Elliptic solutions of the Toda lattice with constraint of type B and deformed Ruijsenaars-Schneider system

TL;DR

This paper investigates elliptic solutions of a constrained Toda lattice, linking their pole dynamics to a deformed Ruijsenaars-Schneider system, and proposes an extension to a field theory.

Contribution

It derives equations of motion for elliptic solutions of a constrained Toda lattice and connects them to a deformed Ruijsenaars-Schneider system, including a new spectral curve analysis.

Findings

Pole dynamics described by deformed Ruijsenaars-Schneider system

Established a Manakov triple representation for the system

Proposed an extension to a field theory for elliptic solutions

Abstract

We study elliptic solutions of the recently introduced Toda lattice with the constraint of type B and derive equations of motion for their poles. The dynamics of poles is given by the deformed Ruijsenaars-Schneider system. We find its commutation representation in the form of the Manakov triple and study properties of the spectral curve. By studying more general elliptic solutions (elliptic families), we also suggest an extension of the deformed Ruijsenaars-Schneider system to a field theory.

Equations

$$\left\{\begin{array}[]{l}\displaystyle{\partial_{t_{1}}\log\Bigl{(}v(x)v(x+\eta)\Bigr{)}=\frac{f(x+\eta)}{v(x+\eta)}}-\frac{f(x)}{v(x)},\\ \\ \partial_{t_{2}}v(x)-\partial_{t_{1}}f(x)=2v^{2}(x)\Bigl{(}v(x-\eta)-v(x+\eta)\Bigr{)}.\end{array}\right.$$

$$v(x)=\frac{\tau (x+\eta )\tau (x-\eta )}{{\tau }^2(x)},f(x)=v(x){\partial }_{t_1}\log \frac{\tau (x+\eta )}{\tau (x-\eta )}.$$

$${\tau }^{\mathrm{Toda}}(x)=\tau (x)\tau (x-\eta ).$$

$$\ddot{x}_{i}+\sum_{j\neq i}^{N}\dot{x}_{i}\dot{x}_{j}\Bigl{(}\zeta(x_{ij}+\eta)+\zeta(x_{ij}-\eta)-2\zeta(x_{ij})\Bigr{)}+g(U_{i}^{-}-U_{i}^{+})=0,$$

$$U_i^{\pm }=\prod _{j\ne i}\frac{\sigma (x_{ij}\pm 2\eta )\sigma (x_{ij}\mp \eta )}{\sigma (x_{ij}\pm \eta )\sigma (x_{ij})},x_{ij}=x_i-x_j$$

$$\partial_{t}\psi(x)=v(x)\Bigl{(}\psi(x+\eta)-\psi(x-\eta)\Bigr{)},$$

$$\Psi (x)=\frac{\tau (x+\eta )}{\tau (x)}\psi (x+\eta ).$$

$${\partial }_t\Psi (x-\eta )=\Psi (x)+b(x)\Psi (x-\eta )-u^{-}(x)\Psi (x-2\eta ),$$

$$b(x)={\partial }_t\log \frac{\tau (x)}{\tau (x-\eta )},u^{-}(x)=\frac{\tau (x-2\eta )\tau (x+\eta )}{\tau (x-\eta )\tau (x)}.$$

$$\tau (x)=C\prod _{j=1}^N\sigma (x-x_j).$$

$$b(x)=\sum_{j}\dot{x}_{j}\Bigl{(}\zeta(x-x_{j}-\eta)-\zeta(x-x_{j})\Bigr{)},$$

$$u^{-}(x)=\prod _j\frac{\sigma (x-x_j-2\eta )\sigma (x-x_j+\eta )}{\sigma (x-x_j-\eta )\sigma (x-x_j)}$$

$$\Phi (x,\lambda )=\frac{\sigma (x+\lambda )}{\sigma (\lambda )\sigma (x)}{e}^{-\zeta (\lambda )x}.$$

$$\Psi (x)=k^{x/\eta }\sum _{i=1}^N{c}_i\Phi (x-x_i,\lambda ),$$

$$b=k^{2\omega /\eta }{e}^{2(\zeta (\omega )\lambda -\zeta (\lambda )\omega )},b^{\prime }=k^{2{\omega }^{\prime }/\eta }{e}^{2(\zeta ({\omega }^{\prime })\lambda -\zeta (\lambda ){\omega }^{\prime })}.$$

$$\begin{array}[]{l}\displaystyle{k^{-1}\sum_{i}\dot{c}_{i}\Phi(x\!-\!x_{i}\!-\!\eta)-k^{-1}\sum_{i}c_{i}\dot{x}_{i}\Phi^{\prime}(x\!-\!x_{i}\!-\!\eta)}\\ \\ \displaystyle{\phantom{aaaaa}=\,\sum_{i}c_{i}\Phi(x-x_{i})+k^{-1}\sum_{j}\dot{x}_{j}\Bigl{(}\zeta(x\!-\!x_{j}\!-\!\eta)-\zeta(x\!-\!x_{j})\Bigr{)}\sum_{i}c_{i}\Phi(x\!-\!x_{i}\!-\!\eta)}\\ \\ \displaystyle{\phantom{aaaaaaaaaaaaaaa}-k^{-2}\prod_{j}\frac{\sigma(x-x_{j}-2\eta)\sigma(x-x_{j}+\eta)}{\sigma(x-x_{j}-\eta)\sigma(x-x_{j})}\sum_{i}c_{i}\Phi(x\!-\!x_{i}\!-\!2\eta),}\end{array}$$

$$c_i-k^{-1}{\dot{x}}_i\sum _j{c}_j\Phi (x_{ij}-\eta )-k^{-2}\sigma (2\eta )U_i^{-}\sum _j{c}_j\Phi (x_{ij}-2\eta )=0,$$

