Linearization, separability and Lax pairs representation of $a_{4}^{\left(2\right)}$ Toda lattice

Bruce Lionnel LIETAP NDI University of Maroua
Faculty of Sciences, Department of Mathematics and Computer Sciences
P.O. Box 814 Maroua, Cameroon [email protected] , Djagwa DEHAINSALA University of NDjamena
Faculty of Exact and Applied Sciences, Department of Mathematics
1 route de Farcha, PO Box 1027 NDjamena, Chad [email protected] and Joseph DONGHO University of Maroua
Faculty of Sciences, Department of Mathematics and Computer Sciences
P.O. Box 814 Maroua, Cameroon [email protected]

Abstract.

The aim of this work is focused on linearizing and found the Lax Pairs of the algebraic complete integrability (a.c.i) Toda lattice associated with the twisted affine Lie algebra $a_{4}^{\left(2\right)}$ . Firstly, we recall that our case of a.c.i is a two-dimensional algebraic completely integrable systems for which the invariant (real) tori can be extended to complex algebraic tori (abelian surfaces). This implies that the geometry can be used to study this system. Secondly, we show that the lattice is related to the Mumford system and we construct an explicit morphism between these systems, leading to a new Poisson structure for the Mumford system. Finally, we give a new Lax equation for this Toda lattice and we construct an explicit linearization of the system.

Key words and phrases:

Toda lattice, integrable system, linearization, Lax representation.

1991 Mathematics Subject Classification:

34G20,34M55,37J35

This work was completed with the support of the Pr Joseph Dongho and Dr Djagwa Dehainsala.

Corresponding author: University of Maroua-Cameroon & University of NDjamena-Chad.

1. Introduction

Many integrable systems from classical mechanics admit a complexification, where phase space and time are complexified, and the geometry of the (complex) momentum map is the best possible complex analogue of the geometry that appears in the Liouville Theorem. Namely, in many relevant examples the generic complexified fiber is an affine part of an Abelian variety (a compact algebraic torus) and the integrable vector fields are translation invariant, when restricted to any of these tori. Such integrable systems are call them algebraic completely integrable systems, following the original definition of Adler and van Moerbeke.

Integrable systems have been integrated classically in terms of quadratures, usually through a sequence of very ingenious algebraic manipulations especially tailored to the problem. More recently, it was realized that whenever a system could be represented as a family of Lax pairs. the system could be linearized on the Jacobian of a spectral curve, defined by the characteristic polynomial of one of the matrices in the Lax pair.

To show that a Hamiltonian system linearizes on an Abelian variety, one may either construct a Lax representation of the differential equation depending on an extra-parameter and linearize on the Jacobian of the curve specified by its characteristic equation, or one may complete the complexified invariant manifolds by using the Laurent solutions of the differential equations. The latter method allows us in addition to identify the nature of the invariant manifolds and of the solutions of the system: in most examples the isospectral manifolds and the invariant manifolds are different.

In the previous work [3], we have prove that the $a_{4}^{\left(2\right)}$ is a two-dimensional integrable system. This system satisfies the linearization criterion [[1], theorem 6.41] and it is an algebraic completely integrable in the Adler-van Moerbeke sense. This system has a smooth hyperelliptic curve of genus two. According to Vanhaecke [7] and Mumford’s description of hyperelliptic Jacobians (see [[5], Section 3.1]), like $\Gamma$ is a hyperelliptic curve of genus two then the Riemann surface $\overline{\Gamma}$ is embedded in its jacobian such that $Jac(\overline{\Gamma})\ \Gamma$ is isomorphic to the space of pairs of polynomials $(u(\lambda);v(\lambda))$ . $u(\lambda)$ is a monic of degree two and $v(\lambda)$ less than two. $f(\lambda)-v^{2}(\lambda)$ is divisible by $u(\lambda)$ .

The aim of this paper is how we can linearize and find the Toda lattice $a_{4}^{(2)}$ Lax pair or Lax representation? To prove this, we construct an explicit map from the generic fiber $\mathbb{F}_{c}$ into the Jacobian of the Riemann surface $\overline{\Gamma}_{c}$ . After we find the kummer surface of $Jac(\mathcal{K}_{c})$ , $u(\lambda),v(\lambda)$ and $f(\lambda)$ .

This paper is organized as follows. In section 2, preliminaries of this work, we give the basic notions of linearising, separating variables and Lax representation. In section 3, main part of the paper, we show that the a.c.i $a_{4}^{\left(2\right)}$ Toda lattice is related to the Mumford system and we construct an explicit morphism between these systems, leading to a new Poisson structure for the Mumford system. Finally, we give a new Lax equation with spectral parameter for this Toda lattice and we construct an explicit linearization of the system.

2. Preliminaries

Let $\mathbb{C}^{n}$ denote a complex vector space of dimension $n$ .

Definition 2.1.

[2] A lattice in $\mathbb{C}^{n}$ is a discrete subgroup of maximal rank in $\mathbb{C}^{n}$ . It is a free abelian group of rank $2$ .

A lattice $\Lambda$ in $\mathbb{C}^{n}$ acts in a natural way on the vector space $\mathbb{C}^{n}$ and the quotient $\mathbb{T}^{n}=\mathbb{C}^{n}/\Lambda$ is called a complex torus.

In the theory of linear algebraic groups there is the notion of a torus. Such a torus is an affine group, whereas a complex torus is compact.

Definition 2.2.

[2] An abelian variety is a complex torus admitting a positive line bundle or equivalently a projective embedding.

Abelian varieties over the complex numbers are special complex tori, that is, quotients of finite-dimensional complex vector spaces modulo a lattice of maximal rank.

The Riemann Relations are necessary and sufficient conditions for a complex torus to be an abelian variety. They were introduced by Riemann in the special case of a Jacobian variety of a curve.

Let $\mathbb{T}^{n}=\mathbb{C}^{n}/\Lambda$ be a complex torus.

Definition 2.3.

[2] A positive line bundle on $\mathbb{T}^{n}$ is by definition a line bundle on $\mathbb{T}^{n}$ whose first Chern class is a positive definite hermitian form on $\mathbb{C}^{n}$ .

A polarization on $\mathbb{T}^{n}$ is by definition the first Chern class $H=c_{1}(L)$ of a positive line bundle $L$ on $\mathbb{T}^{n}$ .

By abuse of notation we sometimes consider the line bundle $L$ on $\mathbb{T}^{n}$ itself as a polarization. The type of $L$ is called the type of the polarization. A polarization is called principal if it is of type $(1,\cdot\cdot\cdot,1)$ .

Definition 2.4.

[2] An abelian variety is a complex torus $\mathbb{T}^{n}$ admitting a polarization $H=c_{1}(L)$ . The pair $(\mathbb{T}^{n},H)$ is called a polarized abelian variety.

According to [2], let $\Gamma$ be a smooth projective curve of genus $g$ over the field of complex numbers. the $g$ -dimensional $\mathbb{C}$ -vector space $H^{0}(\omega_{\Gamma})$ of holomorphic $1$ -forms on $\Gamma$ . The homology group $H^{1}(\Gamma,\mathbb{Z})$ is a free abelian group of rank $2g$ . For convenience we use the same letter for (topological) $1$ -cycles on $\Gamma$ and their corresponding classes in $H^{1}(\Gamma,\mathbb{Z})$ . By Stoke’s theorem any element $\gamma\in H^{1}(\Gamma,\mathbb{Z})$ yields in a canonical way a linear form on the vector space $H^{0}(\omega_{\Gamma})$ , which we also denote by:

\begin{array}[]{cccc}\gamma:&H^{0}(\omega_{\Gamma})&\longrightarrow&\mathbb{C}\\ &\omega&\longmapsto&\int_{\gamma}\omega\end{array}

Definition 2.5.

