=\mathcal{L}^{*}=\mathcal{K}^{\perp}+\mathcal{N}^{\perp}$ and $\mathcal{K}^{\perp}\approx\mathcal{N}^{*}$($\equiv$ the dual of $\mathcal{N}$) paired with $\mathcal{N}$ via an induced form $\left\langle\left\langle,\right\rangle\right\rangle$ inherits the coadjoint symplectic structure of Kostant and Kirillov; its Poisson bracket between functions $H_{1}$ and $H_{2}$ on $\mathcal{N}^{*}$ reads

[TABLE]

Let $V\subset\mathcal{N}^{*}$ be an invariant manifold under the above coadjoint action of $\mathcal{N}$ on $\mathcal{N}^{*}$ and let $\mathcal{A}(V)$ be the algebra of functions defined on a neighborhood of $V$, invariant under the coadjoint action of $\mathcal{L}$ (which is distinct from the $\mathcal{N-N}^{*}$ action). Then the functions $H$ in $\mathcal{A}(V)$ lead to commuting vector fields of the Lax isospectral form,

[TABLE]

$pr_{\mathcal{K}}$* projection onto $\mathcal{K}$.*

The recent paper [8] gives the most general form of the Adler-Kostant-Symes theorem. This theorem produces dynamical Hamiltonian systems having many commuting integrals; some precise results are known for interesting classes of orbits in both the case of finite and infinite dimensional Lie algebras. The finite-dimensional Lie algebras usually lead to noncompact systems, and the infinite-dimensional ones to compact systems. Any finite dimensional Lie algebra $\mathcal{L}$ with bracket $\left[,\right]$ and Killing form $\left\langle,\right\rangle$ leads to an infinite dimensional formal Laurent series extension

[TABLE]

with bracket

[TABLE]

and ad-invariant, symmetric forms

[TABLE]

depending on $k\in\mathbb{Z}$. The forms $\left\langle,\right\rangle_{k}$ are non degenerate if $\left\langle,\right\rangle$ is so. Let $\mathcal{L}_{p,q}$ $(p\leq q)$ be the vector subspace of $\mathcal{L}$, corresponding to powers of $h$ between $p$ and $q$. A first interesting class of problems is obtained by taking $\mathcal{L}=\mathcal{G}l(n,\mathbb{R})$ and by putting the form $\left\langle,\right\rangle_{1}$ on the Kac-Moody extension. Then we have the decomposition into Lie subalgebras

[TABLE]

with $\mathcal{K=K}^{\perp}$, $\mathcal{N=N}^{\perp}$ and $\mathcal{K=N}^{*}$. Another class is obtained by choosing any semi-simple Lie algebra $L$. Then the Kac-Moody extension $\mathcal{L}$ equipped with the form $\left\langle,\right\rangle=\left\langle,\right\rangle_{0}$ has the natural level decomposition

[TABLE]

Let $B^{+}=\sum_{i\geq 0}L_{i}$, $B^{-}=\sum_{i<0}L_{i}$. Then the product Lie algebra $\mathcal{L\times L}$ has the following bracket and pairing

[TABLE]

It admits the decomposition into $\mathcal{K}+\mathcal{N}$ with

[TABLE]

[TABLE]

[TABLE]

where $pr_{0}$ denotes projection onto $L_{0}$. Then from the last theorem , the orbits in $\mathcal{N}^{*}\mathcal{=}K^{\perp}$ possesses a lot of commuting Hamiltonian vector fields of Lax form. We state the following theorem [2, 3, 60] :

Theorem 3

a) The invariant manifold $V_{m}$ , $m\geq 1$ in $\mathcal{K=N}^{*}$ , defined as

[TABLE]

with $diag\left(A_{m-1}\right)=0$, has a natural symplectic structure. The functions

[TABLE]

on $V_{m}$ for good functions $f$ lead to complete integrable commuting Hamiltonian systems of the form

[TABLE]

where

[TABLE]

and their trajectories are straight line motions on the Jacobian of the curve $\mathcal{C}$ of genus $\left(n-1\right)\left(nm-2\right)/2$ defined by (2.2). The coefficients of this polynomial provide the orbit invariants of $V_{m}$ and an independent set of integrals of the motion (of particular interest are the flows where $j=m,k=m+1$ which have the following form

[TABLE]

the flow depends on $f$ through the relation $\beta_{i}=f^{\prime}\left(\alpha_{i}\right)$ only).

b) (The van Moerbeke-Mumford linearization method) : The N-invariant manifolds

[TABLE]

has a natural symplectic structure and the functions $H(l_{1},l_{2})=f(l_{1})$ on $V_{-j,k}$ lead to commuting vector fields of the Lax form

[TABLE]

their trajectories are straight line motions on the Jacobian of a curve defined by the characteristic polynomial of elements in $V_{-j,k}$ thought of as functions of $h$, where $\nabla H(l)\in\mathcal{N}$ is the gradient of $H$ thought of as a function on $\mathcal{L}$.

Using the van Moerbeke-Mumford linearization method [60], Adler and van Moerbeke [3] showed that the linearized flow could be realized on the Jacobian variety $Jac(\mathcal{C})$ (or some subabelian variety of it) of the algebraic curve (spectral curve) $\mathcal{C}$ associated to (2.1). We then construct an algebraic map from the complex invariant manifolds of these dynamical systems to the Jacobian variety $Jac(\mathcal{C})$ of the curve $\mathcal{C}$. Therefore all the complex flows generated by the constants of the motion are straight line motions on these Jacobian varieties, i.e., the linearizing equations are given by

[TABLE]

where $\omega_{1},\ldots,\omega_{g}$ span the $g$-dimensional space of holomorphic differentials on the curve $\mathcal{C}$ of genus $g$.

Example 1

For $m=1$, i.e., for $V_{1}$, we choose

[TABLE]

In this case, the flow described by equation (2.3) (where $\alpha_{i}$ and $\beta_{i}$ can be taken arbitrarily) is reduced to the study of the Euler-Arnold equations for the geodesic flow on $SO(n)$,

[TABLE]

for a left-invariant diagonal metric $\Sigma\lambda_{ij}X_{ij}$. The natural phase space for this motion is an orbit defined in $SO(n)$ by $[n/2]$ orbit invariants. By Theorem 3, the problem is completely integrable and the trajectories are straight lines on $Jac(\mathcal{C})$ of dimension $(n-2)(n-1)/2$ and more specifically, on the Prym variety $Prym(\mathcal{C}/\mathcal{C}_{0})\subset Jac(\mathcal{C})$ of dimension $(n(n-1)/2-[w/2])/2$ induced by the natural involution $\mathcal{C}\longrightarrow\mathcal{C}$, $(z,h)\longmapsto(-z,-h)$, on $\mathcal{C}$ as a result of $X\in so(n)$; $\mathcal{C}_{0}$ is the curve obtained by identifying $(z,h$) with $(-z,-h)$.

Example 2

For $m=2$, i.e., $V_{2}$, if one chooses

[TABLE]

which can also be considered as a rank 2 perturbation of the diagonal matrix $\alpha$ [45, 46, 2, 3], then equation (2.3) reduces to

[TABLE]

This equation can be reduced to the following nonlinear dynamical system :

[TABLE]

where

[TABLE]

which for $f(z)=\ln z$, i.e., $\beta_{i}=\frac{1}{\alpha_{i}}$, we obtain the problem of Jacobi geodesic flow on the ellipsoid :

[TABLE]

expressing the motion of the tangent line ${x+sy:s\in\mathbb{R}}$ to the ellipsoid in the direction $y$ of the geodesic. For $f(z)=\frac{1}{2}z^{2}$, i.e., $\beta_{i}=\alpha_{i}$, we get the Neumann motion [49] of a point on the sphere $S^{n-1}$, $|x|=1$, under the influence of the force $-\alpha x$. From theorem 3, both motions are straight lines on $Jac(\mathcal{C})$, where $\mathcal{C}$ turns out to be hyperelliptic of genus $n-1$ (much lower than the generic one) ramified at the following $2n$ points : some point at $\infty$, the $n$ points $\alpha_{i}$ and $n-1$ other points $\lambda_{i}$ of geometrical significance, based on the observation that generically a line in $\mathbb{R}^{n}$ touches $n-1$ confocal quadrics. To be precise, the set of all common tangent lines to $n-1$ confocal quadrics

[TABLE]

where $Q_{u}(x,y)=\langle(u-\alpha)^{-1}x,y\rangle$, can be parameterized by the quotient of the Jacobian of the hyperelliptic curve $\mathcal{C}$ by an Abelian group $G$. The group is generated by the discrete action obtained by flipping the signs of $x_{k}$ and $y_{k}$ and some trivial one-dimensional action. Letting $h\rightarrow 0$ in the matrix $A$ and excising the largest eigenvalue from this matrix leads to a new isospectral symmetric matrix

[TABLE]

and a flow

[TABLE]

where the spectrum of $L$ is given by the $n-1$ branch points $\lambda_{i}$ above and zero. It follows that the tangent line $\{x+sy:s\in\mathbb{R}\}$ to the ellipsoid remains tangent to $n-2$ other confocal quadrics and the corresponding $n-1$ eigenfunctions of $L$ provide the orthogonal set of normals to the $n-1$ quadrics at the points of tangency, hence recovering a theorem of Chasles. The close relationship between Jacobi’s and Neumann’s problems, which in fact live on the same orbits, was implemented by Knörrer [29], who showed that the normal vector to the ellipsoid moves according to the Neumann problem [49], when the point moves according to the geodesic. These facts, as investigated also by Knörrer [28] and others; the set of all $n-1$ dimensional linear subspaces in the intersection of two quadrics

[TABLE]

in $\mathbb{P}^{2n-1}$ is the Jacobian of the curve $\mathcal{C}$ defined above. This is done by observing that the set of linear subspaces in the above quadrics is the same as the set of $(n-2)$-dimensional linear subspaces tangent to $n-1$ quadrics

[TABLE]

which is dual to the set of tangents to the confocal quadrics. The Neumann problem is also strikingly related to the Korteweg-de Vries equation and various other nonlinear dynamical systems (see [11]).

Example 3

For another example of $V_{2}$ in theorem 3 , we consider the Lagrange top (i.e., a symmetric top with a constant vertical gravitational force acting on its center of mass and leaving the base point of its body symmetry axis fixed) which evolves on an orbit of type $V_{2}$; $n=3$,

[TABLE]

where $\Gamma\in so(3)\simeq\mathbb{R}^{3}$ is the unit vector in the direction of gravity and $M\in so(3)\simeq\mathbb{R}^{3}$ is the angular momentum in body coordinates with regard to the fixed point; moreover $c=(\lambda+\mu)\Upsilon$ where $\Upsilon\in so(3)\simeq\mathbb{R}^{3}$ expresses the coordinates of the center of mass and where $(\lambda+\mu,\lambda+\mu,2\lambda)$ is the inertia tensor in diagonalized form. The situation then leads to a linear flow on an elliptic curve (see [54] and, for higher-dimensional generalizations [53]).

Example 4

As an example of $V_{ij}$ in theorem 3, b) (see [60, 2, 3]), we consider the periodic infinite band matrix $M$ of period $n$ having $j+h+1$ diagonals; the spectrum of $M$ is defined by the points $(z,h)\in\mathbb{C}^{2}$ such that

[TABLE]

Let $M_{h}$ be the square matrix obtained from $M$ and let $\mathcal{C}$ be the curve defined by $\det(M_{h}-zI)=0$. Then the set of infinite band matrices with $j+k+1$ diagonals, in higher dimensions many partial results seem to lead to rigidity. In fact, it was shown that a discrete 2-dimensional Laplacian cannot be deformed, given its periodic spectrum; the proof can be summarized by the observation that the Picard variety of most algebraic surfaces is trivial; the proof that the specific spectral surface defined by the $2$-dimensional Laplacian has trivial Picard variety is based on the technique of toroidal embedding, which reduces cohomological computations to combinatorial questions. Finally, inspired by the dynamical systems, Mumford [48] has given a beautiful description of hyperelliptic Jacobians of dimension $g$. Let $y^{2}=R(z)$ be the monic polynomial of degree $2g+l$ defining the curve $\mathcal{C}$ and let $\theta$ be the thêta divisor. Then $\mbox{Jac}(\mathcal{C})\backslash\theta$ is a variety of polynomials $U$, $V$ with $\mbox{deg }U=g$, $\mbox{deg }V\leq g-1$ and $U$ monic such that $U|B-V^{2}$.