$$U_i^{-}=\prod _{j\ne i}\frac{\sigma (x_{ij}-2\eta )\sigma (x_{ij}+\eta )}{\sigma (x_{ij}-\eta )\sigma (x_{ij})}.$$

$$L_{ij}(k,\lambda )={\dot{x}}_i\Phi (x_{ij}-\eta ,\lambda )+k^{-1}\sigma (2\eta )U_i^{-}\Phi (x_{ij}-2\eta ,\lambda )$$

$$L(k,\lambda )\mathbf{c}=k\mathbf{c}$$

$$\det\Bigl{(}kI-L(k,\lambda\Bigr{)}=0$$

$${\dot{c}}_i=\sum _j{M}_{ij}{c}_j\text{mbox}or\dot{\mathbf{c}}=M\mathbf{c},$$

$$\begin{array}[]{l}\displaystyle{M_{ij}(k,\lambda)=\dot{x}_{i}(1-\delta_{ij})\Phi(x_{ij},\lambda)+k^{-1}\sigma(2\eta)U_{i}^{+}\Phi(x_{ij}-\eta,\lambda)}\\ \\ \phantom{aaaaaaaaaaaaa}-\displaystyle{\delta_{ij}\Bigl{(}\sum_{k}\dot{x}_{k}\zeta(x_{ik}+\eta)-\sum_{k\neq i}\zeta(x_{ik})\Bigr{)}}.\end{array}$$

$$\Bigl{(}\dot{L}+[L,M]\Bigr{)}{\bf c}=0.$$

$$A_{ij}^0=(1-{\delta }_{ij})\Phi (x_{ij}),A_{ij}=\Phi (x_{ij}-\eta ),B_{ij}=\Phi (x_{ij}-2\eta )$$

$$\begin{array}[]{l}\dot{X}_{ij}=\delta_{ij}\dot{x}_{i},\quad U^{\pm}_{ij}=\delta_{ij}U_{i}^{\pm},\\ \\ \displaystyle{Z_{ij}^{\pm}=\delta_{ij}\Bigl{(}\sum_{k}\dot{x}_{k}\zeta(x_{ik}\pm\eta)-\sum_{k\neq i}\dot{x}_{k}\zeta(x_{ik})\Bigr{)}},\\ \\ \displaystyle{D_{ij}^{-}=\delta_{ij}\sum_{k\neq i}\Bigl{(}\zeta(x_{ik}-2\eta)+\zeta(x_{ik}+\eta)-\zeta(x_{ik}-\eta)-\zeta(x_{ik})\Bigr{)}},\\ \\ \displaystyle{S_{ij}^{-}=\delta_{ij}\sum_{k\neq i}\dot{x}_{k}\Bigl{(}\zeta(x_{ik}-2\eta)+\zeta(x_{ik}+\eta)-\zeta(x_{ik}-\eta)-\zeta(x_{ik})\Bigr{)}.}\end{array}$$

$$\begin{array}[]{l}L=\dot{X}A+gk^{-1}U^{-}B,\\ \\ M=\dot{X}A^{0}+gk^{-1}U^{+}A-Z^{+},\end{array}$$

$$\begin{array}[]{l}\dot{L}+[L,M]=\Bigl{(}\ddot{X}+(Z^{+}\!+\!Z^{-})\dot{X}+g(U^{-}\!-\!U^{+})\Bigr{)}A+gk^{-1}(U^{-}\!-\!U^{+})A(L-kI)\\ \\ \phantom{aaaaaaaaaaaaaaaaaaaaaa}+\,W_{0}+gk^{-1}W_{1}+g^{2}k^{-2}W_{2},\end{array}$$

$$\begin{array}[]{l}W_{0}=\dot{X}^{2}A^{\prime}-\dot{X}A^{\prime}\dot{X}+\dot{X}A\dot{X}A^{0}-\dot{X}A^{0}\dot{X}A-\dot{X}AZ^{+}-\dot{X}Z^{-}A,\\ \\ \begin{array}[]{l}\!\!W_{1}=U^{-}\dot{X}DB-U^{-}(S^{-}\!-\!Z^{+})B+U^{-}\dot{X}B^{\prime}-U^{-}B^{\prime}\dot{X}+\dot{X}AU^{+}A\\ \\ \phantom{aaaaaaaaaaaaa}+U^{-}B\dot{X}A^{0}-\dot{X}A^{0}U^{-}B-U^{-}BZ^{+}-U^{-}A\dot{X}A,\end{array}\\ \\ W_{2}=U^{-}BU^{+}A-U^{+}AU^{-}B-(U^{-}-U^{+})AU^{-}B.\end{array}$$

$$\begin{array}[]{l}\Bigl{(}U^{-}BU^{+}A-U^{+}AU^{-}B\Bigr{)}_{ij}\\ \\ \begin{array}[]{l}\displaystyle{=\,\Phi(x_{ij}-3\eta)\sum_{k}\Bigl{[}U_{i}^{-}U_{k}^{+}\Bigl{(}\zeta(x_{ik}-2\eta)+\zeta(x_{kj}-\eta)-\zeta(x_{ij}-3\eta+\lambda)+\zeta(\lambda)\Bigr{)}}\\ \\ \phantom{aaaaaaaaa}-U_{i}^{+}U_{k}^{-}\Bigl{(}\zeta(x_{ik}-\eta)+\zeta(x_{kj}-2\eta)-\zeta(x_{ij}-3\eta+\lambda)+\zeta(\lambda)\Bigr{)}\Bigr{]}.\end{array}\end{array}$$

Full text

1. 1 Introduction
2. 2 The pole ansatz
3. 3 The spectral curve
4. 4 The $\Psi$-function as the Baker-Akhiezer function on the spectral curve
5. 5 Elliptic families and field generalization of the deformed RS system
6. 6 Open problems
7. Appendix A: The Weierstrass and Krichever functions
8. Appendix B: Proof of the matrix identity
9. Acknowledgments
10. References

1 Introduction

The investigation of dynamics of poles of singular solutions to nonlinear integrable equations was initiated in the seminal paper [2], where elliptic and rational solutions to the Korteweg-de Vries and Boussinesq equations were studied. These studies were continued in [3, 4, 5], where it was shown that poles of elliptic solutions to the Kadomtsev-Petviashvili (KP) equation move as particles of the integrable Calogero-Moser (CM) many-body system [6]-[9].