[2] the Jacobian variety or simply the Jacobian of $\Gamma$ , denote by $Jac(\Gamma)$ is a complex torus of dimension $g$ such that

Jac(\Gamma):=H^{0}(\omega_{\Gamma})^{\ast}/H^{1}(\Gamma,\mathbb{Z})

Definition 2.6.

[2] A theta divisor of the Jacobian $Jac(\Gamma)$ is any divisor on $Jac(\Gamma)$ such that the line bundle $\mathcal{O}_{Jac(\Gamma)}(\Theta)$ defines the canonical polarization.

Definition 2.7.

[4] A system of ordinary differential equations over $\mathbb{R}$ is called algebraic complete integrable (a.c.i.) when it is completely integrable and the complexified invariant manifolds complete into algebraic tori (Abelian varieties), whose (complexified) commuting flows extend holomorphically.

According to [4], Let $\mathbb{T}^{n}=\mathbb{C}^{n}/\Lambda$ be a complex algebraic torus, (Abelian variety) with an origin $0$ chosen. Let $i$ be the inverse morphism which coincides with the $(-1)$ -reflection about $0$ .

Definition 2.8.

[4] The Kumrner variety of $\mathbb{T}^{n}$ , denoted by $\mathcal{K}_{c}$ , is the quotient of $\mathbb{T}^{n}$ by the action of the group $(1,i)$ .

The Kummer variety bears the moduli information and has the advantage of possessing a lower degree of embedding in projective space. According to [4], let $\mathcal{D}$ be a divisor on $\mathbb{T}^{n}$ . Denote by $\mathcal{L}(\mathcal{D})$ the invertible sheaf associated to $\mathcal{D}$ .

	$\displaystyle\mathcal{L}(\mathcal{D})$	$\displaystyle=$	{ the vector space of functions $f$ such that
			$\displaystyle(f)=\mbox{divisor of zeroes-divisor of poles}\geq-\mathcal{D}\}$

According to [7] Let $\Gamma$ be a smooth curve of genus $g$ . We define two divisor $\mathcal{D}$ and $\mathcal{D}^{\prime}$ in $Div(\Gamma)$ , the divisor group of $\Gamma$ , to be linearly equivalent, $\mathcal{D}\sim_{l}\mathcal{D}^{\prime}$ , if and only if there exists a meromorphic function $f$ on $\Gamma$ .

According to [4], let $\mathcal{D}$ be an ample divisor on $\mathbb{T}^{n}$ . We denote by $\mathcal{C(D)}$ the set of all divisors $\mathcal{D}^{\prime}$ on $\mathbb{T}^{n}$ such that there are two positive numbers $n,n^{\prime}$ and $n\mathcal{D}$ is algebraically equivalent to $n^{\prime}\mathcal{D}^{\prime}$ .

Definition 2.9.

[1] A compact Riemann surface for which the Kodaira map is not an embedding is called a hyperelliptic Riemann surface (a compact Riemann surfaces of genus $1$ being called an elliptic Riemann surface), while any curve whose (compact) Riemann surface is hyperelliptic is called a hyperelliptic curve (one speaks of an elliptic curve in the genus $1$ case).

3. Separability and linearization of two-dimensional Toda lattice $a_{4}^{(2)}$

3.1. Linearization procedure

According to [1], since $Jac(\Gamma)$ is a principally polarized Abelian variety of dimension $g$ , the Lefschetz Theorem implies that it can be embedded in $\mathbb{P}^{3^{g}-1}$ , by using the sections of $\left[3\Theta\right]$ . However, the sections of $\left[2\Theta\right]$ never embed $Jac(\Gamma)$ in projective space, but rather they embed its Kummer variety $K_{c}(\Gamma)$ in projective space. An important particular case is that of the Kummer surface $K_{c}(\Gamma)$ , where $\Gamma$ is a hyperelliptic Riemann surface of genus $2$ . The line bundle $\left[2\Theta\right]$ that corresponds to twice the principal polarization on $Jac(\Gamma)$ has in this case $4$ independent sections and the associated Kodaira map, which maps $Jac(\Gamma)$ into $\mathbb{P}^{3}$ , factors through $K_{c}(\Gamma)$ , realizing the Kummer surface as a surface in $\mathbb{P}^{3}$ .

Being two-dimensional the image is given by a single equation; to compute the degree of this equation, we use the fact that this degree is given by $\displaystyle\int_{K_{c}(\Gamma)}\omega$ , where $\omega$ is associated $(1,1)$ -form ofthe standard Kahler structure on $\mathbb{P}^{3}$ . Clearly this is twice the volume of $K_{c}(\Gamma)$ , which itself is half the volume of the Jacobi surface (with the polarization of type $(1,1)$ ).

In the two-dimensional case, the invariant manifolds complete into Abelian surfaces by adding one (or several) curves to the affine surfaces. In this case, Vanhaecke proposed in [7] a method which leads to an explicit linearization of the vector field of the a.c.i. system. The computation of the first few terms of the Laurent solutions to the differential equations enables us to construct an embedding of the invariant manifolds in the projective space $\mathbb{P}^{N}$ . From this embedding, one deduces the structure of the divisors $\mathcal{D}_{c}$ to be adjoined to the generic affine in order to complete them into Abelian surfaces $\mathbb{T}_{c}$ . Thus, the system is a.c.i.. The different steps of the algorithm of Vanhaecke are given by:
case 1

a)

If one of the components of $\mathcal{D}_{c}$ is a smooth curve $\Gamma_{c}$ of genus two, compute the image of the rational map $\varphi_{[2\Gamma_{c}]}:\mathbb{T}_{c}^{2}\rightarrow\mathbb{P}^{3}$ which is a singular surface in $\mathbb{P}^{3}$ , the Kummer surface $\mathcal{K}_{c}$ of jacobian $Jac(\Gamma_{c})$ of the curve $\Gamma_{c}$ .
b)

Otherwise, if one of the components of $\mathcal{D}_{c}$ is a $d:1$ unramified cover $\mathcal{C}_{c}$ of a smooth curve $\Gamma_{c}$ of genus two, the map $p:\mathcal{C}_{c}\rightarrow\Gamma_{c}$ extends to the map $\widetilde{p}:\mathbb{T}_{c}^{2}\rightarrow Jac(\Gamma_{c})$ . In this case, let $\mathcal{C}_{c}$ denote the (non complete) linear system $\widetilde{p}[2\Gamma_{c}]\subset[2\mathcal{C}_{c}]$ which corresponds to the complete linear system $[2\mathcal{C}_{c}]$ and compute now the Kummer surface $\mathcal{C}_{c}$ of $Jac(\Gamma_{c})$ as image of $\varphi_{\varepsilon_{c}}:\mathbb{T}_{c}^{2}\rightarrow\mathbb{P}^{3}$ .
c)

Otherwise, change the divisor at infinity so as to arrive in case (a) or (b). This can always be done for any irreducible Abelian surface.

case 2. Choose a Weierstrass point $W$ on the curve $\Gamma_{c}$ and coordinates $(z_{0}:z_{1}:z_{2}:z_{3})$ for $\mathbb{P}^{3}$ such $\varphi_{[2\Gamma_{c}]}(W)=(0:0:0:1)$ in case 1.(a) and $\varphi_{\varepsilon_{c}}(W)=(0:0:0:1)$ in case 1.(b). Then this point will be a singular point (node) for the Kummer surface $\mathcal{K}_{c}$ whose equation is $\small{p_{2}(z_{o};z_{1};z_{2})z^{2}_{3}+p_{3}(z_{o};z_{1};z_{2})z_{3}+p_{4}(z_{o};z_{1};z_{2})=0}$

where the $p_{i}$ are polynomials of degree $i$ . After a projective transformation which fixes $(0:0:0:1)$ , we may assume that $p_{2}(z_{o};z_{1};z_{2})=z^{2}_{1}-4z_{0}z_{2}$ .