As mentioned before, in an unifying approach Griffiths [18] has found necessary and sufficient conditions on $B$ for the Lax flow (2.1) to be linearizable on the Jacobi variety of its spectral curve, without reference to Kac-Moody Lie algebras, so that the flow of the Lax form (2.1) can be linearized on the Jacobian variety $\mbox{Jac}(\mathcal{C})$ for $\mathcal{C}$ defined by (2.2). Suppose that for every $p(z,h)$ belonging to the curve $\mathcal{C}$, with $\dim\ker(A-zI)=1$, (i.e., the corresponding eigenspace of $A$ is one-dimensional) and generated by a vector $v(t,p)\in V$ where $V\simeq\mathbb{C}^{n}$ is an $n$-dimensional vector space. There is then a family of holomorphic mappings which send $(z,h)\in\mathcal{C}$ to $\ker(A-zI)$:

[TABLE]

called the eigenvector map associated to the Lax equation. We set

[TABLE]

where $d=\mbox{deg }f_{t}(\mathcal{C})$; $\mathcal{O}_{\mathbb{P}V}(1)$ is the hyperplane line bundle on $\mathbb{P}V$ and $\mbox{Pic}^{d}(\mathcal{C})$ the Picard variety of $\mathcal{C}$, i.e., let us recall that it is the set of straight bundles of degree $d$ on $\mathcal{C}$. By continuity, the degree of $L_{t}$ does not vary with time $t$. Let H be the hyperplane class of $\mathbb{P}V$. We have

[TABLE]

This expression is the Poincaré dual of the class $[\mathcal{C}]$ of $\mathcal{C}$ and coincides with the degree of $\mathcal{C}$. Hence $\mbox{deg }L_{t}=\mbox{deg}(\mathcal{C})$. While $t$ varies, $L_{t}$ moves in $\mbox{Pic}^{d}(\mathcal{C})$. Therefore, if we fix a line bundle $L_{0}\in\mbox{Pic}^{d}(\mathcal{C})$, the line bundle $L_{0}^{-1}\otimes L_{t}$ moves in the Jacobian variety

[TABLE]

i.e., the mapping $L\longmapsto L_{0}^{-1}\otimes L$ induces a morphism $\mbox{Pic}^{d}(\mathcal{C})\simeq\mbox{Jac}(\mathcal{C})$. The motion of the line bundle $L_{0}^{-1}\otimes L_{t}$ depends on the choice of the matrix $B$ and a question arises : determine necessary and sufficient conditions on the matrix $B$ so that the flow

[TABLE]

can be linearized on the Jacobian variety $\mbox{Jac}(\mathcal{C})$. As we have pointed out, Griffiths has found necessary and sufficient conditions of a cohomological nature on $B$ that the flow $t\longmapsto L_{t}\in\mbox{Jac}(\mathcal{C})$, be linear. His method is based on the observation that the tangent space to any deformation lies in a suitable cohomology group and that on algebraic curves, higher cohomology can always be eliminated using duality theory. In fact, applying more or less standard cohomological techniques from deformation theory [9], we may give necessary and sufficient conditions that the map $t\longmapsto L_{t}$ be linear.

Let

[TABLE]

be a non-constant holomorphic map where $\mathcal{C}$ is a given smooth algebraic curve and $X$ is a complex manifold. We define the normal sheaf of $\mathcal{C}$ in $X$ by the exact sequence

[TABLE]

with $\Theta_{\mathcal{C}}$, $\Theta_{X}$ are the respective tangent sheaves and $f_{\ast}$ is the differential of $f$. Then the Kodaira-Spencer tangent space [9] to the moduli space of the map (2.5) is given by $H^{0}(\mathcal{C},N_{f})$. If

[TABLE]

is a deformation of (2.5), then $\dot{f}\in H^{0}(\mathcal{C},N_{f})$ the corresponding infinitesimal deformation at $t=0$, i.e., in local product coordinates $(z,t)$ on $\cup_{t}\mathcal{C}_{t}$ and $w=(w^{1},w^{2},...,w^{n})$ of $X$, $f_{t}$ is given by $(t,\xi)\longmapsto w(t,\xi)$, then the section $\dot{f}\in H^{0}(\mathcal{C},N_{f})$ is locally given by $\left.\frac{\partial w(t,\xi)}{\partial t}\right|_{t=0}\mbox{modulo }\frac{\partial w(0,\xi)}{\partial z}$. The corresponding cohomological sequence of (1.6) is

[TABLE]

Here $H^{1}(\Theta_{\mathcal{C}})$ is the tangent space to the moduli space of $\mathcal{C}$ as an abstract curve and

[TABLE]

is the tangent to the family of curves $\{\mathcal{C}_{t}\}$. Thus the tangent space to deformations of (2.5) where the curve $\mathcal{C}$ remains fixed, is given by

[TABLE]

Since the isospectral curve $\mathcal{C}$ is independent of $t$, this is the situation that we are interested in.

In the following, we take again the vector space $V$ of dimension $n$ and assume that $X=\mathbb{P}V$ (projective space) and consider the Euler sequence

[TABLE]

This is an exact sequence of vector bundles, so that it remains exact after pulling back to $\mathcal{C}$ via $f^{*}$ (combining this with (2.6)). We have then a diagram of exact sequences ($L=f^{*}\mathcal{O}_{\mathbb{P}V}(1)$) :

[TABLE]

The associated cohomology diagram contains the following piece :

[TABLE]

Consider the family of holomorphic maps $f_{t}:\mathcal{C}\longrightarrow\mathbb{P}V$. Locally choose a coordinate $\xi$ on $\mathcal{C}$ and a position vector mapping $(t,\xi)\longmapsto v(t,\xi)\in V\backslash\{0\}$, i.e., a local lift $v_{t}$ of $f_{t}$ to $V\backslash\{0\}$, such that

[TABLE]

Notice that $v_{t}$ is a time-dependent map $\mathcal{C}\longrightarrow V\backslash\{0\}$. This lift is not canonical and exists only locally, but we are going to use it to define an object denoted $\dot{v}$ which will be independent of the lift and therefore will be globally well defined. Since $\mathcal{O}_{\mathbb{P}V}$ is the tautological bundle of $\mathbb{P}V$, the fibre of $f^{*}\mathcal{O}_{\mathbb{P}V}(-1)$ at a point $p\in\mathcal{C}$ may be identified with the space $\mathbb{C}v_{t}(p)$, which defines the maps $f^{*}\mathcal{O}_{\mathbb{P}V}(-1)V\otimes\mathcal{O}_{\mathcal{C}}$ and

[TABLE]

where $v_{0}$ coincides with the application $v$ mentioned in the previous diagram. If $\widetilde{v}$ is another lift given by

[TABLE]

then we have

[TABLE]

Set

[TABLE]

The latter quantity is well defined of the representative position mapping of $v$, i.e., since the inclusion

[TABLE]

is locally given by $\mathcal{O}_{\mathcal{C}}\ni\phi\longmapsto\phi.v$, it follows that

[TABLE]

is a well-defined and independent of the choice of the lift. Then we have $j(\dot{v})=f$. We are interested in the tangent vector

[TABLE]

Theorem 4

We have

[TABLE]

where $\dot{v}$ is the infinitesimal variation of $f_{t}:\mathcal{C}\longrightarrow\mathbb{P}V$ and in particular, $\dot{L}=0$ if and only if $\dot{v}=\tau(w)$ for some $w\in H^{0}(V\otimes L)$ where $\tau$ is the map in diagram above.

We write

[TABLE]

where we have regarded $h$ as an affine coordinate on the projective line $\mathbb{P}^{1}$ which is the base of the covering $\pi:\mathcal{C}\longrightarrow\mathbb{P}^{1}$, while $h_{0}$, $h_{1}$ are homogeneous coordinates. Recall that $B(t,h)\in H^{0}(\mathcal{C},\mbox{Hom}(V,V(N)))$ where $V$ is the sheaf of sections of the trivial bundle $\mathcal{C}\times V$, $V(D)=V\otimes\mathcal{O}_{\mathcal{C}}(D)$ (Here $B(t,h)$ is a holomorphic section of the bundle $\mbox{Hom}(V,V)\otimes\mathcal{O}_{\mathcal{C}}(N)$, $\mathcal{O}_{\mathcal{C}}(N)=\pi^{*}\mathcal{O}_{\mathbb{P}^{1}}(N)$, i.e., we are viewing $h=[h_{0},h_{1}]$ as a homogeneous coordinate on $\mathbb{P}^{1}$ pulled up to $\mathcal{C}$). Let $D=(h_{0}^{N})$, be the divisor $N.\pi^{-1}(\infty)$ on the curve $\mathcal{C}$. Therefore $B/h_{0}^{N}\in H^{0}(\mathcal{C},\mbox{Hom}(V,V(D)))$, $v\in H^{0}(V\otimes L)$ where $V(D)\cong V(N)$ are the sections of $V\otimes\mathcal{O}_{\mathcal{C}}(D)$ (Here $B/h_{0}^{N}$ is a matrix in $\mbox{Hom}(V,V)$ with meromorphic functions in $H^{0}(\mathcal{C},\mathcal{O}_{\mathcal{C}}(D))$ as entries, i.e., we are viewing $h_{1}/h_{0}$ as a function in $H^{O}(\mathcal{C},\mathcal{O}_{\mathcal{C}}(D))$). Hence $\left(\frac{B}{h_{0}^{N}}\right).v\in H^{0}(\mathcal{C},V\otimes L(D))$ and the cohomological interpretation of Lax equation is given by

Theorem 5

We have

[TABLE]

and $\dot{L}=0$, if and only if there is a meromorphic function $\varphi\in H^{O}(\mathcal{C},\mathcal{O}_{\mathcal{C}}(D))$ such that $\displaystyle{\frac{B}{h_{0}^{N}}.v+\varphi v}\in H^{0}(\mathcal{C},V\otimes L(D))$ is holomorphic.

Near the point $p=(h,z)\in\mathcal{C}$, differentiating with regard to $t$ the eigenvalue problem $Av(t,p)=zv(t,p)$, leads to $\dot{A}v+A\dot{v}=z\dot{v}$. Using the Lax equation : $\dot{A}=[B,A]$, we obtain $A(\dot{v}-Bv)=z(\dot{v}-Bv)$. Since generically the eigenvalues have multiplicity $1$, we have

[TABLE]

for a some $\lambda$, or what is the same $Bv=\dot{v}+\lambda_{j}v$, where $\lambda_{j}$ is the principal part of the Laurent series expansion of $\lambda$ at $p$. Then given the curve $\mathcal{C}$ defined by (2.2) and $p\in\mathcal{C}$, Griffiths defines

[TABLE]

and shows that the Lax flow can be linearized on the Jacobian variety $\mbox{Jac}(\mathcal{C})$ if and only if for every $p\in(h)_{\infty}$ (divisor of the poles of $h$), we have

[TABLE]

Equation (2.1) is invariant under the substitution

[TABLE]

which shows that $B$ is not unique and that its natural place is somewhere in a cohomology group. Let

[TABLE]

be a polynomial of degree $n$. Let

[TABLE]

(where $h$ is seen as a meromorphic function) be a positive divisor on $\mathcal{C}$ and let $z_{j}$ a local coordinate around $p_{j}$. $B$ must be interpreted as an element of $H^{0}(\mathcal{C},\mbox{Hom}(V,V(\mathcal{D}))$ where $V$ is the sheaf of sections of the trivial bundle $\mathcal{C}\times V$ and $V(\mathcal{D})=V\otimes\mathcal{O}_{\mathcal{C}}(\mathcal{D})$. A section of $\mathcal{O}_{\mathcal{D}}(\mathcal{D})$ is written

[TABLE]

it is a principal part (Laurent tail) centered on $p_{j}$. The Mittag-Leffler problem can be formulated as follows : given a principal part $\varphi_{j}$, find conditions for a function $\varphi\in H^{0}(\mathcal{C},\mathcal{O}_{\mathcal{C}}(\mathcal{D}))$ such that $\varphi-\varphi_{j}$ is holomorphic around $p_{j}$. The answer is provided by the following theorem :

Theorem 6

Let $\mathcal{D}=\sum_{j}a_{j}p_{j}$. Given Laurent tail $\{\varphi_{j}\}$, then there exist $\varphi\in H^{0}(\mathcal{C},\mathcal{O}_{\mathcal{C}}(\mathcal{D}))$ such that $\varphi-\varphi_{j}$ is holomorphic near $p_{j}$ if and only if

[TABLE]

for every holomorphic differential $\omega$ on $\mathcal{C}$.