The method suggested by Krichever [5] for elliptic solutions of the KP equation consists in substituting the solution not in the KP equation itself but in the auxiliary linear problem for it (this implies a suitable pole ansatz for the wave function). This method allows one to obtain the equations of motion together with their Lax representation. Later, the power of this method was demonstrated in other important examples. In particular, dynamics of poles of elliptic solutions to the 2D Toda lattice [10] was studied by this method in [11], see also [12]. It was proved that the poles move as particles of the integrable Ruijsenaars-Schneider (RS) many-body system [13, 14] which is a relativistic generalization of the CM system. Another important example is related to the B-version of the KP equation (BKP) [15]-[18]. Elliptic solutions to the BKP equation were studied by Krichever’s method in [19], where equations of motion for the poles were obtained, together with their commutation representation in the form of the Manakov’s triple [20].

Recently, a new integrable hierarchy of the Toda type was suggested [21]. It is the Toda hierarchy with the constraint of type B. The first member of the hierarchy is the following system of equations for two unknown functions $v$, $f$ depending on a space variable $x$ and two time variables $t_{1}$, $t_{2}$:

[TABLE]

Here $\eta$ is a parameter having the meaning of the lattice spacing in the $x$-direction. Let us note that a similar hierarchy was suggested earlier in the paper [22] as an integrable discretization of the Novikov-Veselov equation but the close connection with the Toda lattice was not mentioned there.

Solutions to the Toda lattice with the constraint of type B can be expressed in terms of the tau-function $\tau(x)$ as follows:

[TABLE]

As is shown in [21], the tau-function $\tau(x)$ of the Toda lattice with the constraint of type B is related to the tau-function $\tau^{\rm Toda}(x)$ of the 2D Toda lattice as

[TABLE]

This relation should be compared with the relation between tau-functions of the KP and BKP hierarchies: the former is the square of the latter. In the Toda case, it is not the square but the arguments of the two factors are shifted by $\eta$; in the limit $\eta\to 0$ they become the same.

An interesting problem is to find dynamics of poles of elliptic solutions to the Toda equation of type B. As in the Toda case, one considers solutions which are elliptic functions of $x$, with poles $x_{i}$ depending on the time $t=t_{1}$. This problem was recently addressed in [23], where equations of motion for the poles were obtained from the condition that the first auxiliary linear problem for the equations (1.1) has meromorphic solutions. The corresponding many-body system turns out to be a deformation of the RS system. The equations of motion are:

[TABLE]

where dot means the time derivative,

[TABLE]

and $\sigma(x)$, $\zeta(x)$ are the Weierstrass $\sigma$- and $\zeta$-functions. We use them throughout the paper. Their definitions and properties are given in Appendix A. It is evident that the deformation parameter $g$ can be eliminated from the formulas by re-scaling the time variable as $t\to g^{-1/2}t$ (at $g=0$ (1.4) becomes the RS system). In what follows we fix $g$ to be $g=\sigma(2\eta)$. With this choice of $g$, equations (1.4) are exactly the dynamical equations for poles of elliptic solutions to the Toda lattice with the constraint of type B. The deformed RS system was further studied in [24], where the complete set of integrals of motion was found.

However, the method of [23] does not allow one to find any commutation representation of equations of motion and this remained to be an unsolved problem. This is the problem that we address in the present paper. Using an appropriate pole ansatz for the $\Psi$-function (the solution to the auxiliary linear problem), we apply the method of [5] to the Toda lattice with the constraint of type B. In this way, we obtain the commutation representation of equations (1.4) in the form of Manakov’s triple. We also study properties of the spectral curve and show that the $\Psi$-function is the Baker-Akhiezer function on the spectral curve. We show that the spectral curve admits a holomorphic involution with two fixed points, as it should be for algebraic-geometrical solutions to the Toda hierarchy with the constraint of type B.

It is known that integrable models of the CM and RS type admit extensions to field theories (“field analogues”) in which the coordinates of particles $x_{i}$ become “fields” $x_{i}(x,t)$ depending not only on the time $t$ but also on a space variable $x$. Equations of motion of these more general models can be obtained as equations for poles of more general elliptic solutions (called elliptic families in [25]) to nonlinear integrable equations. In this more general case the solutions are elliptic functions of a linear combination $\displaystyle{u=\sum_{k}\beta_{k}t_{k}}$ of higher times $t_{k}$ of the hierarchy, their poles $u_{i}(x,t)$ being functions of the space and time variables $x,t$ (for example, in the KP/CM case $x=t_{1}$, $t=t_{2}$). The equations of motion, together with their commutation representation, can be obtained by an appropriate pole ansatz. The equations of motion of the field analogues of the CM or RS systems were obtained by this method in [25] and [26] respectively (see also [27], where elliptic families of solutions to the constrained Toda lattice with constraint of type C [28] were discussed). In this paper we apply this method to the Toda hierarchy with the constraint of type B and obtain the field generalization of the deformed RS system. Note that the equations of motion were derived in [29] by a different method, which does not allow one to find a commutation representation for them. Here we reproduce that result and provide the commutation representation which is a field extension of the Manakov triple.