case 3. Finally, let $s_{1}$ and $s_{2}$ be the roots of the quadractic equation $z_{0}s^{2}+z_{1}s+z_{2}=0$ , whose discriminant is $p^{2}(z_{o};z_{1};z_{2})$ , with the $z_{i}$ expressed in terms of the original variables. Then the differential equations describing the vector field of the system are rewritten by direct computation in the classical Weierstrass form

(3.1)

\begin{array}[]{l}\frac{\dot{s_{1}}}{\sqrt{f(s_{1})}}+\frac{\dot{s_{2}}}{\sqrt{f(s_{2})}}=\alpha_{1}dt\\ \frac{s_{1}\dot{s_{1}}}{\sqrt{f(s_{1})}}+\frac{s_{2}\dot{s_{2}}}{\sqrt{f(s_{2})}}=\alpha_{2}dt\end{array}

where $\alpha_{1}$ and $\alpha_{2}$ depend on the torus. From it, the symmetric functions $s_{1}+s_{2}:=-\frac{z_{1}}{z_{0}}$ , $s_{1}s_{2}:=\frac{z_{2}}{z_{0}}$ and the original variables can be written in terms of the Riemann theta function associated to the curve $y^{2}=f(x)$ .

3.2. A.C.I of $a_{4}^{\left(2\right)}$ Toda lattice

In this section, we recall, according to [3], some results relating the two-dimensional $a_{4}^{\left(2\right)}$ Toda lattice. It is well known that this system is a.c.i.

The Toda lattice, introduced by Morikazu Toda in $1967$ [6], is a simple model for a one-dimensional crystal in solid-state physics. It is famous because it is one of the first examples of a completely integrable nonlinear system. It is described by a chain of particles with nearest-neighbor interaction, and its dynamics are governed by the Hamiltonian

H\left(p,q\right)=\displaystyle\sum_{n\in\mathbb{Z}}\left(\frac{p^{2}\left(n,t\right)}{2}+V\left(q\left(n+1,t\right)-q\left(n,t\right)\right)\right),

and the equations of motion

\left\{\begin{array}[]{ll}\frac{d}{dt}p\left(n,t\right)=-\frac{\partial H\left(p,q\right)}{\partial q\left(n,t\right)}=e^{-\left(q\left(n,t\right)-q\left(n-1,t\right)\right)}-e^{-\left(q\left(n+1,t\right)-q\left(n,t\right)\right)}\\ \frac{d}{dt}q\left(n,t\right)=\frac{\partial H\left(p,q\right)}{\partial p\left(n,t\right)}=p\left(n,t\right)\end{array}\right.

where $q\left(n,t\right)$ is the displacement of the $n$ -th particle from its equilibrium position, and $p\left(n,t\right)$ is its momentum (with mass $m=1$ ), and the Toda potential is given by $V\left(r\right)=e^{-r}+r-1$ . The classical Toda lattice is a system of particles with unit mass, connected by exponential springs. Its equations of motion derived from the Hamiltonian.

(3.2)

H=\frac{1}{2}\sum_{j=1}^{n}p_{j}^{2}+\sum_{j=1}^{n-1}e^{q_{j}-q_{j+1}}.

where $q_{j}$ is the position of the j-th particle and $p_{j}$ is its amount of movement. This type of Hamiltonian was considered first by Morikazu Toda [6]. The equation (3.2) is known as the finite classic no periodic Toda lattice to distinguish other versions of various forms of the system. The periodic version of (3.2) is given by

H=\frac{1}{2}\sum_{j=1}^{n}p_{j}^{2}+\sum_{j=1}^{n}e^{q_{j}-q_{j+1}},q_{n+1}=q_{1}.

where the equations of motion are given by

\dot{p}_{j}=-\frac{\partial H}{\partial q_{j}}=e^{\left(q_{j-1}-q_{j}\right)}-e^{\left(q_{j}-q_{j+1}\right)}\mbox{ and }\dot{q}_{j}=\frac{\partial H}{\partial p_{j}}=p_{j},1\leq j\leq n.

The differential equations of the periodic Toda lattice $a_{4}^{\left(2\right)}$ are given on the five dimensions hyperplane $\mathcal{H}=\{\left(x_{0},x_{1},x_{2},y_{0},y_{1},y_{2}\right)\in\mathbb{C}^{6}|y_{0}+2y_{1}+2y_{2}=0\}\mbox{ of }\mathbb{C}^{6}$ by

\left\{\begin{array}[]{l}\dot{x}=x.y\\ \dot{y}=Ax\end{array}\right.

where $x=\left(x_{0},x_{1},x_{2}\right)^{\top}$ , $y=\left(y_{0},y_{1},y_{2}\right)^{\top}$ and $A$ is the Cartan matrix of the twisted affine Lie algebra $a_{4}^{\left(2\right)}$ given in [1] by

\left(\begin{array}[]{ccc}2&-2&0\\ -1&2&-2\\ 0&-1&2\\ \end{array}\right)

and $\varepsilon=\left(1,2,2\right)^{\top}$ is the normalized null vector of $A^{\top}$ . The equations of motion of the Toda lattice $a_{4}^{\left(2\right)}$ are given in [1] by :

(3.3)

\begin{array}[]{llllll}\dot{x_{0}}=x_{0}y_{0}&&&&&\dot{y_{0}}=2x_{0}-2x_{1}\\ \dot{x_{1}}=x_{1}y_{1}&&&&&\dot{y_{1}}=-x_{0}+2x_{1}-2x_{2}\\ \dot{x_{2}}=x_{2}y_{2}&&&&&\dot{y_{2}}=-x_{1}+2x_{2}\end{array}

We denote by $\mathcal{V}_{1}$ the vector field defined by the above differential equations (3.3). Then $\mathcal{V}_{1}$ is the Hamiltonian vector field, with Hamiltonian function $F_{2}=y_{0}^{2}+4y_{2}^{2}-4x_{0}-8x_{1}-16x_{2}$
with respect to the Poisson structure $\{\cdot,\cdot\}$ defined by the following skew-symmetric matrix

(3.4)

J=\frac{1}{8}\left(\begin{array}[]{cccccc}0&0&0&4x_{0}&-2x_{0}&0\\ 0&0&0&-2x_{1}&2x_{1}&-x_{1}\\ 0&0&0&0&-x_{2}&x_{2}\\ -4x_{0}&2x_{1}&0&0&0&0\\ 2x_{0}&-2x_{1}&x_{2}&0&0&0\\ 0&x_{1}&-x_{2}&0&0&0\end{array}\right)

This Poisson structure is given on $\mathbb{C}^{6}$ ; the function $F_{0}=y_{0}+2y_{1}+2y_{2}$ is a Casimir, so that the hyperplane $\mathcal{H}$ is a Poisson subvariety. The rank of this Poisson structure $\{\cdot,\cdot\}$ is $0$ on the three-dimensional subspace $\{x_{0}=x_{1}=x_{2}=0\}$ ; the rank is $2$ on the three four-dimensional subspaces: $\{x_{0}=x_{1}=0\}$ , $\{x_{0}=x_{2}=0\}$ and $\{x_{1}=x_{2}=0\}$ . Thus, for all points of $\mathcal{H}$ except the four subspaces above the rank is $4$ . The vector field $\mathcal{V}_{1}$ admits also the following two constants of motion:

(3.5)

\begin{array}[]{l}F_{1}=x_{0}x_{1}^{2}x_{2}^{2}\\ F_{2}=y_{0}^{2}+4y_{2}^{2}-4x_{0}-8x_{1}-16x_{2}\\ F_{3}=\left(y_{0}^{2}-4x_{0}\right)\left(y_{2}^{2}-4x_{2}\right)-4x_{1}\left(y_{0}y_{2}-4x_{2}-x_{1}\right)\end{array}

$F_{1}$ is a Casimir for $\{\cdot,\cdot\}$ , and the function $F_{3}$ generates a second Hamiltonian vector field $\mathcal{V}_{2}$ , which commutes with $\mathcal{V}_{1}$ , given by the differential equations

(3.6)

\begin{array}[]{l}x_{0}^{{}^{\prime}}=x_{0}y_{2}\left(y_{0}y_{2}-2x_{1}\right)-4x_{0}x_{2}y_{0}\\ x_{1}^{{}^{\prime}}=-x_{1}y_{1}y_{2}\left(y_{1}+y_{2}\right)-x_{1}^{2}y_{1}+x_{1}\left(x_{0}y_{2}+2x_{2}y_{0}\right)\\ x_{2}^{{}^{\prime}}=x_{2}\left(y_{1}+y_{2}\right)\left(\left(y_{1}+y_{2}\right)y_{2}+x_{1}\right)+x_{0}x_{2}y_{0}\\ y_{0}^{{}^{\prime}}=2\left(2x_{1}x_{2}+x_{0}y_{2}^{2}\right)+x_{1}\left(2x_{1}-y_{0}y_{2}\right)-8x_{0}x_{2}\\ y_{1}^{{}^{\prime}}=-x_{0}y_{2}^{2}+2x_{2}\left(3x_{0}-x_{1}\right)+y_{0}y_{2}\left(x_{1}+x_{2}\right)-2x_{1}^{2}+x_{2}y_{0}y_{1}\\ y_{2}^{{}^{\prime}}=x_{1}y_{2}\left(y_{1}+y_{2}\right)+x_{1}^{2}-x_{2}\left(y_{1}+y_{2}\right)-2x_{2}x_{0}\end{array}

Hence the system (3.3) is completely integrable in the Livouille sense. It can be written as a Hamiltonian vector fields

\dot{z}=J\frac{\partial H}{\partial z},z=\left(z_{1},\cdots,z_{6}\right)^{\top}=\left(x_{0},x_{1},x_{2},y_{0},y_{1},y_{2}\right)^{\top}

where $H=F_{2}.$ the Hamiltonian structure is defined by the following Poisson bracket

\{F,H\}=\left\langle\frac{\partial F}{\partial z},J\frac{\partial H}{\partial z}\right\rangle=\displaystyle\sum_{i,k=1}^{6}J_{ik}\frac{\partial F}{\partial z_{i}}\frac{\partial H}{\partial z_{k}}

where $\frac{\partial H}{\partial z}=\left(\frac{\partial H}{\partial x_{0}},\frac{\partial H}{\partial x_{1}},\frac{\partial H}{\partial x_{2}},\frac{\partial H}{\partial y_{0}},\frac{\partial H}{\partial y_{1}},\frac{\partial H}{\partial y_{2}}\right)^{\top}$ and $J$ is an antisymmetric matrix.

The vector field $\mathcal{V}_{2}$ admits the same constants of motion (3.5) and is in involution with $\mathcal{V}_{1}$ therefore $\{F_{2},F_{3}\}=0$ . The involution $\sigma$ defined on $\mathbb{C}^{6}$ by

\sigma\left(x_{0},x_{1},x_{2},y_{0},y_{1},y_{2}\right)=\left(x_{0},x_{1},x_{2},-y_{0},-y_{1},-y_{2}\right)

preserves the constants of motion $F_{1},F_{2}$ and $F_{3}$ , hence leave the fibers of the momentum map $F$ invariant. This involution can be restricts to the hyperplane $\mathcal{H}$ .

Lemma 3.1.

[3] The system of differential equation (3.3) of the vector field $\mathcal{V}_{1}$ has three distincts families of homogeneous Laurent solutions with weights depending on four $\left(dim\mathcal{H}-1\right)$ free parameters.

The set of regular values of the momentum map $\mathbf{F}$ is the Zariski open subset $\Omega$ defined by

\begin{array}[]{cl}\Omega=&\left\{c=\left(c_{1},c_{2},c_{3}\right)\in\mathbb{C}^{3}\mid c_{1}\neq 0\mbox{ and }\right.\\ &\left.256\left(3200000c_{1}^{2}+2000c_{3}^{2}c_{2}c_{1}-225c_{3}c_{2}^{3}c_{1}+c_{3}^{5}\right)+1728c_{2}^{5}c_{1}-32c_{3}^{4}c_{2}^{2}+c_{3}^{3}c_{2}^{4}\neq 0\right\}.\end{array}

At a generic point $c=\left(c_{1},c_{2},c_{3}\right)\in\mathbb{C}^{3}$ , the fiber on $c\in\Omega$ of $\mathbf{F}$ is therefore:

\mathbb{F}_{c}:=\mathbb{F}^{-1}\left(c\right)=\bigcap_{i=1}^{3}\{m\in\mathcal{H}:F_{i}(m)=c_{i}\}

Hence we have the following result which prove that Toda lattice $a_{4}^{\left(2\right)}$ is a completely integrable system in the Liouville sense.

Proposition 3.2.

[3] For $c\in\Omega$ , the fiber $\mathbb{F}_{c}$ over $c$ of the momentum $F$ is a smooth affine variety of dimension $2$ and the rank of the Poisson structure (3.4) is maximal and equal to $4$ at each point of $\mathbb{F}_{c}$ ; moreover the vector fields $\mathcal{V}_{1}$ and $\mathcal{V}_{2}$ are independent at each point of the fiber $\mathbb{F}_{c}$ .

Proposition 3.3.

[3] $\left(\mathcal{H},\{\cdot,\cdot\},\mathbf{F}\right)$ is a completely integrable system describing the Toda lattice $a_{4}^{\left(2\right)}$ where $\mathbf{F}=\left(F_{1},F_{2},F_{3}\right)$ and $\{\cdot,\cdot\}$ are given respectively by (3.5) and (3.4) with commuting vector fields (3.3) and (3.6).

The algebraic complete integrability of the $a_{4}^{\left(2\right)}$ Toda lattice was established in [3] by the following theorem

Theorem 3.4.