The residue of $B$, denoted by $\rho(B)\in H^{0}(\mathcal{C},\mathcal{O}_{\mathcal{D}}(\mathcal{D})$, is the collection of Laurent tails $\{\lambda_{j}\}$ given above (recall that $\lambda_{j}$ is the principal part of the Laurent series expansion of $\lambda$ at $p$). We shall say that the flow $L_{t}$ is linear if there exists a complex number $a$ such that

[TABLE]

The Griffiths theorem is as follows :

Theorem 7

We have

[TABLE]

Let $\mbox{Im res}\subset H^{0}(\mathcal{C},\mathcal{O}_{\mathcal{D}}(\mathcal{D})$ be the Laurent tails of meromorphic functions in $H^{0}(\mathcal{C},\mathcal{O}_{\mathcal{D}}(\mathcal{D})$. Then the flow $L_{t}$ (2.4) in $\mbox{Pic}^{d}(\mathcal{C})$ is linear if and only if

[TABLE]

The condition (2.8) is equivalent to

[TABLE]

If this is satisfied, then the linear flow on $\mbox{Jac}(\mathcal{C})$ is given by the bilinear map

[TABLE]

Example 5

Let $J=\mbox{diag}(\lambda_{1},...,\lambda_{n})$, $\lambda_{j}>0$, be the matrix representing the tensor of inertia of a $n$-dimensional rigid body in a principal axis system and $\Omega(t)\in so(n)$ the skew-symmetric matrix associated to the angular velocity vector of the rigid body in the usual way. Define $M=\Omega J+J\Omega\in so(n)$ the equations of motion of the rigid body can be written as

[TABLE]

These equations are Hamiltonian on each adjoint orbit of $so(n)$ defined by initial conditions with Hamiltonian

[TABLE]

By Manakov’s trick [43], these equations are equivalent to a Lax equation with parameter

[TABLE]

Hence

[TABLE]

is the divisor with $p_{i}$ being the $n$ distinct points lying over $h=\infty$. If $z_{i}=h^{-1}$ is a local coordinate on $\mathcal{C}$ near $p_{i}$, then from equation (2.7) with $B=\Omega+Jh$, we obtain

[TABLE]

Since $\lambda_{i}$ are constant, one has $\rho(\dot{B})=0$ so the flow is linear on $Jac(\mathcal{C}$). Since

[TABLE]

we have $P(h,z)=(-1)^{n}P(-h,-z)$ and there is an involution of the spectral curve

[TABLE]

We note that $\Omega$ moves on an adjoint orbit $\mathcal{O}_{\nu}\subset so(n)$ and to linearize the flow in question we need $\frac{1}{2}\dim\mathcal{O}_{\nu}$ integrals of motion that are in involution where for general $\nu$,

[TABLE]

Let $g(\mathcal{C})$ be the genus of the spectral curve $\mathcal{C}$ and $g(\mathcal{C}_{0})$ the genus of the quotient $\mathcal{C}_{0}=\mathcal{C}/\sigma$ of $\mathcal{C}$ by the involution $\sigma$. Since

[TABLE]

then by the Riemann-Hurwitz formula,

[TABLE]

Associated to the double covering $\mathcal{C}\longrightarrow\mathcal{C}_{0}$ is the Prym variety $Prym(\mathcal{C}/\mathcal{C}_{0})$ and since $\sigma(\rho(B))=-\rho(B)$, the flow in question actually occurs on this complex torus. From (2.11) it follows that

[TABLE]

On the other hand, comparing with (2.10) we obtain

[TABLE]

and we see that the motion of the free rigid linearizes on a torus $Prym(\mathcal{C}/\mathcal{C}_{0})$ of exactly the right dimension.

3 Continued fraction, orthogonal polynomials, the spectrum of Jacobi matrices and difference operators

A Jacobi matrix is a doubly infinite matrix $(a_{ij})$ for $i,j\in\mathbb{Z}$ such that : $a_{ij}=0$ if $|i-j|$ is large enough. We show that the set of these matrices forms an associative algebra and consequently a Lie algebra by anti-symmetrization. Consider the Jacobi matrix

[TABLE]

where all the $b_{j}$ are real and all the $a_{j}$ positive, and the associated continued $\Gamma$-fraction,

[TABLE]

where $a_{0}$ is a positive real number. By cutting off the $\Gamma$-fraction $\varphi(z)$ at the $k$-th term, we obtain the $k$-th Padé approximant $\displaystyle{\frac{A_{k}(z)}{B_{k}(z)}}$ of $\varphi(z)$, i.e.,

[TABLE]

The degree of the polynomial $A_{k}(z)$ is $k-1$, while the degree of $B_{(}z)$ is $k$. Moreover, $\varphi(z)$ admits formal series expansion in a neighborhood of the pole $z=0$, in the following form

[TABLE]

Note that the characteristic polynomial $B_{k}$ of the $\Gamma$-Jacobi matrix

[TABLE]

is the last term of the second order recursion

[TABLE]

The polynomials $A_{k}(z)$, $B_{k}(z)$ form a pair of linearly independent solutions of a second order finite difference equation (the eigenvectors of the Jacobi matrix from which we remove the first row and column) :

[TABLE]

with the boundary conditions :

[TABLE]

We have also the relation :

[TABLE]

From the classical theory, the polynomials $B_{k}$ form an orthogonal system with respect to a Stieltjes measure $d\sigma(x)$ on the real axis,

[TABLE]

Conversely, if a family of polynomials $P_{n}(x)$ is orthogonal for $d\sigma(x)$, then $P_{n}(x)$ satisfies the following recurrence relation:

[TABLE]

where $\lambda_{j}>0$, $\mu$ and $\gamma_{j}>0$ are constants. Moreover, if we consider the continued fraction

[TABLE]

and realize an equivalent transformation

[TABLE]

we reconstruct the $\Gamma$-fraction corresponding to $d\sigma(x)$ (where we can put $\frac{\gamma_{j}}{\lambda_{j}\lambda{j+1}}=a_{j}^{2}$ and $\frac{\mu_{j}}{\lambda_{j}}=b_{j}$). It follows that there is a one-to-one correspondence between the set of Jacobi matrices and that of all the orthogonal polynomial systems on $\mathbb{R}$. In fact, if we choose the orthogonal polynomials

[TABLE]

as the basis of the vector space consisting of all polynomials then the Jacobi matrix represents the multiplication by $x$.

As an example of $V_{-j,k}$ (theorem 3, b)), consider the infinite Jacobi matrix (symmetric, tridiagonal and $N$-periodic) :

[TABLE]

The matrix $A$ is $N$-periodic when

[TABLE]

We denote by $f=(...,f_{-1},f_{0},f_{1},...)$ the (infinite) column vector and by $D$ (shift operator) the operator passage of degree +1, $Df_{i}=f_{i+1}$. Since the matrix $A$ is $N$-periodic, we have

[TABLE]

Reciprocally, this relation of commutation means that $N$ is the period of $A$. Let

[TABLE]

be the finite Jacobi matrix (symmetric tridiagonal and $N$-periodic). The determinant of the matrix

[TABLE]

is

[TABLE]

where $(z,h)\in\mathbb{C}\times\mathbb{C}^{*}$, $\alpha=\prod_{i=1}^{N}a_{i}$ and $P(z)$ is a polynomial of degree $N$ with real coefficients :

[TABLE]

Let $\mathcal{C}$ be the Riemann surface defined by

[TABLE]

We suppose that $\alpha\neq 0$. From the equation

[TABLE]

we derive the following relation :

[TABLE]

Note that $\mathcal{C}$ is a hyperelliptic curve branched in $2N$ points given by the roots of the polynomial $P(z)=\pm 2\alpha$, and admits two points at infinity $\mathcal{P}$ and $\mathcal{Q}$; the point $\mathcal{P}$ covering the case $z=\infty$, $h=\infty$ while the point $\mathcal{Q}$ is relative to the case $z=\infty$, $h=0$. (The hyperelliptic involution on $\mathcal{C}$ maps $(z,h)$ into $(z,h^{-1})$ and the curve $\mathcal{C}$ may be singular). According to the Riemann-Hurwitz formula, the genus of $\mathcal{C}$ is $g=N-1$. The meromorphic function $h$ has neither zero nor poles except in the neighborhood of $z=\infty$. When $z\nearrow\infty$, we have on the sheet +,

[TABLE]

which shows that $h$ has a pole of order $N$. Similarly, when $z\nearrow\infty$, we have on the sheet -,

[TABLE]

and therefore $h$ has a zero of order $N$. Therefore the divisor $(h)$ of the function $h$ on $\mathcal{C}$ is

[TABLE]

where $\mathcal{P}$ and $\mathcal{Q}$ are the two points covering $\infty$ on the sheets + and - respectively. The map

[TABLE]

is an antiholomorphic involution. In other words, the map $\sim:p\longmapsto\widetilde{p}$ is such that : $\widetilde{\mathcal{P}}=\mathcal{Q}$. When $|h|=1$, the finite matrix $A(h)$ is self-adjoint and therefore admits a real spectrum. Hence, the set of fixed points of this involution denoted by $\mathcal{C}^{\sim}$ is determined by

[TABLE]

Note that this set divides $\mathcal{C}$ into two distinct regions $\mathcal{C}_{+}$ and $\mathcal{C}_{-}$. More precisely, we have

[TABLE]

so $\mathcal{C}=\mathcal{C}_{+}\cup\mathcal{C}^{\sim}\cup\mathcal{C}_{-}$. The first region $\mathcal{C}_{+}$ contains the point $\mathcal{P}$ while the second $\mathcal{C}_{-}$ contains the point $\mathcal{Q}$. In fact $\mathcal{C}^{\sim}$ can be seen as the frontier of $\mathcal{C}_{+}$ and $\mathcal{C}_{-}$, So $\mathcal{C}^{\sim}$ is homologous to zero. Moreover, the involution $\sim$ extends to an involution $*$ on the field of meromorphic functions as follows:

[TABLE]

and on the differential space as follows :

[TABLE]

Hence, we have $h^{*}=h^{-1}$ and $z^{*}=z$. The condition that the matrices $A$ and $D^{N}$ have an eigenvector in common is parameterized by the Riemann surface $\mathcal{C}$ (2.6), let $f=(...,f_{-1},f_{0},f_{1},...)$ such an eigenvector. In the following, appropriate standardization is used by selecting $f_{0}\equiv 1$, from where $F_{N}=h$. Let us therefore $\overline{f}=(f_{1},f_{2},...,f_{N-1})^{\top}$. Since $\overline{f}$ satisfies

[TABLE]

then we have

[TABLE]

where $\Delta_{k,l}$ is the $(k,l)$-cofactor of $(A(h)-zI)$, that is to say,

[TABLE]

is the $(k,l)-\mbox{minor of }(A(h)-zI)$, i.e., the determinant of the $N-1$ submatrix obtained by removing the $k^{th}$-line and the $l^{th}$-column of the matrix $(A(h)-zI)$). In particular, $f$ can be expressed as a rational function in $z$ and $h$,

[TABLE]

According to matrix (3.4), we note that

[TABLE]

where $(-z)^{N-2}+\cdots,$ is a polynomial of degree $N-2$. Similarly,

[TABLE]

is a polynomial of degree $N-1$. To determine the divisor structure of $f_{k}$, one proceeds as follows : for $f_{1}$, we have

[TABLE]

and for the other $f_{k}$, we consider first the matrix (3.4) shifted by one, i.e.,

[TABLE]

Hence,

[TABLE]

and as above, we have

[TABLE]

which implies that

[TABLE]

And in general, we get

[TABLE]

Let $\mathcal{D}$ be a minimal positive divisor on $\mathcal{C}$ such that :

[TABLE]

It is shown that the degree of $\mathcal{D}$ is

[TABLE]

We show that the divisor $\mathcal{D}$ is regular with respect to $\mathcal{P}$ and $\mathcal{Q}$, i.e., such that

[TABLE]

The proof consists in showing first that the divisor $\mathcal{D}$ is general. Recall that a positive divisor $\mathcal{D}$ of degree $g$ on $\mathcal{C}$ is general if $(\omega_{j}(p_{j}))\neq 0$, $p_{j}\in\mathcal{C}$, $1\leq j\leq g$ where $(\omega_{1},...,\omega_{g})$ is a normalized base of differential forms on $\mathcal{C}$. It is shown that $\mathcal{C}$ is general if and only if $\dim\mathcal{L}(\mathcal{D})=1$ (where $\mathcal{L}(\mathcal{D})$ denotes the set of meromorphic functions $f$ on $\mathcal{C}$ such that : $(f)+\mathcal{D}\geq 0$) or if and only if $\dim\Omega(-\mathcal{D})=0$ where $\Omega(\mathcal{D})$ denotes the set of meromorphic differential forms $\omega$ on $\mathcal{C}$ such that the divisor $(\omega)+\mathcal{D}\geq 0$. Consider an integer $k>g-2$, then we deduce from the Riemann-Roch theorem

[TABLE]

Since $\dim\Omega(-\mathcal{D}-k\mathcal{P})=0$, because a holomorphic differential can have at most $2g-2$ zeroes, then

[TABLE]