The paper is organized as follows. In section 2 we introduce the pole ansatz for the solution of the auxiliary linear problem and obtain the equations of motion together with their commutation representation. Properties of the spectral curve are studied in Section 3. Section 4 is devoted to analytic properties of the $\Psi$-function on the spectral curve, which characterize it as the Baker-Akhiezer function. In Section 5 we consider elliptic families of solutions to the Toda hierarchy with constraint of type B and obtain equations of motion for the field analogue of the deformed RS system together with their commutation representation. Section 6 contains concluding remarks. There are also two appendices. In Appendix A the definitions and properties of the special functions used in the main text are presented. Appendix B is devoted to the proof of an important matrix identity for elliptic Cauchy matrices.

2 The pole ansatz

The Toda lattice with constraint of type B follows from compatibility conditions for auxiliary linear problems. The first of them is the differential-difference equation [21]

[TABLE]

where $t=t_{1}$ and $v(x)$ is expressed through the tau-function $\tau(x)$ as in (1.2). For our purposes it is convenient to pass to another gauge and rewrite equation (2.1) in terms of the function

[TABLE]

The equation for $\Psi$ reads:

[TABLE]

where

[TABLE]

For elliptic solutions of the Toda lattice of type B, we have:

[TABLE]

The zeros $x_{j}$ are assumed to be all distinct. It follows that the coefficient functions

[TABLE]

[TABLE]

in the equation (2.3) are elliptic (double-periodic with periods $2\omega$, $2\omega^{\prime}$). Therefore, one can find double-Bloch solutions $\Psi(x)$, i.e., solutions such that $\Psi(x+2\omega)=b\Psi(x)$, $\Psi(x+2\omega^{\prime})=b^{\prime}\Psi(x)$ with some Bloch multipliers $b,b^{\prime}$. Any non-trivial double-Bloch function (i.e., not just an exponential function) must have poles. The simplest non-trivial double-Bloch function having one pole in the fundamental domain is the function

[TABLE]

It depends on the spectral parameter $\lambda$. We call it the Krichever function since he introduced it for analysis of elliptic solutions as early as in 1980 [5]. The main properties of the Krichever function are listed in Appendix A. We use the following pole ansatz for $\Psi$:

[TABLE]

where the coefficients $c_{i}$ do not depend on $x$. The parameters $k,\lambda$ are spectral parameters which are going to be connected by equation of the spectral curve. Using the quasiperiodicity properties (A7) of the Krichever function, we see that $\Psi$ given by (2.7) is indeed a double-Bloch function with Bloch multipliers

[TABLE]

In what follows we often suppress the second argument of $\Phi$ writing simply $\Phi(x)=\Phi(x,\lambda)$. The substitution of the pole ansatz into (2.3), yields:

[TABLE]

where dot means the $t$-derivative and $\Phi^{\prime}(x):=\partial_{x}\Phi(x,\lambda)$. Both sides have poles at the points $x=x_{i}$ and $x=x_{i}+\eta$ (possible poles at $x=x_{i}+2\eta$ in the last terms cancel by zeros of the numerator). The second order poles at $x=x_{i}+\eta$ cancel identically. Identification of the first order poles at $x=x_{i}$ gives the equations

[TABLE]

where

[TABLE]

Below we will also encounter the function $U_{i}^{+}$ which differs from $U_{i}^{-}$ by the change $\eta\to-\eta$. Introducing the $N\!\times\!N$ matrix $L=L(k,\lambda)$ with matrix elements

[TABLE]

and the vector ${\bf c}=(c_{1},\ldots,c_{N})^{T}$, we can write (2.9) as

[TABLE]

which implies that

[TABLE]

(here and below $I$ is the unity matrix). This is the equation of the spectral curve. A point of the curve is a pair $P=(k,\lambda)$ with $k$, $\lambda$ satisfying equation (2.13). Properties of the spectral curve will be discussed below. Identification of the first order poles at $x=x_{i}+\eta$ in (2.8) gives the equations

[TABLE]

where the matrix $M=M(k,\lambda)$ is given by

[TABLE]

The system of equations (2.12), (2.14) is overdetermined. The compatibility condition is

[TABLE]

In order to deal with the compatibility condition it is convenient to introduce matrices

[TABLE]

and diagonal matrices

[TABLE]

We will also use the matrices $A_{ij}^{\prime}=\Phi^{\prime}(x_{ij}-\eta)$, $B_{ij}^{\prime}=\Phi^{\prime}(x_{ij}-2\eta).$ In this notation, the matrices $L$ and $M$ read

[TABLE]

where $g=\sigma(2\eta)$.

The calculation of the left hand side of (2.16) yields:

[TABLE]

where

[TABLE]

A direct calculation using the identities (A8), (A9), shows that $W_{0}=0$. The calculation of $W_{2}$, with the help of (A9), gives:

[TABLE]

Using the fact that the sum of residues of the elliptic function

[TABLE]

is equal to zero, we see that

[TABLE]

so $W_{2}=0$. The calculation which shows that $W_{1}=0$ is most involved. In this calculation, one should use the identity which follows from the fact that the sum of residues of the elliptic function

[TABLE]

is equal to zero (this function has simple poles at $x=x_{k}$ and $x=x_{k}-\eta$ and a second order pole at $x=x_{i}-\eta$).

As a result, we have the matrix identity

[TABLE]

where $P$ is the matrix

[TABLE]

and

[TABLE]

The compatibility condition (2.16) states that $P=0$, i.e,

[TABLE]

which are equations of motion (1.4) of the deformed RS system.

We have obtained a commutation representation of the equations of motion in the form of the Manakov triple:

[TABLE]

Introducing ${\cal L}(k,\lambda)=L(k,\lambda)-kI$ and

[TABLE]

we can write it in the form

[TABLE]

so the matrices ${\cal L},M^{+},M^{-}$ form the Manakov triple.

Note that the matrix $R$ in (2.25) is traceless:

[TABLE]

since $\displaystyle{\sum_{i}(U^{-}_{i}-U^{+}_{i})}$ is proportional to the sum of residues of the elliptic function

[TABLE]

The fact that the matrix $R$ is traceless is used in the next section for the proof that the spectral curve is an integral of motion.