[3] Let $\left(\mathcal{H},\{\cdot,\cdot\},\mathbf{F}\right)$ be an integrable system describing the Toda lattice $a_{4}^{\left(2\right)}$ where $\mathbf{F}=\left(F_{1},F_{2},F_{3}\right)$ and $\{\cdot,\cdot\}$ are given respectively by (3.5) and (3.4) with commuting vector fields (3.3).

i)

$\left(\mathcal{H},\{\cdot,\cdot\},\mathbf{F}\right)$ is a weight homogeneous algebraical completely integrable system.

ii)

For $c\in\Omega$ , the fiber $\mathbb{F}_{c}$ of its momentum map is completed in an abelian surface $\mathbb{T}^{2}_{c}$ (the Jacobian of the hyperelliptic curve (of genus two) $\overline{\Gamma_{c}}^{\left(2\right)}$ ) by the addition of a singular divisor $\mathcal{D}_{c}$ composed of three irreducible components: $\mathcal{D}_{c}^{\left(0\right)}$ defined by:

\begin{array}[]{ll}\Gamma_{c}^{\left(0\right)}:&16d^{2}a^{8}-\left(256d^{3}+8d^{2}c_{2}\right)a^{6}+\left(1536d^{2}+96dc_{2}+8c_{3}+c_{2}^{2}\right)d^{2}a^{4}-\left(\left(8\left(8c_{3}+48dc_{2}+c_{2}^{2}\right.\right.\right.\\ &\left.\left.\left.+512d^{2}\right)d+2c_{2}c_{3}\right)d^{2}+64c_{1}\right)a^{2}+\left(8d\left(c_{2}c_{3}+16dc_{3}+64d^{2}c_{2}+512d^{3}+2dc_{2}^{2}\right)+c_{3}^{2}\right)d^{2}=0\end{array}

and $\mathcal{D}_{c}^{\left(1\right)}$ defined by:

\Gamma_{c}^{\left(1\right)}:256ad^{3}-\left(\left(4a^{2}-c_{2}\right)^{2}-16c_{3}\right)d^{2}-+64c_{1}=0,

two singular curves of respective genus $3$ and $4$ and one smooth curve and $\mathcal{D}_{c}^{\left(2\right)}$ defined by

\Gamma_{c}^{\left(2\right)}:e^{4}a^{4}-\left(8c_{1}+c_{2}e^{2}\right)a^{2}e^{2}-64e^{5}+4e^{2}c_{1}c_{2}+4c_{3}e^{4}+16c_{1}^{2}=0.

of genus $2$ and isomorphic to $\overline{\Gamma_{c}^{\left(2\right)}}$ . The curves intercept each other as indicated in figure:

Figure: Curves completing the invariant surfaces $\mathbb{F}_{c}$ of the Toda lattice $a_{4}^{\left(2\right)}$ in abelian surfaces where $\mathcal{D}_{i}$ is the curve $\mathcal{D}_{c}^{\left(i\right)}$ .

4. Linearization and Lax pairs of the $a_{4}^{\left(2\right)}$ Toda lattice

The involution $(-1)$ on the abelian surface give a singular surface, his Kummer surface. Here we give an equation of Kummer surface lie with the Jacobi surface $\mathcal{T}_{c}^{2}=Jac(\overline{\Gamma}_{c})$ where $\overline{\Gamma}_{c}$ is a hyperelliptic Riemann surface of genus $2$ define above. The surface $\mathcal{T}_{c}^{2}$ is an abelian principal polarisation and the section of the line bundle $\left[2\mathcal{D}_{c}^{(2)}\right]$ embed his Kummer surface in the projectif space $\mathcal{P}^{6}$

Consider the functions which have a double pole on one of component of divisor $\mathcal{D}_{c}$ , namely $\mathcal{D}_{c}^{(2)}$ and no pole on the other.

Now, we find a basis function on $\mathcal{H}$ which has a double pole in $t$ when we substitute the principal balance $x(t;m_{2})$ and no poles when the other principal balances are substituted. Using $x(t,m_{0})$ , $x(t,m_{1})$ and $x(t,m_{2})$ give in [3], we obtain a basis of these functions constituate by the functions $\theta_{i}$ give in the following table:

k	$\dim\mathcal{F}^{k}$	$\dim\mathcal{H}^{k}$	$\dim\mathcal{Z}^{k}_{\rho}$	$\sharp dep$	$\zeta^{k}$	indep. functions
0	1	1	1	0	1	$\theta_{0}$
1	2	0	0	0	0	-
2	6	1	2	1	1	$\theta_{1}$
3	10	0	0	0	0	-
4	20	2	4	3	1	$\theta_{2}$
5	30	0	0	0	0	-
6	50	2	6	5	1	$\theta_{3}$
7	70	0	0	0	0	-
8	105	3	8	8	0	-

(4.1)

\begin{array}[]{cl}\theta_{0}=&1\\ \theta_{1}=&x_{2}\\ \theta_{2}=&x_{1}x_{2}+4x_{2}^{2}-y_{2}^{2}x_{2}\\ \theta_{3}=&x_{1}x_{2}^{2}\end{array}

The four functions $\theta_{i}$ are the line bundle section $\left[2\mathcal{D}_{c}^{(2)}\right]$ .
Hence we can formulate the following result:

Proposition 4.1.

The Koidara map which correspond to these functions:

\begin{array}[]{cccc}\psi_{c}:&Jac(\overline{\Gamma}_{c})&\longrightarrow&\mathcal{P}^{3}\\ &m=(x_{0},x_{1},x_{2},y_{0},y_{2})&\longmapsto&(\theta_{0}(m):\theta_{1}(m):\theta_{2}(m):\theta_{3}(m)),\end{array}

applied the Jacobi surface $\mathcal{T}_{c}^{2}=Jac(\overline{\Gamma}_{c})$ on his Kummer surface, which is a singular quartic in the projective space $\mathcal{P}^{3}$ . The basis $(\theta_{0}:\theta_{1}:\theta_{2}:\theta_{3})$ is taking convenably.

Proof.

By substitute the balance $x(t,m_{2})$ in the $\theta_{i}$ , $i=0,...,3$ functions and taking the coefficients of $t^{-2}$ of Laurent series $\theta_{i}(t,m_{2})$ , the map $\psi_{c}$ induce on $\Gamma_{c}$ a map

\begin{array}[]{cccc}\psi_{c}^{(2)}:&(a,e)&\longmapsto&\left(0:1:\frac{1}{4e^{2}}\left(a^{2}e^{2}-c_{2}e^{2}-4c_{1}\right):e\right)\end{array}.

Consider a Weierstrass point on $\overline{\Gamma}_{c}$ $\infty:a=\varsigma^{-1}\mbox{ }\mbox{ , }e=\frac{1}{64}\left(\varsigma^{-4}-c_{2}\varsigma^{-2}+4c_{3}+O\left(\varsigma^{6}\right)\right)$ . we obtain

(4.4)

\displaystyle\begin{array}[]{ccl}\psi_{c}^{(2)}(\infty)&=&\lim\limits_{\varsigma\rightarrow 0}\left(0:64\varsigma^{4}:16\varsigma^{2}-16c_{2}\varsigma^{4}+O(\varsigma^{6}):1-c_{2}\varsigma^{2}+O(\varsigma^{4})\right)\\ &=&\left(0:0:0:1\right)\end{array}

hence a basis $(\theta_{0}:\theta_{1}:\theta_{2}:\theta_{3})$ is take convenably. ∎

Consider the constants of motion

(4.5)

\begin{array}[]{cl}F_{1}=&x_{0}x_{1}^{2}x_{2}^{2}=c_{1}\\ F_{2}=&y_{0}^{2}+4y_{2}^{2}-4x_{0}-8x_{1}-16x_{2}=c_{2}\\ F_{3}=&(y_{0}^{2}-4x_{0})(y_{2}^{2}-4x_{2})-4x_{1}(y_{0}y_{2}-4x_{2}-x_{1})=c_{3}\end{array}

and eliminating the variables $(x_{0},x_{1},x_{2},y_{0},y_{2})$ in the principals balances $x(t,m_{0})$ , $x(t,m_{1})$ and $x(t,m_{2})$ in [3] we obtain:

(4.6)

x_{0}=\frac{c_{1}\theta_{1}^{2}}{\theta_{3}^{2}}\mbox{ }\mbox{ , }\mbox{ }x_{1}=\frac{\theta_{3}}{\theta_{1}^{2}}\mbox{ }\mbox{ , }\mbox{ }x_{2}=\theta_{1}