Moreover, $\mathcal{L}(\mathcal{D}+j\mathcal{P})$ is strictly larger than $\mathcal{L}(\mathcal{D}+(j-1)\mathcal{P})$, because $f_{j}$ belongs to the first space and not to the second. Therefore by lowering the index $j$ down to [math], it follows that $\dim\mathcal{L}(\mathcal{D})=1$, which shows that $\mathcal{D}$ is general. Let’s show now that $\mathcal{D}$ is regular. It suffices to proceed by induction. Since $\dim\mathcal{L}(\mathcal{D})=1$ and $\mathcal{L}(\mathcal{D}-\mathcal{Q})\subsetneqq\mathcal{L}(\mathcal{D})$ (i.e., $f_{0}=1\notin\mathcal{L}(\mathcal{D}-\mathcal{Q})$; the function $f_{0}=1$ belongs to the second space but not the first), then

[TABLE]

Assume that

[TABLE]

then by the Riemann-Roch theorem

[TABLE]

implies equality since $f_{k+1}$ belongs to the first space. Since $f_{k+1}$ belongs to $\mathcal{L}(\mathcal{D}+(k+1)\mathcal{P}-(k+1)\mathcal{Q})$ but not to $\mathcal{L}(\mathcal{D}+(k+1)\mathcal{P}-(k+2)\mathcal{Q})$, we have that

[TABLE]

Consider the differential of $F$ (2.34) while taking into account that $z$ appears only on the diagonal of the matrix $A(h)-zI$. We have

[TABLE]

and either

[TABLE]

We have

[TABLE]

Or

[TABLE]

so

[TABLE]

and consequently

[TABLE]

From this we deduce that $\omega^{*}=\omega$. In addition, $\omega\geq 0$ on $\mathcal{C}^{\sim}$. (Indeed, on $\mathcal{C}^{\sim}$ we have

[TABLE]

Let $h=\rho e^{i\theta}$. Note that in all finite number points, $h$ is a local parameter on $\mathcal{C}$ while $\theta$ is a local parameter on $\mathcal{C}^{\sim}$. Like

[TABLE]

at these points and by continuity at all points). We also have a relation which shows that the scalar product between $f_{k}$ and $f_{l}$ is

[TABLE]

That is, the functions $f_{k}$, $k\in\mathbb{Z}$, are orthogonal to $\mathcal{C}^{\sim}$ with respect to $\omega$. We deduce from these properties that the divisor of $\omega$ is

[TABLE]

for the involution $\sim$ introduced previously. Given a matrix of the form $A$ (2.3), we have obtained a series of data $\{\mathcal{C},z,h,\mathcal{D},\omega\}$.

What is remarkable is that the reverse is also true (for further information, see [60]) :

Theorem 8

There is a one-to-one correspondence between the following sets of data :

a) Let $a_{i},b_{i}\in\mathbb{C}$, $a_{i}\neq 0$, where

[TABLE]

An infinite $N$-periodic matrix

[TABLE]

modulo conjugation by $N$-periodic diagonal matrices with real entries.

b) A curve $\mathcal{C}$ (possibly singular) of genus $N-1$ with two points $\mathcal{P}$ and $\mathcal{Q}$ on $\mathcal{C}$, a divisor $\mathcal{D}$ of degree $N-1$ on $\mathcal{C}$ and two meromorphic functions $h$ and $z$ on $\mathcal{C}$ such that :

[TABLE]

where $S$ is a positive divisor not containing the points $\mathcal{P}$ and $\mathcal{Q}$. The curve $\mathcal{C}$ is equipped with an antiholomorphic involution

[TABLE]

for which

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

such that : $\mathcal{P}\in\mathcal{C}_{+}$ and $\mathcal{Q}\in\mathcal{C}_{-}$. By using the involution $\sim$, we introduce an involution $*$ acting on the space of all functions on $\mathcal{C}$ in a way

[TABLE]

and on the differential space as follows :

[TABLE]

then $h^{*}=h^{-1}$, and $z^{*}=z$. The divisor of a differential form $\omega$ on $\mathcal{C}$ is

[TABLE]

For any difference operator $X$, we define

[TABLE]

Let $\mathcal{M}$ be the vector space of infinite $N$-periodic matrices $A$ such that for some $K$, $c_{ij}=0$ if $|i-j|>K$. On $\mathcal{M}$, we introduce the following scalar product :

[TABLE]

We say that a functional $F$ is differentiable if there exists a matrix $\frac{\partial F}{\partial C}$ in $\mathcal{M}$ such that for every $D$,

[TABLE]

Note that we also have

[TABLE]

Define the following bracket between two differentiable functionals $F$ and $G$ on $\mathcal{M}$,

[TABLE]

$\{,\}$ satisfies the Jacobi identity. Let $P(A,S,S^{-1})$ be a polynomial in $S+S^{-1}$ and $A$ with real coefficients. Consider the following Lax equation:

[TABLE]

When the matrix $A(t)$ deforms with $t$, then only the divisor $\mathcal{D}$ varies while $\{\mathcal{C},z,h,\mathcal{P},\mathcal{Q}\}$ remain fixed. As we have already shown, the coefficients of $z^{i}h^{j}$ in equation (2.5) are invariants of this motion. The divisor $\mathcal{D}(t)$ evolves linearly on the Jacobian manifold $\mbox{Jac}(\mathcal{C})$. Any linear flow over $\mbox{Jac}(\mathcal{C})$ is equivalent to Equation (2.8) and is a Hamiltonian flow with respect to the above (Poisson) bracket. In particular, the flow

[TABLE]

is written in terms of the (Poisson) bracket as follows :

[TABLE]

The (Poisson) bracket of two functional of the form $Tr\left(S^{-k}A^{l+1}\right)$ is zero, which means that we have a set of integrals in involution. Let $(\omega_{1},...,\omega_{g})$ be a holomorphic differential basis on the hyperelliptic curve $\mathcal{C}$. We have

[TABLE]

and let

[TABLE]

Since the order of the zeros of $\omega_{k}$ at the points at infinity $\mathcal{P}$, $\mathcal{Q}$ is equal to $g-k$, then $c_{k}=0$ for $k<g-j+1$ and $c_{k}\neq 0$ for $k=g-j+1$. Therefore, a complete set of flows is given by the functions $z,z^{2},...,z^{g}$ and the flow which leaves invariant the spectrum of $A$ and $X$ is given by a polynomial $P(z)$ of degree at most equal to $g$ :

[TABLE]

where $P(A)^{+}$ (resp. $-P(A)^{-}$) is the upper triangular part of $P(A)$ (lower), including the diagonal of $P(A)$. The (Poisson) bracket between two functional $F$ and $G$ can still be written in the form

[TABLE]

where $\frac{\partial F}{\partial a}$ and $\frac{\partial F}{\partial b}$ are the column vectors whose elements are given by $\frac{\partial F}{\partial a_{i}}$ and $\frac{\partial F}{\partial b}_{i}$ respectively while $J$ is the antisymmetric matrix Of order $2n$ defined by

[TABLE]

The symplectic structure is given by

[TABLE]

Flaschka variables [13] :

[TABLE]

applied to the form (3.9) with $x_{N+1}=0$, leads to the symplectic structure

[TABLE]

used by Moser [45, 46] during the study of the dynamical system of $N-1$ particules moving freely on the real axis under the influence of the exponential potential. See also the example below concerning the study of Toda lattice. We have

[TABLE]

where $\beta_{2},...,\beta_{N}$ are the $g$ invariant, functionally independent and in involution. These invariants are given by the $g=N-1$ points chosen from the spectrum of $A_{1}$ and $A_{-1}$, i.e., by the branch points of the hyperelliptic curve $\mathcal{C}$ or by the quantities $TrA^{k}$, $2\leq k\leq N$.

With the Jacobi matrix, we can associate an operator $T$ on a separable Hilbert space $E$ as follows,

[TABLE]

where $(e_{1},...)$ is an orthonormal basis in $E$. The operator $T$ is symmetric. Indeed, for any two finite vectors $u$ and $v$, we have $\langle Tu,v\rangle=\langle u,Tv\rangle$, according to the symmetry of the Jacobi matrix. Moreover, if the Carleman’s condition :

[TABLE]

is satisfied, then the operator $T$ is self-adjoint and its spectrum is simple with $e_{0}$ a generating element. In this case, the information about the spectrum of $T$ is contained in function,

[TABLE]

defined at $z\notin\sigma(T)$ where $\sigma(x)=\langle I_{x}e_{0},e_{0}\rangle$ and $I_{x}$ is the resolution of the identity of the operator $T$. Recall that the infinite continued fraction converges if the limit (2.2) exists. If the operator $T$ is self-adjoint, then the continued fraction $\varphi(z)$ converges uniformly in any closed bounded domain of $z$ without common points with real axis, to the analytic function defined by (3.10). If the support of $d\sigma(x)$ is bounded, then the sequence $\left(\frac{A_{k}(z)}{B_{k}(z)}\right)$ converges uniformly to a holomorphic function near $z=\infty$. Moreover, if a Jacobi matrix is bounded, i.e., if there exists $\rho>0$ such that,

[TABLE]

then the associated $\Gamma$-fraction converges uniformly on the following domain $\{z:|z|\geq\rho\}$ and the support of $d\sigma(x)$ is included in $[-\rho,\rho]$. In the case of a periodic Jacobi matrix, this one is obviously bounded and therefore the associated $\Gamma$-fraction converges near $z=\infty$. In addition, the function $\varphi(z)$ is written in the form (3.10) (Cauchy-Stieltjes transform of $d\sigma(x)$), which shows that $\varphi(z)$ has zero of first order at $z=\infty$ and for any point $z$ belonging the upper-half plane, the imaginary part of $\varphi(z)$ is non positive.

We will now extend the Jacobi matrix $\Gamma$ to the infinite symmetric, tridiagonal and $N$-periodic Jacobi matrix $A$(3.3) and use the results obtained previously. We consider $\varphi(z)$(3.1) as being the associated $N$-periodic $\Gamma$-fraction. The latter converges near the infinite point $z=\infty$. After an analytic prolongation, the function $\varphi(z)$ coincides with $a_{0}f_{1}$ where $f_{1}$ is a meromorphic function on the genus $N-1$ hyperelliptic curve $\mathcal{C}$(3.6). The latter is branched at the $2N$ real zeroes $\xi_{1}$, $\xi_{2}$,…,$\xi_{2N}$ of the polynomial $P^{2}(z)-4\alpha^{2}$. The interval $[\xi_{2j-1},\xi_{2j}]$, $1\leq j\leq N$, is called the stable band and the interval $[\xi_{2j},\xi_{2j+1}]$, $1\leq j\leq N-1$, is called the unstable band.

Theorem 9

Each zero $\sigma_{1}<\sigma_{2}<\cdots<\sigma_{N-1}$ of $\Delta_{k,l}$ (3.7), belongs to the $j$-th finite unstable band $[\lambda_{2j},\lambda_{2j+1}]$, $1\leq j\leq N-1$.