3 The spectral curve

The equation of the spectral curve is

[TABLE]

The time evolution $L\to L(t)$ of our “Lax matrix” is not isospectral. Nevertheless, the characteristic polynomial, $\det(kI-L)$, is an integral of motion, so the spectral curve does not depend on time. Indeed,

[TABLE]

[TABLE]

where we have used (2.25) and the fact that the matrix $R$ is traceless. The characteristic polynomial $\det(kI-L(k,\lambda))$ is a generating function for integrals of motion (strictly speaking, it is not polynomial but Laurent polynomial in $k$).

To study properties of the spectral curve, it is convenient to pass to the gauge-transformed Lax matrix $\tilde{L}=e^{-\eta\zeta(\lambda)}H^{-1}LH$, where $H$ is the diagonal matrix with matrix elements $H_{ii}=e^{-\zeta(\lambda)x_{i}}$, and to the spectral parameter $z=ke^{-\eta\zeta(\lambda)}$. Then the equation of the spectral curve is

[TABLE]

where

[TABLE]

Let us denote

[TABLE]

It is the generating function of integrals of motion. The equation of the spectral curve is

[TABLE]

The calculation of the determinant in (3.4) [24] shows that $Q(z,\lambda)$ is given by

[TABLE]

where $J_{k}$ are integrals of motion of the deformed RS system. They are given by [24]

[TABLE]

where

[TABLE]

Here the summation is over subsets ${\cal I}$ and ${\cal I}^{\prime}$ of the set ${\cal N}=\{1,\ldots,N\}$ such that ${\cal I}\cap{\cal I}^{\prime}=\emptyset$ and

[TABLE]

[TABLE]

In particular,

[TABLE]

The characteristic equation $Q(z,\lambda)=0$ defines a Riemann surface $\tilde{\Gamma}$ which is a $2N$-sheet covering of the $\lambda$-plane. Any point of it is the pair $(z,\lambda)$, where $z,\lambda$ are connected by equation (3.5). There are $2N$ points above each point $\lambda$. It is easy to see from the right hand side of (3.6) that the Riemann surface $\tilde{\Gamma}$ is invariant under the simultaneous transformations

[TABLE]

The factor of $\tilde{\Gamma}$ over the transformations (3.12) is an algebraic curve $\Gamma$ which covers the elliptic curve ${\cal E}$ with periods $2\omega,2\omega^{\prime}$. It is the spectral curve of the deformed RS model.

It is clear from (3.6) that the spectral curve $\Gamma$ admits a holomorphic involution $\iota$ with two fixed points. Indeed, the equation $Q(z,\lambda)=0$ is invariant under the involutive transformation

[TABLE]

as is easily seen from (3.6). Therefore, the fixed points can lie above the points $\lambda_{*}$ such that $\lambda_{*}=2N\eta-\lambda_{*}$ modulo the lattice with periods $2\omega,2\omega^{\prime}$, i.e. $\lambda_{*}=N\eta-\omega_{\alpha}$, where $\omega_{\alpha}$ is either [math] or one of the three half-periods $\omega_{1}=\omega$, $\omega_{2}=\omega^{\prime}$, $\omega_{3}=\omega+\omega^{\prime}$. Substituting this into the equation of the spectral curve, we conclude that the fixed points are $Q_{1}=(1,N\eta)$, $Q_{2}=(-1,N\eta)$ and there are no fixed points above $\lambda_{*}=N\eta-\omega_{\alpha}$ with $\omega_{\alpha}\neq 0$.

The genus $h$ of the spectral curve $\Gamma$ can be found using the following argument. Let us apply the Riemann-Hurwitz formula to the covering $\Gamma\to{\cal E}$. We have $2h-2=\nu$, where $\nu$ is the number of ramification points of the covering, which are zeros on $\Gamma$ of the function $\partial Q(z,\lambda)/\partial z$. Since the number of zeros of a function on $\Gamma$ is equal to the number of its poles (counted with multiplicities), we can find $\nu$ as the number of poles of $\partial Q/\partial z$. It is clear that poles of this function may lie only above the points $\lambda=0$ and $\lambda^{\iota}=0$, where we denote $\lambda^{\iota}=2N\eta-\lambda$. As $\lambda\to 0$, we see from (3.6) that one root of the Laurent polynomial $Q(z,\lambda)$ tends to $\infty$ while other $2N-1$ roots remain finite and non-zero. Similarly, as $\lambda\to 2N\eta$, one root of the Laurent polynomial $Q(z,\lambda)$ tends to [math] while other $2N-1$ roots remain non-zero. Therefore, we can write

[TABLE]

and

[TABLE]

where the functions $h_{j}(\lambda)$ are regular at $\lambda=0$ and $h_{N}(0)=J_{1}$. From (3.14) and (3.15) we see that one sheet above $\lambda=0$ (the one where $z=\infty$) and one sheet above $\lambda=2N\eta$ (where $z=0$) are distinguished. The points $P_{\infty}=(\infty,0)$ and $P_{0}=(0,2N\eta)$ are marked points of the spectral curve. Taking the $z$-derivative of (3.14), we see that the function $\partial Q/\partial z$ has a pole of order $N-1$ on the distinguished sheet above $\lambda=0$ and simple poles on the other $N-1$ sheets. From (3.15) we see that the function $\partial Q/\partial z$ has a pole of order $N+1$ on the distinguished sheet above $\lambda=2N\eta$ and simple poles on the other $N-1$ sheets. So the total number of poles is $\nu=4N-2$, hence $h=2N$.