Using the second equations of (4.1) and (4.5), we obtain

(4.7)

y_{0}^{2}=\frac{1}{\theta_{1}^{2}\theta_{3}^{2}}\left(4c_{1}\theta_{1}^{4}+4\theta_{3}^{3}+\theta_{1}\theta_{3}^{2}\left(c_{2}\theta_{1}+4\theta_{2}\right)\right)\mbox{ }\mbox{ }\mbox{ }\mbox{ , }\mbox{ }\mbox{ }y_{2}^{2}=\frac{1}{\theta_{1}^{2}}\left(4\theta_{1}^{3}-\theta_{1}\theta_{2}+\theta_{3}\right)

Rewriting the last equation of (4.5) on the follow form

4x_{1}y_{0}y_{2}=\left((y_{0}^{2}-4x_{0})(y_{2}^{2}-4x_{2})-c_{3}\right)+4x_{1}\left(4x_{2}+x_{1}\right)

we obtain a Kummer surface of Jac( $\overline{\Gamma}_{c}$ ). It can be put in the follow form

(4.8)

\left(\left(c_{2}+16\theta_{1}\right)^{2}-16\left(16\theta_{2}+4c_{2}\theta_{1}+c_{3}\right)\right)\theta_{3}^{2}+2\theta_{3}f_{3}\left(\theta_{1},\theta_{2}\right)+f_{4}\left(\theta_{1},\theta_{2}\right)=0

where $f_{3}$ is a polynomial of degre $3$ , $f_{4}$ of degre $4$ in $\theta_{1}$ and $\theta_{2}$ given by

	$\displaystyle f_{3}\left(\theta_{1},\theta_{2}\right)$	$\displaystyle=$	$\displaystyle-\left(c_{2}+16\theta_{1}\right)\left(\theta_{2}\left(\theta_{1}c_{2}+4\theta_{2}\right)+c_{3}\theta_{1}^{2}\right)-64c_{1}$
	$\displaystyle f_{4}\left(\theta_{1},\theta_{2}\right)$	$\displaystyle=$	$\displaystyle\left(c_{3}\theta_{1}^{2}+4\theta_{2}^{2}\right)^{2}-\theta_{1}\left(-2\theta_{1}^{2}c_{2}\theta_{2}c_{3}+256c_{1}\theta_{1}^{2}-\theta_{1}c_{2}^{2}\theta_{2}^{2}-64c_{1}\theta_{2}-8\theta_{2}^{3}c_{2}\right)$

Hence we have the following results:

Proposition 4.2.

A quartic equation of the Kummer surface of Jac( $\overline{\Gamma}_{c}$ ), in terms of $\theta_{i}$ is given by

\left(\left(c_{2}+16\theta_{1}\right)^{2}-16\left(16\theta_{2}+4c_{2}\theta_{1}+c_{3}\right)\right)\theta_{3}^{2}+2\theta_{3}f_{3}\left(\theta_{1},\theta_{2}\right)+f_{4}\left(\theta_{1},\theta_{2}\right)=0

where $f_{3}$ is a polynomial of degre $3$ , $f_{4}$ of degre $4$ in $\theta_{1}$ and $\theta_{2}$ given by

	$\displaystyle f_{3}\left(\theta_{1},\theta_{2}\right)$	$\displaystyle=$	$\displaystyle-\left(c_{2}+16\theta_{1}\right)\left(\theta_{2}\left(\theta_{1}c_{2}+4\theta_{2}\right)+c_{3}\theta_{1}^{2}\right)-64c_{1}$
	$\displaystyle f_{4}\left(\theta_{1},\theta_{2}\right)$	$\displaystyle=$	$\displaystyle\left(c_{3}\theta_{1}^{2}+4\theta_{2}^{2}\right)^{2}-\theta_{1}\left(-2\theta_{1}^{2}c_{2}\theta_{2}c_{3}+256c_{1}\theta_{1}^{2}-\theta_{1}c_{2}^{2}\theta_{2}^{2}-64c_{1}\theta_{2}-8\theta_{2}^{3}c_{2}\right)$

Theorem 4.3.

The vector field $\mathcal{V}_{1}$ 3.3 extends to a linear vector field on the abelian surface $\mathbb{T}^{2}_{c}$ and the Jacobi form for the differentials equation can be written as

\left\{\begin{array}[]{l}\frac{\dot{\lambda_{1}}}{\sqrt{f(\lambda_{1})}}+\frac{\dot{\lambda_{2}}}{\sqrt{f(\lambda_{2})}}=0\\ \frac{\lambda_{1}\dot{\lambda_{1}}}{\sqrt{f(\lambda_{1})}}+\frac{\lambda_{2}\dot{\lambda_{2}}}{\sqrt{f(\lambda_{2})}}=\frac{1}{2i}dt\end{array}\right.

with $f(\lambda)=\lambda_{i}^{5}+2c_{2}\lambda_{i}^{4}+\left(8c_{3}+c_{2}^{2}\right)\lambda_{i}^{3}+8c_{2}c_{3}\lambda_{i}^{2}+16c_{3}^{2}\lambda_{i}-16384c_{1}$ and $v^{2}=f(\lambda)$ is birational equivalent to the hyperelliptic curve of genus two $\mathcal{K}_{c}$

Proof.

Consider coefficient of $\theta_{3}^{2}$ in equation (4.8) with the variables $x_{i}$ and $y_{i}$

\Delta=\left(c_{2}+16x_{2}\right)^{2}-4\left(4x_{2}\left(-16y_{2}^{2}+64x_{2}+16x_{1}+4c_{2}\right)+4c_{3}\right)

Let $u(\lambda)$ an unitary polynomial in $\lambda$ such that the discriminant is $\Delta$ , hence we have:

	$\displaystyle u(\lambda)$	$\displaystyle=$	$\displaystyle\lambda^{2}+\left(c_{2}+16x_{2}\right)\lambda+4x_{2}\left(-16y_{2}^{2}+64x_{2}+16x_{1}+4c_{2}\right)+4c_{3}$
		$\displaystyle=$	$\displaystyle\lambda^{2}+\left(y_{0}^{2}+4y_{2}^{2}-4x_{0}-8x_{1}\right)\lambda+\left(4x_{1}-2y_{0}y_{2}\right)^{2}-16x_{0}y_{2}^{2}$

Lets $\lambda_{1}$ and $\lambda_{2}$ roots of polynomial $f(\lambda)$ , we have:

(4.9)

\lambda_{1}+\lambda_{2}=-16x_{2}-c_{2}\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ , }\mbox{ }\mbox{ }\mbox{ }\lambda_{1}\lambda_{2}=4x_{2}\left(-16y_{2}^{2}+64x_{2}+16x_{1}+4c_{2}\right)+4c_{3}

that imply,with respect with $\mathcal{V}_{1}$

(4.10)

\dot{\lambda}_{1}+\dot{\lambda}_{2}=-16x_{2}y_{2}\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ , }\mbox{ }\mbox{ }\mbox{ }\dot{\lambda}_{1}\lambda_{2}+\lambda_{1}\dot{\lambda}_{2}=-16x_{2}\left(-y_{2}\left(y_{0}^{2}-4x_{0}\right)+2x_{1}y_{0}\right)

Let $v\left(\lambda\right)$ a polynom define,up to a multiplicative constante, by :

(4.13)