The function $\varphi(z)$ can be expressed (see below) by means of Abelian integrals on the hyperelliptic curve $\mathcal{C}$ (3.6). For $N=1$, $B_{k}(x)$ is the $k$-th Tschebyscheff polynomial of the second type. For $N>1$, Kato [25, 26] has found a new phenomenon related to discrete measures. We have seen that

[TABLE]

belonging to $\mathcal{L}(\mathcal{D}+\mathcal{P}-\mathcal{Q})$. Then, we have

Theorem 10

The function $\varphi(z)$ can explicitly written by means of Abelian integrals on the hyperelliptic curve $\mathcal{C}$ (3.6) as follows,

[TABLE]

where,

[TABLE]

and

[TABLE]

The differentials obtained in the previous section,

[TABLE]

($a$ and $b$ are constants) are positive mesures on each stable band $[\xi_{2j-1},\xi_{2j}]$. Therefore, the expression (3.11) means that $\varphi(z)$ can be obtained by the Cauchy-Stieltjes transform of

[TABLE]

as follows,

[TABLE]

The function $\varphi(z)$ belongs to $\mathcal{L}(\mathcal{D}^{\prime}+\mathcal{P}-\mathcal{Q})$ where $\mathcal{D}^{\prime}=\sigma_{1}^{+}+\cdots+\sigma_{N-1}^{+}$ is contained in $\mathcal{C}_{+}=\{p\in\mathcal{C}:|h|>1\}$ (see previous section). From expression (3.11), we have

[TABLE]

where $j_{1}<j_{2}<...<j_{l}$ denote the numbers for which $\mbox{Res}(\varphi(z),\sigma_{j}^{-})>0$ and $j_{l+1}<j_{l+2}<...<j_{N-1}$ the numbers for which $\mbox{Res}(\varphi(z),\sigma_{j}^{-})=0$. Hence,

[TABLE]

Example 6

The Toda lattice equations [57] (discretized version of the Korteweg-de Vries equation) describe the motion of $n$ particles with exponential restoring forces and are governed by the following Hamiltonian

[TABLE]

Here the nonlinear dynamical system is described by the following Hamiltonian equations :

[TABLE]

Flaschka variables [13] :

[TABLE]

can be used to express the symplectic structure $\omega$ (3.9) in terms of $x_{j}$ and $y_{j}$ as follows,

[TABLE]

then

[TABLE]

where $x_{j}^{*}\equiv x_{j}-x_{1}$ and $y_{j}^{*}\equiv y_{j}$. We will study the integrability of this problem with the Griffiths approach. There are two cases :

$(i)$* The non-periodic case, i.e.,*

[TABLE]

where the masses are arranged on a line. In term of the Flaschka variables above, Toda’s equations take the following form

[TABLE]

with $a_{N+1}=a_{1}$ and $b_{N+1}=b_{1}$. To show that this system is completely integrable, one should find $N$ first integrals independent and in involution each other. From the second equation, we have

[TABLE]

and we normalize the $b_{i}$’s by requiring that $\sum_{j=1}^{N}b_{j}=0$ (applying this fact to (3.9), leads to the original symplectic form $\omega=\frac{1}{2}\sum_{j=2}^{N}dx_{j}\wedge dy_{j}$). This is a first integral for the system and to show that it is completely integrable, we must find $N-1$ other integrals that are functionally independent and in involution. We further define $N\times N$ matrices $A$ and $B$ with

[TABLE]

[TABLE]

Then the proposed system is equivalent to the Lax equation $\dot{A}=[B,A]$. From theorem 1, we know that the quantities

[TABLE]

are first integrals of motion. To be more precise

[TABLE]

Notice that $I_{1}$ is the first integral already know. Since these $N$ first integrals are shown to be independent and in involution each other, the system in question is thus completely integrable.

$(ii)$* The periodic case, i.e.,*

[TABLE]

the connected masses will be arranged on a circle. We show that in this case, the spectrum of the periodic Jacobi matrix

[TABLE]

remains invariant in time. The matrix $B(h)$ depending on the spectral parameter $h$, has the form

[TABLE]

and the rest follows from the general theory. Note that if $a_{j}(0)\neq 0$, then $a_{j}(t)\neq 0$ for all $t$. Since $A^{\top}(h)=A(h^{-1})$, then

[TABLE]

Therefore, the application

[TABLE]

is an involution on the spectral curve $\mathcal{C}$. We choose

[TABLE]

Note that here the matrix $A$ is meromorphic (whereas previously we considered it to be a polynomial in $h$) but we will see that we can adopt the theory explained in this section, to this situation too . We have

[TABLE]

Let us assume that $\prod_{j=1}^{N-1}a_{j}\neq 0$ and pose

[TABLE]

In $\mathbb{P}^{2}(\mathbb{C})$, the affine algebraic curve of equation $Q(h,z)=0$, is singular at infinity for $n\geq 4$. We will compute the genus of normalization $\mathcal{C}$ of this curve. Note that $\mathcal{C}$ is a double covering of $\mathbb{P}^{1}(\mathbb{C})$ branched into $2N$ points coinciding with the fixed points of involution $\sigma$ (3.12), that is, points where $h=\pm 1$. According to the Riemann-Hurwitz formula, the genus $g$ of the curve $\mathcal{C}$ is

[TABLE]

Consider the covering $\mathcal{C}\longrightarrow\mathbb{P}^{1}(\mathbb{C})$ below and set

[TABLE]

where $\mathcal{P}$ and $\mathcal{Q}$ are located on two separate sheets. From the equation $Q(h,z)=0$, the divisor of $h$ is

[TABLE]

In that case, the divisor $\mathcal{D}$ is written

[TABLE]

hence $B\in H^{0}(\mathcal{C},\mbox{Hom}(V,V(\mathcal{D}))$. The residue $\rho(B)\in H^{0}(\mathcal{C},\mathcal{O}_{\mathcal{D}}(\mathcal{D})$ satisfies the conditions of theorem 7, and consequently the linear flow is given by the application (2.9). To compute the residue $\rho(B)$ of $B$, we will determine a set of holomorphic eigenvectors, using the van Moerbeke-Mumford method described above. Let us calculate the residue in $\mathcal{Q}$ and the result will be similarly deduced in $\mathcal{P}$. Let $\mathcal{E}=\sum_{j=1}^{g}r_{j}$ be a general divisor of degree $g$ such that :

[TABLE]

According to the Riemann-Roch theorem,

[TABLE]

hence

[TABLE]

Let

[TABLE]

be a base with $f_{N}=h$. We can choose a vector $v$ of the following form $v=(f_{1},...,f_{N})^{\top}$, such that $v$ is an eigenvector of $A$, i.e.,

[TABLE]

Hence, $V=h^{-1}v$ is a holomorphic eigenvector. Without restricting generality, we take $N=3$. The system $Av=zv$, is written explicitly

[TABLE]

By multiplying each equation of this system by $h^{-1}$, everything becomes holomorphic except the last equation, i.e., $a_{3}f_{1}=z+\mbox{Taylor}$. Recall that the residue $\rho(B)$ of $B$ is the section of $\mathcal{O}_{\mathcal{D}}(\mathcal{D})$ induced by $\lambda$ in the equation (2.7): $Bv=\dot{v}+\lambda v$. In other words,

[TABLE]

and therefore

[TABLE]

We deduce that $\rho(B)=h^{-1}z$, and $\rho(\dot{B})=0$. The same conclusion holds for the residue in $\mathcal{P}$. Consequently, the flow in question linearizes on the Jacobian variety of $\mathcal{C}$.

4 Algebraically integrable systems

Consider the system of nonlinear differential equations

[TABLE]

where $f_{1},...,f_{n}$ are functions of $n+1$ complex variables $t,z_{1},...,z_{n}$ and which apply a domain of $\mathbb{C}^{n+1}$ into $\mathbb{C}$. The Cauchy problem is the search for a solution $\left(z_{1}(t),...,z_{n}(t)\right)$ in a neighborhood of a point $t_{0}$, satisfying the initial conditions :

[TABLE]

The system (4.1) can be written in vector form in $\mathbb{C}^{n}$,

[TABLE]

In this case, the Cauchy problem will be to determine the solution $z(t)$ such that

[TABLE]

When the functions $f_{1},...,f_{n}$ are holomorphic in the neighborhood of the $\left(t_{0},z_{1}^{0},...,z_{n}^{0}\right)$, then the Cauchy problem admits a holomorphic solution and only one. A question arises : can the Cauchy problem admits some non-holomorphic solution in the neighborhood of point $\left(t_{0},z_{1}^{0},...,z_{n}^{0}\right)?$ When the functions $f_{1},...,f_{n}$ are holomorphic, the answer is negative. Other circumstances may arise for the Cauchy problem concerning the system of differential equations (4.1), when the holomorphic hypothesis relative to the functions $f_{1},...,f_{n}$ is no longer satisfied in the neighborhood of a point. In such a case, it can be seen that the behavior of the solutions can take on the most diverse aspects. In general, the singularities of the solutions are of two types : mobile or fixed, depending on whether or not they depend on the initial conditions. Important results have been obtained by Painlevé [50]. Suppose that the system (4.1) is written in the form

[TABLE]

where

[TABLE]

[TABLE]

polynomials with several indeterminate $z_{1},...,z_{n}$ and algebraic coefficients in $t$. There are two cases :

(i) the fixed singularities are constituted by four sets of points. The first is the set of singular points of the coefficients $A_{i_{1},\ldots,i_{n}}^{\left(k\right)}\left(t\right)$, $B_{j_{1},\ldots,j_{n}}^{\left(k\right)}\left(t\right)$ intervening in the polynomials $P_{k}\left(t,z_{1},...,z_{n}\right)$ and $Q_{k}\left(t,z_{1},...,z_{n}\right)$. In general this set contains $t=\infty$. The second set consists of the points $\alpha_{j}$ such that : $Q_{k}\left(t,z_{1},...,z_{n}\right)=0$, which occurs if all the coefficients $B_{j_{1},\ldots,j_{n}}^{\left(k\right)}\left(t\right)$ vanish for $t=\alpha_{j}$. The third is the set of points $\beta_{l}$ such that for some values $\left(z_{1^{\prime}},...,z_{n^{\prime}}\right)$ of $\left(z_{1},...,z_{n}\right)$, we have $P_{k}\left(\beta_{l},z_{1^{\prime}},...,z_{n^{\prime}}\right)=Q_{k}\left(\beta_{l},z_{1^{\prime}},...,z_{n^{\prime}}\right)=0$. Then the second members of the above system are presented in the indeterminate form $\frac{0}{0}$ at the points $\left(\beta_{l},z_{1^{\prime}},...,z_{n^{\prime}}\right)$. Finally, the set of points $\gamma_{n}$ such that there exist $u_{1},...,u_{n}$, for which $R_{k}\left(\gamma_{n},u_{1},...,u_{n}\right)=S_{k}\left(\gamma_{n},u_{1},...,u_{n}\right)=0$, where $R_{k}$ and $S_{k}$ are polynomials in $u_{1},...,u_{n}$ obtained from $P_{k}$ and $Q_{k}$ by setting $z_{1}=\frac{1}{u_{1}},\ldots,z_{n}=\frac{1}{u_{n}}$. Each of these sets contains only a finite number of elements. The system in question has a finite number of fixed singularities.

(ii) the mobile singularities of solutions of this system are algebraic mobile singularities: poles and (or) algebraic critical points. There are no essential singular points for the solution $\left(z_{1},...,z_{n}\right)$.

Considering the system of differential equations (4.1), we can find sufficient conditions for the existence and uniqueness of meromorphic solutions. The existence and uniqueness for the solution of the Cauchy problem concerning the system (4.1), can be obtained using the method of indeterminate coefficients. The solution will be explained in the form of a Laurent series. Such a solution is formal because we obtain it by performing on various series, which we assume a priori convergent, various operations whose validity remains to be justified. The problem of convergence will therefore arise. The result will therefore be established as soon as we have verified that these series are convergent. This will be done using the majorant method [14, 7, 21]. In the following, we will consider the Cauchy problem concerning the normal system (4.1) where $f_{1},...,f_{n}$ do not depend explicitly on $t$, i.e.,

[TABLE]

We suppose that $f_{1},...,f_{n}$ are rational functions in $z_{1},...,z_{n}$ and that the system (4.2) is weight-homogeneous, i.e., there exist positive integers $l_{1},...,l_{n}$ such that

[TABLE]

for each non-zero constant $\alpha$. In other words, the system (4.2) is invariant under the transformation

[TABLE]

Note that if the determinant $\det\left(z_{j}\frac{\partial f_{i}}{\partial z_{j}}-\delta_{ij}f_{i}\right)_{1\leq i,j\leq n}$, is not identically zero, then the choice of the numbers $s_{1},...,s_{n}$ is unique. In what follows, we will assume that $t_{0}=z_{0}=0$, which does not affect the generality of the results.

Theorem 11

Suppose that

[TABLE]

($k_{i}\in\mathbb{Z}$, some $k_{i}>0$) is the formal solution (Laurent series), obtained by the method of undetermined coefficients of the weight-homogeneous system (4.2). Then the coefficients $z_{i}^{(0)}$ satisfy the nonlinear equation

[TABLE]

where $1\leq i\leq n$, while $z_{i}^{(1)},z_{i}^{(2)},...$ each satisfy a system of linear equations of the form

[TABLE]

where $z^{(k)}=(z_{1}^{(k)},...,z_{n}^{(k)})^{\top}$ and

[TABLE]

is the Jacobian matrix. Moreover, the formal series (4.3) are convergent.

The series (4.3) is the only meromorphic solution in the sense that this solution results from the fact that the coefficients $z_{i}^{(k)}$ are determined unequivocally with the adopted method of calculation. The result of the previous theorem applies to the following quasi-homogeneous differential equation of order $n$ :

[TABLE]

$f$ being a rational function in $\displaystyle{z,\frac{dz}{dt},...,\frac{d^{n-1}z}{dt^{n-1}}}$ and

[TABLE]

Indeed, the differential equation above reduces to a system of $n$ first order differential equations by setting

[TABLE]

We thus obtain

[TABLE]

Such a system constitutes a particular case of the normal system (4.2).

Consider now Hamiltonian dynamical systems of the form

[TABLE]

where $H$ is the Hamiltonian and $J=J(z)$ is a skew-symmetric matrix with polynomial entries in $z$, for which the corresponding Poisson bracket

[TABLE]

satisfies the Jacobi identities.