4 The $\Psi$-function as the Baker-Akhiezer function on the

spectral curve

The coefficients $c_{i}$ in the pole ansatz for the function $\Psi$ (2.7) are functions on the spectral curve $\Gamma$: $c_{i}=c_{i}(t,P)$ ($P=(z,\lambda)$ is a point on the curve). Let us normalize them by the condition $c_{1}(0,P)=1$. After normalization the components $c_{i}(0,P)$ become meromorphic functions on $\Gamma$ outside the marked points $P_{\infty}$ and $P_{0}$ located above $\lambda=0$ and $\lambda=2N\eta$. The location of poles of the $c_{i}$’s depends on the initial data. As we shall see, the function $\Psi$ has essential singularities at the marked points.

Let ${\cal S}(t)$ be the fundamental matrix of solutions to the equation $\partial_{t}{\cal S}=M{\cal S}$, ${\cal S}(0)=I$. It is a regular function of $z,\lambda$ for $\lambda\neq 0,\,2N\eta$. Using the Manakov’s triple representation (2.25), we can write

[TABLE]

Using the relations ${\bf c}(t)={\cal S}(t){\bf c}(0)$ and $M=\dot{\cal S}{\cal S}^{-1}$, we rewrite this equation as

[TABLE]

Equivalently, we can represent it in the form of the differential equation

[TABLE]

for the vector ${\bf b}(t)={\cal S}^{-1}(L-kI){\bf c}(t)$ with the initial condition ${\bf b}(0)=0$. Under mild conditions which are satisfied in our case in the general position, the differential equation with zero initial data has the unique solution ${\bf b}(t)=0$ for all $t>0$. It then follows that ${\bf c}(t)={\cal S}(t){\bf c}(0)$ is the common solution of the equations $\dot{\bf c}=M{\bf c}$ and $L{\bf c}=k{\bf c}$ for all $t>0$. Therefore, the vector ${\bf c}(t,P)$ has the same $t$-independent poles as ${\bf c}(0,P)$.

The number of these poles can be found using the argument that the $N$-particle deformed RS system is equivalent to a reduction of the $2N$-particle RS system in which the particles stick together in pairs with the distance $\eta$ between the particles in each pair. As is known from [11], the number of poles of the $\Psi$-function for the latter system is $2N-1$. The reduction means that the spectral curve becomes special (it admits the involution) but the number of poles of the $\Psi$-function remains the same.

In order to investigate the analytic properties of the $\Psi$-function, it is convenient to pass to the gauge equivalent pair of matrices $\tilde{L}$, $\tilde{M}$, where

[TABLE]

with the diagonal matrix $H_{ij}=\delta_{ij}e^{-\zeta(\lambda)x_{i}}$ as before. The gauge-transformed linear system is

[TABLE]

where $\tilde{\bf c}=H^{-1}{\bf c}$.

Let us first consider what happens above the point $\lambda=0$. Near this point we can expand:

[TABLE]

where $\ell_{ij}$ is some matrix. Therefore, for solutions of the eigenvalue equation $\tilde{L}\tilde{\bf c}=z\tilde{\bf c}$ we should consider separately two cases. One is $\displaystyle{\sum_{j}\tilde{c}_{j}\neq 0}$ at $\lambda=0$, then

[TABLE]

(here we use another normalization than $\tilde{c}_{1}=1$). This case corresponds to the distinguished sheet of the spectral curve (near the point $P_{\infty}$). The other is $\displaystyle{\sum_{j}\tilde{c}_{j}=\lambda+O(\lambda^{2})}$ as $\lambda\to 0$, then we obtain $N-1$ eigenvector of the matrix $\dot{x}_{i}+gz^{-1}U_{i}^{-}+\ell_{ij}$ and the eigenvalue $z$ is finite. This corresponds to all other sheets of the spectral curve above the point $\lambda=0$.

The expansion of the matrix $\tilde{M}$ yields:

[TABLE]

[TABLE]

where $m_{ij}$ is some matrix. On the distinguished sheet we have:

[TABLE]

so $c_{i}(t)\propto e^{tz}$ on the distinguished sheet as $z\to\infty$. On all other sheets we have $\displaystyle{\sum_{j}\tilde{M}_{ij}\tilde{c}_{j}=O(1)}$ as $\lambda\to 0$. Therefore, the $\Psi$-function has the essential singularity at the point $P_{\infty}$ of the form $z^{x/\eta}e^{tz}$, where $z^{-1}$ is chosen as a local parameter around this point. On all other sheets above $\lambda=0$ the $\Psi$-function is regular. Indeed, on these sheets $\displaystyle{\sum_{j}c_{j}=0}$ and so the possible poles at $\lambda=0$ are absent.

The analysis in the neighborhood of the point $\lambda=2N\eta$ is more involved. To perform it, we need a non-trivial matrix identity for elliptic Cauchy matrices. Let us introduce the matrix $G^{(n)}(\lambda)$ with matrix elements

[TABLE]

Note that $G^{(0)}(\lambda)=I$. In Appendix B we prove the following identity:

[TABLE]

In particular, at $m=-n$ this identity states that

[TABLE]

Using the identity (4.4), it is not difficult to see that

[TABLE]

where $D$ is the diagonal matrix

[TABLE]

and

[TABLE]

We see that $L^{+}(z,\lambda)$ has the same structure as $\tilde{L}(z,\lambda)$ and we have

[TABLE]

Therefore, on the distinguished sheet we have

[TABLE]

or

[TABLE]

The action of $\tilde{M}$ to the vector $\tilde{\bf c}$ on all sheets above the point $\lambda=2N\eta$ except the distinguishes one gives regular expressions in $z\to 0$. The action of the singular part of $\tilde{M}$ as $z\to 0$ to the vector $\tilde{\bf c}$ on the distinguished sheet yields:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where in the second equality we have used the identity (4.4). We see that

[TABLE]

on the distinguished sheet, so $c_{i}(t)\propto e^{-z^{-1}t}$ as $z\to 0$. Therefore, the $\Psi$-function has the essential singularity at the point $P_{0}$ of the form $z^{x/\eta}e^{-tz^{-1}}$, where $z$ is chosen as a local parameter around this point. On all other sheets above $\lambda=2N\eta$ the $\Psi$-function is regular.