\displaystyle\begin{array}[]{ccc}v\left(\lambda\right)&=&32i\left[x_{2}y_{2}\lambda+x_{2}\left(y_{2}\left(y_{0}^{2}-4x_{0}\right)-2x_{1}y_{0}\right)\right]\\ &=&-2i\left(\dot{\lambda}_{1}+\dot{\lambda}_{2}\right)\lambda+2i\left(\dot{\lambda}_{1}\lambda_{2}+\lambda_{1}\dot{\lambda}_{2}\right).\end{array}

by substituation (4.9) and (4.10) in (4.5), and by eliminating variables $x_{0},x_{1},x_{2},y_{0}$ and $y_{2}$ , we obtain two quadrics polynoms in $\dot{\lambda}_{i}^{2}$ given by

\dot{\lambda}_{i}^{2}=\frac{\lambda_{i}^{5}+2c_{2}\lambda_{i}^{4}+\left(8c_{3}+c_{2}^{2}\right)\lambda_{i}^{3}+8c_{2}c_{3}\lambda_{i}^{2}+16c_{3}^{2}\lambda_{i}-16384c_{1}}{4\left(\lambda_{1}-\lambda_{2}\right)^{2}}\mbox{ , }i=1,2

verify

(4.14)

\left\{\begin{array}[]{l}\frac{\dot{\lambda_{1}}}{\sqrt{f(\lambda_{1})}}+\frac{\dot{\lambda_{2}}}{\sqrt{f(\lambda_{2})}}=0\\ \frac{\lambda_{1}\dot{\lambda_{1}}}{\sqrt{f(\lambda_{1})}}+\frac{\lambda_{2}\dot{\lambda_{2}}}{\sqrt{f(\lambda_{2})}}=\frac{1}{2i}dt\end{array}\right.

with

f(\lambda)=\lambda_{i}^{5}+2c_{2}\lambda_{i}^{4}+\left(8c_{3}+c_{2}^{2}\right)\lambda_{i}^{3}+8c_{2}c_{3}\lambda_{i}^{2}+16c_{3}^{2}\lambda_{i}-16384c_{1}

and like $v^{2}=f(\lambda)$ then:

$\displaystyle\sqrt{f\left(\lambda_{l}\right)}$	$\displaystyle=$	$\displaystyle v\left(\lambda_{l}\right)$
	$\displaystyle=$	$\displaystyle 2i\left[16x_{2}y_{2}\lambda_{l}+16ix_{2}\left(y_{2}\left(y_{0}^{2}-4x_{0}\right)-2x_{1}y_{0}\right)\right]$
	$\displaystyle=$	$\displaystyle-2i\left(\dot{\lambda}_{1}+\dot{\lambda}_{2}\right)\lambda_{l}+2i\left(\dot{\lambda}_{1}\lambda_{2}+\lambda_{1}\dot{\lambda}_{2}\right)$

hence

\left\{\begin{array}[]{l}\sqrt{f\left(\lambda_{1}\right)}=-2i\left(\dot{\lambda}_{1}+\dot{\lambda}_{2}\right)\lambda_{1}+2i\left(\dot{\lambda}_{1}\lambda_{2}+\lambda_{1}\dot{\lambda}_{2}\right)\\ \sqrt{f\left(\lambda_{2}\right)}=-2i\left(\dot{\lambda}_{1}+\dot{\lambda}_{2}\right)\lambda_{2}+2i\left(\dot{\lambda}_{1}\lambda_{2}+\lambda_{1}\dot{\lambda}_{2}\right)\end{array}\right.\Longrightarrow\left\{\begin{array}[]{l}\sqrt{f\left(\lambda_{1}\right)}=-2i\left(\lambda_{1}-\lambda_{2}\right)\dot{\lambda}_{1}\\ \sqrt{f\left(\lambda_{2}\right)}=2i\left(\lambda_{1}-\lambda_{2}\right)\dot{\lambda}_{2}\end{array}\right.

This show that the Toda is linearising on the Jacobian variety of the curve $\mathcal{K}_{c}$ . It is able to see how $\mathcal{K}_{c}$ and $v^{2}=f(s)$ are related.

\mathcal{K}_{c}:z^{2}=h(t)=t^{5}-2c_{2}t^{4}+\left(8c_{3}+c_{2}^{2}\right)t^{3}-8c_{2}c_{3}t^{2}+16c_{3}^{2}t+16384c_{1}

and

v^{2}=f(\lambda)=\lambda^{5}+2c_{2}\lambda^{4}+\left(8c_{3}+c_{2}^{2}\right)\lambda^{3}+8c_{2}c_{3}\lambda^{2}+16c_{3}^{2}\lambda-16384c_{1}

then we easy verify by taking $\lambda=-t$ that $z=iv$ .

one verifies, by a direct computation, that the expression $f(\lambda)-v^{2}(\lambda)$ is divisible by $u(\lambda)$ with

f(\lambda)=\lambda^{5}+2c_{2}\lambda^{4}+\left(8c_{3}+c_{2}^{2}\right)\lambda^{3}+8c_{2}c_{3}\lambda^{2}+16c_{3}^{2}\lambda-16384c_{1}

Hence $y^{2}=f(\lambda)$ is birational to the affine curve $\Gamma_{c}$ by adding the Weierstrass points at infinity $a=\pm\sqrt{\frac{t^{5}-2c_{2}t^{4}+\left(8c_{3}+c_{2}^{2}\right)t^{3}-8c_{2}c_{3}t^{2}+16c_{3}^{2}t+16384c_{1}}{\left(t^{2}-c_{2}t+4c_{3}\right)^{2}}},\mbox{ }\mbox{ }\mbox{ }e=\frac{1}{64}\left(t^{2}-c_{2}t+4c_{3}\right)$ . ∎

The form 4.14 is ewuivalent to

\frac{d}{dt}\left(\sum^{2}_{k=1}\int_{0_{k}}^{Q_{k}}\overrightarrow{\omega}\right)=\left(\begin{array}[]{c}0\\ 2i\\ \end{array}\right)

where $\overrightarrow{\omega}=\left(\frac{dx}{\sqrt{f(x)}},\frac{xdx}{\sqrt{f(x)}}\right)^{\top}$ is a basis for holomorphic differentials on $\overline{\Gamma}_{c}$ , $Q_{1}:=\left(\lambda_{1},\sqrt{f(\lambda_{1})}\right)$ and $Q_{2}:=\left(\lambda_{2},\sqrt{f(\lambda_{2})}\right)$ two points of $\Gamma_{c}$ and $Q_{1}+Q_{2}=\left(\lambda_{1},\sqrt{f(\lambda_{1})}\right)+\left(\lambda_{2},\sqrt{f(\lambda_{2})}\right)$ viewed as a divisor on the genus $2$ hyperelliptic curve $\Gamma_{c}$ . Thus, by integrating 4.14, we see that the flow of $\overline{\mathcal{V}}_{1}$ is linear on the Jacobian of the curve $\Gamma_{c}$ . By using [[5], Theorem 5.3], one shows that the symmetric functions $\lambda_{1}$ and $\lambda_{2}$ , and hence the original phase variables can be written in terms of theta functions.

Now we also establish a link between the $a_{4}^{\left(2\right)}$ Toda lattice and the Mumford system [5]. By using a method due to Vanhaecke [8], we construct an explicit morphism between these two systems. Thus, we obtain a new Poisson structure for the Mumford system and then derive a new Lax equation for the $a_{4}^{\left(2\right)}$ Toda lattice.