The system (4.4) with polynomial right hand side will be called algebraic complete integrable (in abbreviated form : a.c.i.) in the sense of Adler-van Moerbeke [7, 29, 58] when the following conditions hold.

i) The system admits $n+k$ independent polynomial invariants $H_{1},...,H_{n+k}$ of which $k$ invariants (Casimir functions) lead to zero vector fields

[TABLE]

the $n=(m-k)/2$ remaining ones $H_{k+1}=H$,…,$H_{k+n}$ are in involution (i.e., $\left\{H_{i},H_{j}\right\}=0$),which give rise to $n$ commuting vector fields. For generic $c_{i}$, the invariant manifolds $\overset{n+k}{\underset{i=1}{\bigcap}}\left\{z\in\mathbb{R}^{m}:H_{i}=c_{i}\right\}$ are assumed compact and connected. According to the Arnold-Liouville theorem [6], there exists a diffeomorphism

[TABLE]

and the solutions of the system (4.4) are straight lines motions on these real tori.

ii) The invariant manifolds thought of as lying in $\mathbb{C}^{m}$,

[TABLE]

are related, for generic $c_{i}$, to Abelian varieties $T^{n}=\mathbb{C}^{n}/Lattice$ (complex algebraic tori) as follows :

[TABLE]

where $\mathcal{D}$ is a divisor (codimension one subvarieties) in $T^{n}$. In the natural coordinates $(t_{1},...,t_{n})$ of $T^{n}$ coming from $\mathbb{C}^{n}$, the coordinates $z_{i}=z_{i}(t_{1},...,t_{n})$ are meromorphic and $\mathcal{D}$ is the minimal divisor on $T^{n}$ where the variables $z_{i}$ blow up. Moreover, the flows (4.4) (run with complex time) are straight-line motions on $T^{n}$.

Mumford gave one in his Tata lectures [48], which includes as well the noncompact. Algebraic means that the torus can be defined as an intersection

[TABLE]

involving a large number of homogeneous polynomials $P_{i}$. Condition $i)$ means, in particular, there is an algebraic map

[TABLE]

making the following sums linear in $t$ :

[TABLE]

where $\omega_{1},...,\omega_{n}$ denote holomorphic differentials on some algebraic curves. If the Hamiltonian flow (4.4) is a.c.i., it means that the variables $z_{i}$ are meromorphic on the torus $T^{n}$ and by compactness they must blow up along a codimension one subvariety (a divisor) $\mathcal{D}\subset T^{n}$. By the a.c.i. definition, the flow (4.4) is a straight line motion in $T^{n}$ and thus it must hit the divisor $\mathcal{D}$ in at least one place. Moreover through every point of $\mathcal{D}$, there is a straight line motion and therefore a Laurent expansion around that point of intersection. Hence the differential equations must admit Laurent expansions which depend on the $n-1$ parameters defining $\mathcal{D}$ and the $n+k$ constants $c_{i}$ defining the torus $T^{n}$, the total count is therefore

[TABLE]

parameters. The fact that algebraic complete integrable systems possess $(m-1)$-dimensional families of Laurent solutions, was implicitly used by Kowalewski [32] in her classification of integrable rigid body motions. Such a necessary condition for algebraic complete integrability can be formulated as follows [5] :

Theorem 12

If the Hamiltonian system (4.4) (with invariant tori not containing elliptic curves) is algebraic complete integrable, then each $z_{i}$ blows up after a finite (complex) time, and for every $z_{i}$, there is a family of solutions (4.4) depending on $\dim(\mbox{phase space})-1=m-1$ free parameters. Moreover, the system (4.4) possesses families of Laurent solutions depending on $m-2$, $m-3$,…,$m-n$ free parameters. The coefficients of each one of these Laurent solutions are rational functions on affine algebraic varieties of dimensions $m-1$, $m-2$, $m-3$,…,$m-n$.

The question is whether this criterion is sufficient. The main problem will be to complete the affine variety $\mathcal{A}$, into an Abelian variety. A naive guess would be to take the natural compactification $\overline{\mathcal{A}}$ of $\mathcal{A}$ by projectivizing the equations. Indeed, this can never work for a general reason: an Abelian variety $\widetilde{\mathcal{A}}$ of dimension bigger or equal than two is never a complete intersection, that is it can never be described in some projective space $\mathbb{P}^{n}$ by $n$-dim $\widetilde{\mathcal{A}}$ global polynomial homogeneous equations. In other words, if $\mathcal{A}$ is to be the affine part of an Abelian variety, $\overline{\mathcal{A}}$ must have a singularity somewhere along the locus at infinity. The trajectories of the vector fields (4.4) hit every point of the singular locus at infinity and ignore the smooth locus at infinity. In fact, the existence of meromorphic solutions to the differential equations (4.4) depending on some free parameters can be used to manufacture the tori, without ever going through the delicate procedure of blowing up and down. Information about the tori can then be gathered from the divisor. More precisely, around the points of hitting, the system of differential equations (4.4) admit a Laurent expansion solution depending on $m-1$ free parameters and in order to regularize the flow at infinity, we use these parameters to blowing up the variety $\overline{\mathcal{A}}$ along the singular locus at infinity. The new complex variety obtained in this fashion is compact, smooth and has commuting vector fields on it; it is therefore an Abelian variety. The system (4.4) with $k+n$ polynomial invariants has a coherent tree of Laurent solutions, when it has families of Laurent solutions in $t$, depending on $n-1$, $n-2$,…, $m-n$ free parameters. Adler and van Moerbeke [5] have shown that if the system possesses several families of $(n-1)$-dimensional Laurent solutions (principal Painlevé solutions) they must fit together in a coherent way and as we mentioned above, the system must possess $(n-2)$-, $(n-3)$-,…dimensional Laurent solutions (lower Painlevé solutions), which are the gluing agents of the $(n-1)$-dimensional family. The gluing occurs via a rational change of coordinates in which the lower parameter solutions are seen to be genuine limits of the higher parameter solutions an which in turn appears due to a remarkable propriety of algebraic complete integrable systems; they can be put into quadratic form both in the original variables and in their ratios. As a whole, the full set of Painlevé solutions glue together to form a fiber bundle with singular base. A partial converse to theorem 12, can be formulated as follows [5] :

Theorem 13

If the Hamiltonian system (4.4) satisfies the condition $i)$ in the definition of algebraic complete integrability and if it possesses a coherent tree of Laurent solutions, then the system is algebraic complete integrable and there are no other $m-1$-dimensional Laurent solutions but those provided by the coherent set.

We assume that the divisor is very ample and in addition projectively normal (see [17, 42] for definitions when needed). Consider a point $p\in\mathcal{D}$, a chart $U_{j}$ around $p$ on the torus and a function $y_{j}$ in $\mathcal{L}(\mathcal{D})$ having a pole of maximal order at $p$. Then the vector $(1/y_{j},y_{1}/y_{j},\ldots,y_{N}/y_{j})$ provides a good system of coordinates in $U_{j}$. Then taking the derivative with regard to one of the flows

[TABLE]

are finite on $U_{j}$ as well. Therefore, since $y_{j}^{2}$ has a double pole along $\mathcal{D}$, the numerator must also have a double pole (at worst), i.e., $\dot{y_{i}}y_{j}-y_{i}\dot{y_{j}}\in\mathcal{L}(2\mathcal{D})$. Hence, when $\mathcal{D}$ is projectively normal, we have that

[TABLE]

i.e., the ratios $y_{i}/y_{j}$ form a closed system of coordinates under differentiation. At the bad points, the concept of projective normality play an important role: this enables one to show that $y_{i}/y_{j}$ is a bona fide Taylor series starting from every point in a neighborhood of the point in question. Moreover, the Laurent solutions provide an effective tool for find the constants of the motion. For that, just search polynomials $H_{i}$ of $z$, having the property that evaluated along all the Laurent solutions $z(t)$ they have no polar part. Indeed, since an invariant function of the flow does not blow up along a Laurent solution, the series obtained by substituting the formal solutions (4.3) into the invariants should, in particular, have no polar part. The polynomial functions $H_{i}(z(t))$ being holomorphic and bounded in every direction of a compact space, (i.e., bounded along all principle solutions), are thus constant by a Liouville type of argument. It thus an important ingredient in this argument to use all the generic solutions. To make these informal arguments rigorous is an outstanding question of the subject. Assume Hamiltonian flows to be weight-homogeneous with a weight $l_{i}\in\mathbb{N}$, going with each variable $z_{i}$. Observe that then the constants of the motion $H$ can be chosen to be weight-homogeneous :

[TABLE]

The study of the algebraic complete integrability of Hamiltonian systems, includes several passages to prove rigorously. Here we mention the main passages. We saw that if the flow is algebraically completely integrable, the differential equations (4.4) must admits Laurent series solutions (4.3) depending on $m-1$ free parameters. We must have $k_{i}=l_{i}$ and coefficients in the series must satisfy at the 0thstep non-linear equations,

[TABLE]

and at the $k$thstep, linear systems of equations :

[TABLE]

where

[TABLE]

If $m-1$ free parameters are to appear in the Laurent series, they must either come from the non-linear equations (4.5) or from the eigenvalue problem (4.6), i.e., $L$ must have at least $m-1$ integer eigenvalues. These are much less conditions than expected, because of the fact that the homogeneity $k$ of the constant $H$ must be an eigenvalue of $L$. The formal series solutions are convergent as a consequence of the majorant method. Thus, the first step is to show the existence of the Laurent solutions, which requires an argument precisely every time $k$ is an integer eigenvalue of $L$ and therefore $L-kI$ is not invertible. One shows the existence of the remaining constants of the motion in involution so as to reach the number $n+k$. Then you have to prove that for given $c_{1},...,c_{m},$ the set

[TABLE]

defines one or several $n-1$ dimensional algebraic varieties ("Painlevé" divisor) having the property that $\overset{n+k}{\underset{i=1}{\bigcap}}\left\{z\in\mathbb{C}^{m}:H_{i}=c_{i}\right\}\cup\mathcal{D}$, is a smooth compact, connected variety with $n$ commuting vector fields independent at every point, i.e., a complex algebraic torus $\mathbb{C}^{n}/Lattice$. Therefore, the flows $\,J\frac{\partial H_{k+i}}{\partial z}$ $,...,J\frac{\partial H_{k+n}}{\partial z}$ are straight line motions on this torus (for concrete applications, see for example [5, 6, 7, 19, 36, 38, 58]). Let’s point out that having computed the space of functions $\mathcal{L}(\mathcal{D})$ with simple poles at worst along the expansions, it is often important to compute the space of functions $\mathcal{L}(k\mathcal{D})$ of functions having $k$-fold poles at worst along the expansions. These functions play a crucial role in the study of the procedure for embedding the invariant tori into projective space. From the divisor $\mathcal{D}$, a lot of information can be obtained with regard to the periods and the action-angle variables.

The idea of the Adler-van Moerbeke’s proof [4, 40] (or what can be called the Liouville-Arnold-Adler-van Moerbeke theorem) is closely related to the geometric spirit of the (real) Arnold-Liouville theorem [10]. Namely, a compact complex $n$-dimensional variety on which there exist $n$ holomorphic commuting vector fields which are independent at every point is analytically isomorphic to a $n$-dimensional complex torus $\mathbb{C}^{n}/Lattice$ and the complex flows generated by the vector fields are straight lines on this complex torus.