The established analytic properties of the $\Psi$-function (2.7) allow one to conclude that it is the Baker-Akhiezer function on the spectral curve.

5 Elliptic families and field generalization of the deformed

RS system

In this section we consider more general elliptic solutions of the Toda hierarchy with the constraint of type B, which are elliptic functions of a linear combination $\displaystyle{u=\sum_{k}\beta_{k}t_{k}}$ of higher times $t_{k}$ of the hierarchy. In [25], where elliptic solutions to the KP hierarchy were considered, such solutions were called elliptic families. The dynamics of their poles provide field extensions of the systems of CM and RS type. We are going to obtain the field extension of the deformed RS model in this way.

It is convenient to pass to the lattice space variable $n=x/\eta$ and consider poles of the more general elliptic solutions $u_{i}^{n}$ (zeros of the tau-function) as functions of the space variable $n$ and time variable $t=t_{1}$ (with $\bar{t}_{1}=-t_{1}$). As it follows from the general arguments of [25] (see also [26]), the general form of the tau-function is

[TABLE]

where $c_{1},\,c_{2}$ are constants and the function $\rho^{n}$ does not depend on $u$. We assume that all $u_{j}^{n}$’s are distinct. The linear problem (2.3) reads

[TABLE]

where

[TABLE]

[TABLE]

We assume that $U^{-,n}$ is an elliptic function of $u$, i.e.,

[TABLE]

The pole ansatz for the $\Psi$-function is

[TABLE]

where $\Phi(u,\lambda)$ is the Krichever function (A6). Plugging (5.3), (5.4) and (5.6) into (5.2), we obtain an equality for elliptic functions of $u$. The both sides have poles at the points $u=u_{i}^{n}$ (first order poles) and $u=u_{i}^{n-1}$ (second and first order poles). The second order poles cancel identically. Equating the residues at the poles, one obtains a system of linear equations for the column vector ${\bf c}^{n}=(c_{1}^{n},\ldots,c^{n}_{N})^{T}$. This system can be written in the form

[TABLE]

where $L^{n}$, $M^{n}$, $H^{n}$, $G^{n}$ are $N\!\times\!N$ matrices of the form

[TABLE]

The notation $\sigma^{(^{\prime})}(u_{i}^{n}-u_{k}^{n})$ means that this multiplier at $k=i$ should be omitted. To avoid misunderstanding, we should stress that the matrices $L^{n}$, $M^{n}$ differ from $L,M$ used in Section 2.

The system (5.7) is overdetermined. To obtain the compatibility condition, we take the time derivative of the first equation in (5.7), shift $n\to n+1$ in the second one and equate the results. In this way, we obtain the compatibility condition:

[TABLE]

Note that $L^{n}$, $M^{n}$ are the matrices of the Lax pair for the field analogue of the RS model (see [26]). In [26]), the following matrix identity was proved:

[TABLE]

where $D^{n}$ is the diagonal matrix with diagonal matrix elements

[TABLE]

The proof is based on the identity (A9) for the Krichever function. Using this identity, we represent the first term in (5.9) in the form

[TABLE]

[TABLE]

where

[TABLE]

and

[TABLE]

Let us introduce the matrix $W^{n}$ with matrix elements

[TABLE]

The following matrix identities hold:

[TABLE]

[TABLE]

The proof is based on the identity (A9). This identity implies that the matrix element $(ij)$ of the left hand side of (5.14) equals $\Phi(u_{i}^{n+1}-u_{j}^{n-2},\lambda)$ times sum of residues of the elliptic function

[TABLE]

and thus is equal to zero. The calculations which are necessary to prove (5.15) are more involved but, again, they are based on the identity (A9) (and (A8)). In the calculations, one should use the fact that sum of residues of the elliptic function

[TABLE]

is equal to zero (note that this function has simple as well as double poles).

Using the identities (5.10), (5.14), (5.15), we can represent the left hand side of compatibility condition (5.9) in the form

[TABLE]

However, the expression in the brackets here vanishes due to the first equation in (5.7) and we arrive at the compatibility condition in the form $\tilde{D}_{ii}^{n}=0$ for all $i$. This gives the equations of motion for the lattice “fields” $u_{i}^{n}$:

[TABLE]

where $W_{i}^{\pm,n}$ are given by (5.12). Summing them over all $i$, we obtain the equation

[TABLE]

Equations (5.16) together with (5.17) form a system of $N+1$ differential equations for the $N+1$ fields $u_{j}^{n}=u_{j}^{n}(t)$ ($j=1,\ldots,N$), $\rho^{n}=\rho^{n}(t)$. These equations provide the field extension of the deformed RS system. They were obtained in [29] by a different method. Our new method allows us to provide a commutation representation for them in the form of a field extension of the Manakov triple:

[TABLE]

where $R^{n}$ is the diagonal matrix with matrix elements

[TABLE]

If we set $u_{i}^{n}=x_{i}-n\eta$, equations (5.16), (5.17) reduce to the equations of motion of the deformed RS system for the $x_{i}$’s.

6 Open problems

In conclusion, let us list some open problems and ideas for further research related to the deformed RS system.

• a)

The most important unsolved problem is to find the Hamiltonian structure of the deformed RS model, if any. With the commutation representation at hand, one may hope to apply the general method developed by Krichever to approach this problem. A related question is quantization of the deformed RS system.

• b)

In [29] an integrable time discretization of the deformed RS system was obtained. It is desirable to find a commutation representation of the fully discrete equations of motion. Presumably, the pole ansatz for elliptic solutions of the fully discrete BKP equation should be used.

• c)

It is known that the rational and trigonometric CM and RS models are connected by various duality relations. It would be interesting to find similar duality relations for the deformed RS models and their $\eta\to 0$ degenerations.