According the fact that the expression $f(\lambda)-v^{2}(\lambda)$ is divisible by $u(\lambda)$ such that the above formulas define a point of $Jac(\overline{\Gamma}^{2}_{c})\backslash\Gamma^{2}_{c}$ , there exist a polynomial $w$ in $\lambda$ of degree $3=$ deg $u+1$ . By direct calculation, we obtain:

	$\displaystyle w(\lambda)$	$\displaystyle=$	$\displaystyle\frac{f(\lambda)-v^{2}(\lambda)}{u(\lambda)}$
		$\displaystyle=$	$\displaystyle\lambda^{3}+w_{2}\lambda^{2}+w_{1}\lambda+w_{0},$

where

$\displaystyle w_{0}$	$\displaystyle=$	$\displaystyle 256y_{0}^{2}x_{2}^{2}-1024x_{0}x_{2}^{2}$
$\displaystyle w_{1}$	$\displaystyle=$	$\displaystyle 16x_{1}^{2}+4y_{0}^{2}y_{2}^{2}-32y_{0}^{2}x_{2}-16x_{0}y_{2}^{2}+128x_{0}x_{2}+256x_{2}^{2}+128x_{2}x_{1}-16x_{1}y_{0}y_{2}$
$\displaystyle w_{2}$	$\displaystyle=$	$\displaystyle y_{0}^{2}-8x_{1}-32x_{2}-4x_{0}+4y_{2}^{2}.$

The linearizing variables 4.9 and 4.10 suggest a morphism $\varphi$ from the $a^{(2)}_{4}$ Toda lattice to genus $2$ odd Mumford system:

\left\{\left(\begin{array}[]{cc}v(\lambda)&u(\lambda)\\ w(\lambda)&-v(\lambda)\\ \end{array}\right)\in M_{2}(\mathbb{C}[\lambda])\mbox{ such that }\begin{array}[]{c}deg(u)=2=deg(w)-1\\ deg(v)<2;\mbox{u,w are monic }\end{array}\right\}\cong\mathbb{C}^{7},

where $\mathbb{C}^{7}$ is a phase space of Mumford system. The morphism $\varphi$ is given by:

(4.15)

\begin{array}[]{ccc}(x_{0},x_{1},x_{2},y_{0},y_{2})&\longmapsto&\left\{\begin{array}[]{ll}u(\lambda)=&\lambda^{2}+u_{1}\lambda+u_{0}\\ v(\lambda)=&v_{1}\lambda+v_{0}\\ w(\lambda)=&\lambda^{3}+w_{2}\lambda^{2}+w_{1}\lambda+w_{0}\end{array}\right.\end{array}

with

\begin{array}[]{llllll}u_{0}=&\left(4x_{1}-2y_{0}y_{2}\right)^{2}-16x_{0}y_{2}^{2}&&&v_{0}=&16x_{2}\left(y_{2}\left(y_{0}^{2}-4x_{0}\right)-2x_{1}y_{0}\right)\\ u_{1}=&-\left(y_{0}^{2}+4y_{2}^{2}-4x_{0}-8x_{1}\right)&&&v_{1}=&16x_{2}y_{2}\end{array}

\begin{array}[]{ll}w_{0}=&256y_{0}^{2}x_{2}^{2}-1024x_{0}x_{2}^{2}\\ w_{1}=&16x_{1}^{2}+4y_{0}^{2}y_{2}^{2}-32y_{0}^{2}x_{2}-16x_{0}y_{2}^{2}+128x_{0}x_{2}+256x_{2}^{2}+128x_{2}x_{1}-16x_{1}y_{0}y_{2}\\ w_{2}=&y_{0}^{2}-8x_{1}-32x_{2}-4x_{0}+4y_{2}^{2}\end{array}

Theorem 4.4.

A Lax representation of the vector field $\mathcal{V}_{1}=\mathcal{X}_{F_{1}}$ is given by:

\dot{X}=\left[X(\lambda),Y(\lambda)\right]

by taking

X(\lambda)=\left(\begin{array}[]{cc}v(\lambda)&u(\lambda)\\ w(\lambda)&-v(\lambda)\\ \end{array}\right)\mbox{ and }Y(\lambda)=\left(\begin{array}[]{cc}0&1\\ b(\lambda)&0\\ \end{array}\right)

where $u(\lambda),v(\lambda)$ and $w(\lambda)$ are the polynomials defined above. The coefficient $b(\lambda)=\lambda-32x_{2}$ of the matrix $Y(\lambda)$ is the polynomial part of the rational function $w(\lambda)/u(\lambda)$ .

Acknowledgements

We would like to extend our sincere gratitude to Professor Pol Vanhaecke at University of Poitiers for his particular contributions in providing clarifications and guidance on our research theme, for the enriching exchanges and thoughtful advice he generously offered us throughout this project.

References

[1] Adler, M., Moerbeke,P.V and Vanhaecke, P., Intégrabilité algébrique, géométrie Painlevé et algèbres de Lie, Ergeb. Math. Grenzgeb. 47 (3) (2004) Berlin- Heidelberg : Springer.
[2] Herbert,L. and Birkenhake, C.,Abelian Varieties over the Complex Numbers,A Graduate Course, Grundlehren Text Editions, Springer,https://doi.org/10.1007/978-3-031-25570-0.
[3] Lietap, N.,Dehainsala, D. and Dongho, J., Algebraic complete integrability of the $a_{4}^{(2)}$ Toda lattice, University of maroua,SIGMA 20 (2024), 087, 26 pages, https://doi.org/10.3842/SIGMA.2024.087.
[4] Piovan, L., Algebraically completely integrable systems and Kummer varieties, Brandeis University, Math. Ann. 290, pp.349-403 (1991)
[5] Mumford, D.: Tata Lectures on Theta: 2. Jacobian Theta Functions and Differential Equations, Progr. Math., vol. 43, Boston, MA: Birkhäuser, 1984.
[6] Toda, M., One-Dimensional Dual Transformation, J. Phys. Soc. Japan, 1965, vol. 20, no. 11, pp. 2095A (see also Progr. Theoret. Phys. Suppl., 1966, no. 36, pp. 113-119).
[7] Vanhaecke, P., Linearising two-dimensional integrable systems and the construction of action-angle variables. Math. Z., 1992, vol. 211, no. 2, pp. 265-313.
[8] Vanhaecke, P.,Integrable Systems in the Realm of Algebraic Geometry, 2nd ed., Lecture Notes in Math., vol.1638, Berlin: Springer, 2001.

Linearization, separability and Lax pairs representation of a4(2)a_{4}^{\left(2\right)} Toda lattice

Abstract.

Key words and phrases:

1991 Mathematics Subject Classification:

1. Introduction

2. Preliminaries

Definition 2.1.

Definition 2.2.

Definition 2.3.

Definition 2.4.

Definition 2.5.

Definition 2.6.

Definition 2.7.

Definition 2.8.

Definition 2.9.

3. Separability and linearization of two-dimensional Toda lattice a4(2)a_{4}^{(2)}

3.1. Linearization procedure

3.2. A.C.I of a4(2)a_{4}^{\left(2\right)} Toda lattice

Lemma 3.1.

Proposition 3.2.

Proposition 3.3.

Theorem 3.4.

4. Linearization and Lax pairs of the a4(2)a_{4}^{\left(2\right)} Toda lattice

Proposition 4.1.

Proof.

Proposition 4.2.

Theorem 4.3.

Proof.

Theorem 4.4.

Acknowledgements

References

Linearization, separability and Lax pairs representation of $a_{4}^{\left(2\right)}$ Toda lattice

3. Separability and linearization of two-dimensional Toda lattice $a_{4}^{(2)}$

3.2. A.C.I of $a_{4}^{\left(2\right)}$ Toda lattice

4. Linearization and Lax pairs of the $a_{4}^{\left(2\right)}$ Toda lattice