Theorem 14

Let $\overline{\mathcal{A}}$ be an irreducible variety defined by an intersection

[TABLE]

involving a large number of homogeneous polynomials $P_{i}$ with smooth and irreducible affine part

[TABLE]

Put $\overline{\mathcal{A}}\equiv\mathcal{A}\cup\mathcal{D}$, i.e., $\mathcal{D}=\overline{\mathcal{A}}\cap\{Z_{0}=0\}$ and consider the map

[TABLE]

Let $\widetilde{\mathcal{A}}=f(\overline{\mathcal{A}})=\overline{f(\mathcal{A})}$, $\mathcal{D}=\mathcal{D}_{1}\cup...\cup\mathcal{D}_{r}$, where $\mathcal{D}_{i}$ are codimension $1$ subvarieties and

[TABLE]

Assume that :

$(i)$* $f$ maps $\mathcal{A}$ smoothly and 1-1 onto $f(\mathcal{A})$.*

$(ii)$* There exist $n$ holomorphic vector fields $X_{1},...,X_{n}$ on $\mathcal{A}$ which commute and are independent at every point. One vector field, say $X_{k}$ (where $1\leq k\leq n$), extends holomorphically to a neighborhood of $\mathcal{S}_{k}$ in the projective space $\mathbb{P}^{N}(\mathbb{C})$.*

$(iii)$* For all $p\in\mathcal{S}_{k}$, the integral curve $f(t)\in\mathbb{P}^{N}(\mathbb{C})$ of the vector field $X_{k}$ through $f(0)=p\in\mathcal{S}_{k}$ has the property that*

[TABLE]

This condition means that the orbits of $X_{k}$ through $\mathcal{S}_{k}$ go immediately into the affine part and in particular, the vector field $X_{k}$ does not vanish on any point of $\mathcal{S}_{k}$. Then

$a)$* $\widetilde{A}$ is compact, connected, admits an embedding into $\mathbb{P}^{N}(\mathbb{C})$ and is diffeomorphic to a $n$-dimensional complex torus. The vector fields $X_{1},...,X_{n}$ extend holomorphically and remain independent on $\widetilde{\mathcal{A}}$.*

$b)$* $\widetilde{\mathcal{A}}$ is a Kähler variety, a Hodge variety and in particular, $\mathcal{A}$ is the affine part of an Abelian variety $\widetilde{\mathcal{A}}$.*

Example 7

The periodic $5$-particle Kac-van Moerbeke lattice [23] is given by the following quadratic vector field

[TABLE]

où $(x_{1},...,x_{5})\in\mathbb{C}^{5}$ et $x_{j}=x_{j+5}$. This system forms a Hamiltonian vector field for the Poisson structure

[TABLE]

and admits three independent first integrals

[TABLE]

Let’s show that this system is algebraically completely integrable. We easily check that $H_{1}$ and $H_{2}$ are involution while $H_{3}$ is a Casimir. The system in question is therefore integrable in the Liouville sense. In addition, it is shown [6] that the affine variety

[TABLE]

defined by the intersection of the constants of motion is isomorphic to $\mbox{Jac}(\mathcal{C})\backslash\mathcal{D}$ where

[TABLE]

is a smooth curve of genus $2$ and $\mathcal{D}$ consists of five copies of $\mathcal{C}$ in the Jacobian variety $\mbox{Jac}(\mathcal{C})$. The flows generated by $H_{1}$ et $H_{2}$ are linearized on $\mbox{Jac}(\mathcal{C})$ and the system in question is algebraically completely integrable. The reader interested in the study of this system via various methods can find further information with more detail in [6] as well as in [56].

Example 8

The problem we are going to study now is the generalized periodic Toda systems. Let $e_{0},...,e_{l}$ be linearly dependent vectors in the Euclidean vector space $(\mathbb{R}^{l+1},\langle.|.\rangle)$, $l\geq 1$ , such that they are $l$ to $l$ linearly independent (i.e, for all $j$, the vectors $e_{0},...,\widehat{e_{j}},...,e_{l}$ are linearly independent). Suppose that the non-zero reals $\xi_{0},...,\xi_{l}$ satisfying

[TABLE]

are non-zero sum; that is,

[TABLE]

Let $A=(a_{ij})_{0\leq i,j\leq l}$ be the matrix whose elements are defined by

[TABLE]

We consider the vector field $X_{A}$ on $\mathbb{C}^{2(l+1)}$,

[TABLE]

where $x,y\in\mathbb{C}^{l+1}$ and $x.y=(x_{0}y_{0},...,x_{l}y_{l})$. It has been shown [7] that if $X_{A}$ is an integrable vector field of an irreducibly algebraically completely integrable system, then $A$ is the Cartan matrix of a possibly twisted affine Lie algebra. This system was studied by many authors (see [17] and references therein). Specific detailed results can be found on the technical paper [6] (and also in [7]) about link between the Toda lattice, Dynkin diagrams, singularities and Abelian varieties. The periodic $l+1$ particle Toda lattices are associated to extended Dynkin diagrams. They have $l+1$ polynomial invariants, as many as there are dots in the Dynkin diagram and are integrable Hamiltonian systems. The complex invariant manifold defined by putting these invariants equals to generic constants completes into an Abelian variety by gluing on a specific divisor $\mathcal{D}$. The latter is entirely described by the extended Dynkin diagram : each point of the diagram corresponds to a component of the divisor and each subdiagram determines the intersection of the corresponding divisors. The global geometry of the complex invariant tori (Abelian varieties), such as polarization, divisor equivalences, dimension of certain linear systems, etc., is also entirely given by the extended Dynkin diagram and the linear equivalence between them is expressed in Lie-theoretic terms. More precisely, the divisor $\mathcal{D}$ consists of $l+1$ irreducible components $\mathcal{D}_{j}$ each associated with a root $\alpha_{j}$ of the Dynkin diagram $\Delta$. The intersection of $k$ components $\mathcal{D}_{{j}_{1}},...,\mathcal{D}_{{j}_{k}}$ satisfies the following relation : the intersection multiplicity of the intersection of $k$ components of the divisor equals $\frac{\mbox{order}(W)}{\det(A)}$ where $W$ and $A$ are the Weyl group and the Cartan matrix going with the sub-Dynkin diagram $\alpha_{{j}_{1}},...,\alpha_{{j}_{k}}$ associated with the $k$ components. The intersection of all the divisors is empty and the intersection of all divisors but one is a discrete set of points whose number is explicitly determined. we have the following expression for the set-theoretical number of points in terms of the Dynkin diagram

[TABLE]

where the integers $p_{\alpha}$, are given by the null vector of the Cartan matrix going with $\Delta$. The singularities of the divisor are canonically associated to semi-simple Dynkin diagrams. The singularities of each component occur only at the intersections with other components and their multiplicities at the intersection with other divisors are expressed in terms of how a corresponding root is located in the sub-Dynkin diagram determined by this root and those of the members of the above divisor intersection. The following inclusion holds for the singular locus $\mbox{sing}(\mathcal{D}k)$ of $\mathcal{D}_{k}$ :

[TABLE]

The multiplicity of the singularity of a particular component $\mathcal{D}_{k}$, at its intersection with $m$ other divisors, i.e., $\mbox{sing}(\mathcal{D}_{k})\cap(\mathcal{D}_{j_{1}},...,\mathcal{D}_{j_{m}})$, all $j_{1},...,j_{m}\neq k$, is entirely specified by the way the corresponding root $\alpha_{k}$ sits in the sub-Dynkin diagram $\alpha_{k},\alpha_{j_{1}},...,\alpha_{j_{m}}$. (See [6], for proof of these results as well as other information).

There are many examples of dynamical systems which have the weak Painlevé property that all movable singularities of the general solution have only a finite number of branches and some integrable systems appear as coverings of algebraic completely integrable systems. The invariant varieties are coverings of Abelian varieties and these systems are called algebraic completely integrable in the generalized sense. These systems are Liouville integrable and by the Arnold-Liouville theorem, the compact connected manifolds invariant by the real flows are tori; the real parts of complex affine coverings of Abelian varieties. Most of these systems of differential equations possess solutions which are Laurent series of $t^{1/n}$ ($t$ being complex time) and whose coefficients depend rationally on certain algebraic parameters. In other words, for these systems just replace in the definition of the complete algebraic integrability above the condition ii) by the following : iii) the invariant manifolds $\mathcal{A}$ are related to an $l$-fold cover $\widetilde{T}^{n}$ of $T^{n}$ ramified along a divisor $\mathcal{D}$ in $T^{n}$ as follows : $\mathcal{A}=\widetilde{T}^{n}\backslash\mathcal{D}$.

Example 9

Let us consider the Ramani-Dorizzi-Grammaticos (RDG) series of integrable potentials [52, 16] :

[TABLE]

It can be straightforwardly proven that a Hamiltonian $H$ :

[TABLE]

containing $V$ is Liouville integrable, with an additional first integral :

[TABLE]

The study of cases $m=1$ and $m=2$ is easy. The study of other cases is not obvious. For the case $m=3$, one obtains the Hénon-Heiles system [19]:

[TABLE]

corresponding to a generalized Hénon-Heiles Hamiltonian

[TABLE]

where $A$ is a constant parameters and $y_{1},y_{2},x_{1},x_{2}$ are canonical coordinates and momenta, respectively. The system (4.7) can be written in the form

[TABLE]

where

[TABLE]

The second integral of motion is

[TABLE]

When one examines all possible singularities, one finds that it possible for the variable $y_{1}$ to contain square root terms of the type $\sqrt{t}$, which are strictly not allowed by the Painlevé test. However, these terms are trivially removed by introducing some new variables $z_{1},\ldots,z_{5}$, which restores the Painlevé property to the system (to see further). And reasoning as above, we obtain a new algebraically completely integrable system. The functions $H_{1}\equiv H$ and $H_{2}$ commute :

[TABLE]

The second flow commuting with the first is regulated by the equations :

[TABLE]

The system (4.7) admits Laurent solutions in $\sqrt{t}$, depending on three free parameters : $\alpha$, $\beta$, $\gamma$ and they are explicitly given as follows

[TABLE]

These formal series solutions are convergent as a consequence of the majorant method. By substituting these series in the constants of the motion $H_{1}=b_{1}$ and $H_{2}=b_{2}$, i.e.,

[TABLE]

one eliminates the parameter $\gamma$ linearly, leading to an equation connecting the two remaining parameters $\alpha$ and $\beta$ :

[TABLE]

which is nothing but the equation of an algebraic curve $\mathcal{D}$ along which the $u(t)\equiv(y_{1}(t),y_{2}(t),x_{1}(t),x_{2}(t))$ blow up. To be more precise $\mathcal{D}$ is the closure of the continuous components of

[TABLE]

i.e.,

[TABLE]

The invariant variety

[TABLE]

is a smooth affine surface for generic $(b_{1},b_{2})\in\mathbb{C}^{2}$. The Laurent solutions restricted to the surface $\mathcal{A}$ are parameterized by the curve $\mathcal{D}$. We show that the system (4.7) is part of a new system of differential equations in five unknowns having two cubic and one quartic invariants (constants of motion). By inspection of the expansions (4.8), we look for polynomials in $(y_{1},y_{2},x_{1},x_{2})$ without fractional exponents. Let

[TABLE]

be a morphism on the affine variety $\mathcal{A}$(4.9) where $z_{1},\ldots,z_{5}$ are defined as

[TABLE]

Using the two first integrals $H_{1}$, $H_{2}$ and differential equations (4.7), we obtain a system of differential equations in five unknowns,

[TABLE]

having two cubic and one quartic invariants (constants of motion),

[TABLE]

This new system is completely integrable and can be written as

[TABLE]

where $H=F_{1}$. The Hamiltonian structure is defined by the Poisson bracket

[TABLE]

where

[TABLE]

and

[TABLE]

is a skew-symmetric matrix for which the corresponding Poisson bracket satisfies the Jacobi identities. The second flow commuting with the first is regulated by the equations

[TABLE]

These vector fields are in involution, i.e., $\{F_{1},F_{2}\}=0$, and the remaining one is Casimir, i.e., $J\frac{\partial F_{3}}{\partial z}=0$. The invariant variety

[TABLE]

is a smooth affine surface for generic values of $c_{1},c_{2}$, $c_{3}$. The system (4.11) possesses Laurent series solutions which depend on four free parameters. These meromorphic solutions restricted to the surface $\mathcal{B}$(4.12) can be read off from (4.8) and the change of variable (4.10). Following the method mentioned previously, one find the compactification of $\mathcal{B}$ into an Abelian surface $\widetilde{B}$, the system of differential equations (4.11) is algebraic complete integrable and the corresponding flows evolve on $\widetilde{\mathcal{B}}$. We show that the invariant surface $\mathcal{A}$(4.9) can be completed as a cyclic double cover $\overline{\mathcal{A}}$ of an Abelian surface $\widetilde{\mathcal{B}}$. The system (4.7) is algebraic complete integrable in the generalized sense. Moreover, $\overline{\mathcal{A}}$ is smooth except at the point lying over the singularity of type $A_{3}$ and the resolution $\widetilde{\mathcal{A}}$ of $\overline{\mathcal{A}}$ is a surface of general type. We have shown that the morphism $\varphi$(4.10) maps the vector field (4.7) into an algebraic completely integrable system (4.11) in five unknowns and the affine variety $\mathcal{A}$(4.9) onto the affine part $\mathcal{B}$(4.12) of an Abelian variety $\widetilde{\mathcal{B}}$. This explains (among other) why the asymptotic solutions to the differential equations (4.7) contain fractional powers. All this is summarized as follows [41]:

Theorem 15

The system (4.7) admits Laurent solutions with fractional powers (i.e., contain square root terms of the type $\sqrt{t}$ which are strictly not allowed by the Painlevé test) depending on three free parameters and is algebraic complete integrable in the generalized sense. The morphism $\varphi$ (4.10) (which restores the Painlevé property) maps this system into a new algebraic completely integrable system (4.11) in five unknowns.