• d)

A natural generalization of our result would be the extension of the correspondence with dynamics of poles of elliptic solutions to the whole infinite hierarchies, as it was done for the Toda/RS hierarchies in [30].

• e)

It is known that the field generalization of the RS model is gauge equivalent to a classical spin model on the lattice [26]. It is natural to ask whether any models of the spin chain type are equivalent in this sense to the field extension of the deformed RS model obtained in this paper.

Appendix A: The Weierstrass and Krichever functions

In this appendix we present the definition and main properties of the special functions used in the main text.

Let $\omega$, $\omega^{\prime}$ be complex numbers such that ${\rm Im}(\omega^{\prime}/\omega)>0$. The Weierstrass $\sigma$-function with quasi-periods $2\omega$, $2\omega^{\prime}$ is defined by the following infinite product over the lattice $2\omega m+2\omega^{\prime}m^{\prime}$, $m,m^{\prime}\in\mbox{\Bbb Z}$:

[TABLE]

It is an odd quasiperiodic function with two linearly independent quasi-periods in the complex plane. The expansion around $x=0$ is

[TABLE]

The monodromy properties of the $\sigma$-function under shifts by the quasi-periods are as follows:

[TABLE]

Here $\zeta(x)$ is the Weierstrass $\zeta$-function defined as

[TABLE]

As $x\to 0$,

[TABLE]

The Weierstrass $\wp$-function is defined as $\wp(x)=-\zeta^{\prime}(x)$. It is an even double-periodic function with periods $2\omega,2\omega^{\prime}$ and with second order poles at the points of the lattice $s=2\omega m+2\omega^{\prime}m^{\prime}$ with integer $m,m^{\prime}$. As $x\to 0$, $\wp(x)=x^{-2}+O(x^{2})$.

In the main text we use the Krichever function

[TABLE]

which has a simple pole at $x=0$. The expansion of $\Phi(x,\lambda)$ as $x\to 0$ is

[TABLE]

where $\alpha_{1}=-\frac{1}{2}\,\wp(\lambda)$, $\alpha_{2}=-\frac{1}{6}\,\wp^{\prime}(\lambda)$. The quasiperiodicity properties of the Krichever function are:

[TABLE]

As a function of $\lambda$, $\Phi(x,\lambda)$ is a double-periodic function with periods $2\omega$, $2\omega^{\prime}$.

In the main text we use the following identities for the Krichever function:

[TABLE]

[TABLE]

The first of them directly follows from the definition. To prove the second one, consider the function $\Phi(x,\lambda)\Phi(y,\lambda)/\Phi(x+y,\lambda)$ as a function of $x$. It is an elliptic function of $x$ whose poles and zeros coincide with those of the function $\zeta(x)+\zeta(y)-\zeta(x+y+\lambda)+\zeta(\lambda)$. Therefore, their ratio is a constant which can be found by putting $x$ to some special value.

Appendix B: Proof of the matrix identity

Let $G^{(n)}(\lambda)$ be the $N\!\times\!N$ matrix with matrix elements

[TABLE]

where

[TABLE]

The purpose of this appendix is to prove the following matrix identity:

[TABLE]

The proof is by induction in $N$. A simple calculation shows that (B2) holds at $N=1$. The induction assumption is that it is true for all matrices up to size $N-1$. Let us show that it holds for $N\!\times\!N$ matrices. We will prove that

[TABLE]

is a double-periodic function in $\lambda$ and $x_{i}$ without poles. Whence it is a constant which can be found by putting $x_{i}$ to some particular value. Explicitly, we have:

[TABLE]

First, let us consider this function as a function of $\lambda$. Obviously, it is double-periodic in $\lambda$ with possible simple poles at $\lambda=0$ and $\lambda=-x_{ik}-(m+n-mN)\eta$ if $x_{ik}+(m+n-mN)\eta\neq 0$. The residue is proportional to

[TABLE]

since it is the sum of residues of the elliptic function

[TABLE]

Hence $S_{ik}$ is regular as a function of $\lambda$. At $x_{ik}+(m+n-mN)\eta=0$ we have a double pole with a coefficient proportional to $r$, hence this pole is actually absent. Therefore, we conclude that $S_{ik}$ does not depend on $\lambda$.

Next, consider $S_{ik}$ as a function of $x_{i}$. From (B3) it is clear that it is an elliptic function of $x_{i}$. Let us consider possible poles of this function:

• a)

A possible simple pole at $x_{i}=x_{k}-(m+n-mN)\eta-\lambda$ which depends on $\lambda$. Since $S_{ik}$ does not depend on $\lambda$, this pole is actually absent.

• b)

Possible simple poles at $x_{i}=x_{l}$. The residue is proportional to

[TABLE]

[TABLE]

The first term comes from the sum in (B3) at $j=l$, the second one from $j=i$.

• c)

Possible simple poles at $x_{i}=x_{l}-(m+n)\eta$ with $l\neq k$. The residue is proportional to

[TABLE]

[TABLE]

As soon as it does not depend on $\lambda$ we can put $\lambda=-n\eta$, then we have

[TABLE]

Rearranging, we write this as

[TABLE]

[TABLE]

This is proportional to the matrix element of the product

[TABLE]

for $(N-1)\times(N-1)$ matrices at $\lambda=(n+mN)\eta$ which is the unity matrix by the induction assumption. Therefore, we have:

[TABLE]

since $l\neq k$.

We conclude that $S_{ik}$ is regular as a function of $x_{i}$ and thus it does not depend on $x_{i}$. To find it, we put $x_{i}=x_{l}-n\eta$ for some $l\neq i$. The evaluation of $S_{ik}$ at this point yields $S_{ik}=1$, so the identity (B2) is proved.

Acknowledgments

The research of A.Z. has been supported in part within the framework of the HSE University Basic Research Program.