The case $m=4$, corresponds the Ramani Dorizzi Grammaticos (RDG) system [52, 15],

[TABLE]

corresponding to the Hamiltonian

[TABLE]

This system is integrable in the sense of Liouville, the second first integral (of degree $8$) being

[TABLE]

The first integrals $H_{1}$ and $H_{2}$ are in involution, i.e., $\left\{H_{1},H_{2}\right\}=0$. The system (4.13) is weight-homogeneous with $q_{1},q_{2}$ having weight $1$ and $p_{1},p_{2}$ weight $2$, so that $H_{1}$(4.14) and $H_{2}$(4.15) have weight $4$ and $8$ respectively. When one examines all possible singularities, one finds that it possible for the variable $q_{1}$ to contain square root terms of the type $\sqrt{t}$, which are strictly not allowed by the Painlevé test. However, we will see later that these terms are trivially removed by introducing the variables $z_{1},\ldots,z_{5}$ (4.20) which restores the Painlevé property to the system. Let $\mathcal{B}$ be the affine variety defined by

[TABLE]

for generic $(b_{1},b_{2})\in\mathbb{C}^{2}$. The system (4.13) possesses $3$-dimensional family of Laurent solutions (principal balances) depending on three free parameters $u,v$ and $w$. There are precisely two such families, labelled by $\varepsilon=\pm 1$, and they are explicitly given as follows

[TABLE]

These formal series solutions are convergent as a consequence of the majorant method. By substituting these series in the constants of the motion $H_{1}=b_{1}$ and $H_{2}=b_{2}$, one eliminates the parameter $w$ linearly, leading to an equation connecting the two remaining parameters $u$ and $v$ :

[TABLE]

According to Hurwitz’s formula, this defines a Riemann surface $\Gamma$ of genus $16$. The Laurent solutions restricted to the affine surface $\mathcal{B}$(4.16) are thus parameterized by two copies $\Gamma_{-1}$ and $\Gamma_{1}$ of the same Riemann surface $\Gamma$. Let

[TABLE]

be the morphism defined on the affine variety $\mathcal{B}$(4.16) by

[TABLE]

These variables are easily obtained by simple inspection of the series (4.17). By using the variables (4.20) and differential equations (4.13), one obtains

[TABLE]

This new system on $\mathbb{C}^{5}$ admits the following three first integrals

[TABLE]

The first integrals $F_{1}$ and $F_{2}$ are in involution , while $F_{3}$ is trivial (Casimir function). The invariant variety $\mathcal{A}$ defined by

[TABLE]

is a smooth affine surface for generic values of $(c_{1},c_{2},c_{3})\in\mathbb{C}^{3}$. The system (4.21) is completely integrable and possesses Laurent series solutions which depend on four free parameters $\alpha,\beta,\gamma$ et $\theta$ :

[TABLE]

where $\varepsilon=\pm 1$. The convergence of these series is guaranteed by the majorant method. Substituting these developments in equations (4.22), one obtains three polynomial relations between $\alpha,\beta,\gamma$ and $\theta$. Eliminating $\gamma$ and $\theta$ from these equations, leads to an equation connecting the two remaining parameters $\alpha$ and $\beta$ :

[TABLE]

The Laurent solutions restricted to the surface $\mathcal{A}$ (4.23) are thus parameterized by two copies $\mathcal{C}_{-1}$ and $\mathcal{C}_{1}$ of the same Riemann surface $\mathcal{C}$ (4.25). According to the Riemann-Hurwitz formula, the genus of $\mathcal{C}$ is $7$. We embed these curves in a hyperplane of $\mathbb{P}^{15}(\mathbb{C})$ using the sixteen functions :

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $W(s_{j},s_{k})\equiv\dot{s}_{j}s_{k}-s_{j}\dot{s}_{k}$ (Wronskian) and we show that these curves have two points in common in which $\mathcal{C}_{1}$ is tangent to $\mathcal{C}_{-1}$. The system (4.13) is algebraic complete integrable in the generalized sense. The invariant surface $\mathcal{B}$(4.16) can be completed as a cyclic double cover $\overline{\mathcal{B}}$ of the Abelian surface $\widetilde{\mathcal{A}}$, ramified along the divisor $\mathcal{C}_{1}+\mathcal{C}_{-1}$. Moreover, $\overline{\mathcal{B}}$ is smooth except at the point lying over the singularity (of type $A_{3}$) of $\mathcal{C}_{1}+\mathcal{C}_{-1}$ (double points of intersection of the curves $\mathcal{C}_{1}$ and $\mathcal{C}_{-1}$) and the resolution $\widetilde{\mathcal{B}}$ of $\overline{\mathcal{B}}$ is a surface of general type. We shall resume the proof of these results. Observe that the morphism $\varphi$(4.19) is an unramified cover. The Riemann surface $\Gamma$(4.18) play an important role in the construction of a compactification $\overline{\mathcal{B}}$ of $\mathcal{B}$. Let us denote by $G$ a cyclic group of two elements $\{-1,1\}$ on

[TABLE]

where $\tau=t^{1/2}$ and $U_{\varepsilon}^{j}$ is an affine chart of $\Gamma_{\varepsilon}$ for which the Laurent solutions (4.24) are defined. The action of $G$ is defined by

[TABLE]

and is without fixed points in $V_{\varepsilon}^{j}$. So we can identify the quotient $V_{\varepsilon}^{j}/G$ with the image of the smooth map $h_{\varepsilon}^{j}:V_{\varepsilon}^{j}\longrightarrow\mathcal{B}$ defined by the expansions (4.24). We have

[TABLE]

and

[TABLE]

i.e., $G\times G$ acts separately on each coordinate. Thus, identifying $V_{\varepsilon}^{j}/G^{2}$ with the image of $\varphi\circ h_{\varepsilon}^{j}$ in $\mathcal{A}$. Note that $\mathcal{B}_{\varepsilon}^{j}=V_{\varepsilon}^{j}/G$ is smooth (except for a finite number of points) and the coherence of the $\mathcal{B}_{\varepsilon}^{j}$ follows from the coherence of $V_{\varepsilon}^{j}$ and the action of $G$. Now by taking $\mathcal{B}$ and by gluing on various varieties $\mathcal{B}_{\varepsilon}^{j}\backslash\{\mbox{some points}\}$, we obtain a smooth complex manifold $\widehat{\mathcal{B}}$ which is a double cover of the Abelian variety $\widetilde{\mathcal{A}}$ ramified along $\mathcal{C}_{1}+\mathcal{C}_{-1}$, and therefore can be completed to an algebraic cyclic cover of $\widetilde{\mathcal{A}}$. To see what happens to the missing points, we must investigate the image of $\Gamma\times\{0\}$ in $\cup\mathcal{B}_{\varepsilon}^{j}$. The quotient $\Gamma\times\{0\}/G$ is birationally equivalent to the Riemann surface $\Upsilon$ of genus $7$ :

[TABLE]

[TABLE]

where $y=uv,x=u^{2}$. The Riemann surface $\Upsilon$ is birationally equivalent to $\mathcal{C}$. The only points of $\Upsilon$ fixed under $(u,v)\longmapsto(-u,-v)$ are the points at $\infty$, which correspond to the ramification points of the map

[TABLE]

and coincides with the points at $\infty$ of the Riemann surface $\mathcal{C}$. Then the variety $\widehat{\mathcal{B}}$ constructed above is birationally equivalent to the compactification $\overline{\mathcal{B}}$ of the generic invariant surface $\mathcal{B}$. So $\overline{\mathcal{B}}$ is a cyclic double cover of the Abelian surface $\widetilde{\mathcal{A}}$ ramified along the divisor $\mathcal{C}_{1}+\mathcal{C}_{-1}$, where $\mathcal{C}_{1}$ and $\mathcal{C}_{-1}$ have two points in commune at which they are tangent to each other. The system (4.13) is algebraic complete integrable in the generalized sense. Moreover, $\overline{\mathcal{B}}$ is smooth except at the point lying over the singularity (of type $A_{3}$) of $\mathcal{C}_{1}+\mathcal{C}_{-1}$. In term of an appropriate local holomorphic coordinate system $(X,Y,Z),$ the local analytic equation about this singularity is $X^{4}+Y^{2}+Z^{2}=0$. Let $\widetilde{\mathcal{B}}$ be the resolution of singularities of $\overline{\mathcal{B}},$ $\mathcal{X}(\widetilde{\mathcal{B}})$ be the Euler characteristic of $\widetilde{\mathcal{B}}$ and $p_{g}(\widetilde{\mathcal{B}})$ the geometric genus of $\widetilde{\mathcal{B}}$. Then $\widetilde{\mathcal{B}}$ is a surface of general type with invariants: $\mathcal{X}(\widetilde{\mathcal{B}})=1$ and $p_{g}(\widetilde{\mathcal{B}})=2$. In summary we have [37],

Theorem 16

The system (4.13) admits Laurent solutions ,

[TABLE]

depending on three free parameters : $u,v$ and $w$. These solutions restricted to the surface $\mathcal{B}$(4.16) are parameterized by two copies $\Gamma_{1}$ and $\Gamma_{-1}$ of the Riemann surface $\Gamma$(4.18) of genus $16$. This system (4.13) is algebraic complete integrable in the generalized sense and extends to a new system (4.29) of five differential equations algebraically completely integrable with three quartics invariants (4.22). Generically, the invariant manifold $\mathcal{A}$(4.23) defined by the intersection of these quartics form the affine part of an Abelian surface $\widetilde{\mathcal{A}}$. The reduced divisor at infinity

[TABLE]

is very ample and consists of two components $\mathcal{C}_{1}$ and $\mathcal{C}_{-1}$ of a genus $7$ curve $\mathcal{C}$(4.25). In addition, the invariant surface $\mathcal{B}$ can be completed as a cyclic double cover $\overline{\mathcal{B}}$ of the Abelian surface $\widetilde{\mathcal{A}}$, ramified along the divisor $\mathcal{C}_{1}+\mathcal{C}_{-1}$. Moreover, $\overline{B}$ is smooth except at the point lying over the singularity (of type $A_{3}$) of $\mathcal{C}_{1}+\mathcal{C}_{-1}$ and the resolution $\widetilde{B}$ of $\overline{B}$ is a surface of general type with invariants : $\mathcal{X}(\widetilde{B})=1$ and $p_{g}(\widetilde{B})=2$.

Consider on the Abelian variety $\widetilde{\mathcal{A}}$ the holomorphic $1$-forms $dt_{1}$ and $dt_{2}$ defined by $dt_{i}(X_{F_{j}})=\delta_{ij}$, where $X_{F_{1}}$ and $X_{F_{2}}$ are the vector fields generated respectively by $F_{1}$ and $F_{2}$. Taking the differentials of $\zeta=1/z_{2}$ and $\xi=\frac{z_{1}}{z_{2}}$ viewed as functions of $t_{1}$ and $t_{2}$, using the vector fields and the Laurent series (4.24) and solving linearly for $dt_{1}$ and $dt_{2}$, we obtain the holomorphic differentials

[TABLE]

with

[TABLE]

The zeroes of $\omega_{2}$ provide the points of tangency of the vector field $X_{F_{1}}$ to $\mathcal{C}_{\varepsilon}$. We have

[TABLE]

and $X_{F_{1}}$ is tangent to $\mathcal{H}_{\varepsilon}$ at the point covering $\alpha=\infty$. Note that the reflection $\sigma$ on the affine variety $\mathcal{A}$ amounts to the flip

[TABLE]

changing the direction of the commuting vector fields. It can be extended to the (-Id)-involution about the origin of $\mathbb{C}^{2}$ to the time flip

[TABLE]

on $\widetilde{\mathcal{A}}$, where $t_{1}$ and $t_{2}$ are the time coordinates of each of the flows $X_{{F}_{1}}$ and $X_{{F}_{2}}$. The involution $\sigma$ acts on the parameters of the Laurent solution (3.24) as follows

[TABLE]

interchanges the Riemann surfaces $\mathcal{C}_{\varepsilon}$ and the linear space $\mathcal{L}$ can be split into a direct sum of even and odd functions. Geometrically, this involution interchanges $\mathcal{C}_{1}$ and $\mathcal{C}_{-1}$, i.e., $\mathcal{C}_{-1}=\sigma\mathcal{C}_{1}$.

Remark : However, the case $m=5$, corresponds to a system with an Hamiltonian of the form

[TABLE]

The corresponding Hamiltonian system admits a second first integral :

[TABLE]

and admits three $3$-dimensional families solutions $x$, $y$, which are Laurent series of $t^{1/3}$ : $x=at^{-\frac{1}{3}}$, $x=bt^{-\frac{2}{3}}$, $b^{3}=-\frac{2}{9}$, but for which there are no polynomial $P$ such that $P(x(t),y(t),\dot{x}(t),\dot{y}(t))$ is Laurent series in $t$. This problem needs to be studied and understood.