Toda lattice hierarchy and trigonometric Ruijsenaars-Schneider hierarchy

TL;DR

This paper establishes a connection between the dynamics of poles in solutions of the 2D Toda lattice hierarchy and the Ruijsenaars-Schneider model, extending known particle correspondence to the entire hierarchy through Hamiltonian flows.

Contribution

It extends the pole-particle correspondence from the 2D Toda lattice to its hierarchy, linking pole dynamics to Ruijsenaars-Schneider Hamiltonians at all hierarchical times.

Findings

Pole dynamics governed by hierarchy times match Ruijsenaars-Schneider Hamiltonians.

Hierarchy flows correspond to traces of powers of the Lax matrix.

Extension of particle-model correspondence to entire Toda hierarchy.

Abstract

We consider solutions of the 2D Toda lattice hierarchy which are trigonometric functions of the ``zeroth'' time $t_0=x$. It is known that their poles move as particles of the trigonometric Ruijsenaars-Schneider model. We extend this correspondence to the level of hierarchies: the dynamics of poles with respect to the $m$-th hierarchical time $t_m$ (respectively, $\bar t_m$) of the 2D Toda lattice hierarchy is shown to be governed by the Hamiltonian which is proportional to the $m$-th Hamiltonian $\mbox{tr}\, L^m$ (respectively, $\mbox{tr}\, L^{-m}$) of the Ruijsenaars-Schneider model, where $L$ is the Lax matrix.

Equations

$$\tau(x,{\bf t},\bar{\bf t})=\exp\Bigl{(}-\sum_{k\geq 1}kt_{k}\bar{t}_{k}\Bigr{)}\prod_{i=1}^{N}\Bigl{(}e^{2\gamma x}-e^{2\gamma x_{i}({\bf t},\bar{\bf t})}\Bigr{)},$$

$$H_m=-\frac{\sinh (m\gamma \eta )}{m\gamma \eta }\text{mbox}tr{L}^m,$$

$$L_{ij}=\frac{\gamma \eta e^{\eta p_i}}{\sinh (\gamma (x_i-x_j-\eta ))}\prod _{l\ne i}\frac{\sinh (\gamma (x_i-x_l+\eta ))}{\sinh (\gamma (x_i-x_l))}$$

$$H_1=\sum _i{e}^{\eta p_i}\prod _{l\ne i}\frac{\sinh (\gamma (x_i-x_l+\eta ))}{\sinh (\gamma (x_i-x_l))}$$

$${\bar{H}}_m=-\frac{\sinh (m\gamma \eta )}{m\gamma \eta }\text{mbox}tr{L}^{-m}.$$

$$\mathcal{L}=e^{\eta {\partial }_x}+\sum _{k\ge 0}{U}_k(x)e^{-k\eta {\partial }_x},\bar{\mathcal{L}}=c(x)e^{-\eta {\partial }_x}+\sum _{k\ge 0}{\bar{U}}_k(x)e^{k\eta {\partial }_x},$$

$${\partial }_{t_m}\mathcal{L}=[{\mathcal{A}}_m,\mathcal{L}],{\partial }_{t_m}\bar{\mathcal{L}}=[{\mathcal{A}}_m,\bar{\mathcal{L}}]{\mathcal{A}}_m=({\mathcal{L}}^m{)}_{\ge 0},$$

$${\partial }_{{\bar{t}}_m}\mathcal{L}=[{\bar{\mathcal{A}}}_m,\mathcal{L}],{\partial }_{{\bar{t}}_m}\bar{\mathcal{L}}=[{\bar{\mathcal{A}}}_m,\bar{\mathcal{L}}]{\bar{\mathcal{A}}}_m=({\bar{\mathcal{L}}}^m{)}_{<0},$$

$${\partial }_{t_n}{\mathcal{A}}_m-{\partial }_{t_m}{\mathcal{A}}_n+[{\mathcal{A}}_m,{\mathcal{A}}_n]=0,$$

$${\partial }_{{\bar{t}}_n}{\mathcal{A}}_m-{\partial }_{t_m}{\bar{\mathcal{A}}}_n+[{\mathcal{A}}_m,{\bar{\mathcal{A}}}_n]=0,$$

$${\partial }_{{\bar{t}}_n}{\bar{\mathcal{A}}}_m-{\partial }_{{\bar{t}}_m}{\bar{\mathcal{A}}}_n+[{\bar{\mathcal{A}}}_m,{\bar{\mathcal{A}}}_n]=0.$$

$$\left\{\begin{array}[]{l}\partial_{t_{1}}\Bigl{(}U_{0}(x)+U_{0}(x-\eta)\Bigr{)}=U_{1}(x+\eta)-U_{1}(x-\eta)\\ \\ \partial_{t_{2}}U_{0}(x)=\partial_{t_{1}}\Bigl{(}U_{0}^{2}(x)+U_{1}(x+\eta)+U_{1}(x)\Bigr{)}.\end{array}\right.$$

$$\partial_{t_{2}}\Bigl{(}v(x+\eta)-v(x)\Bigr{)}=\partial_{t_{1}}^{2}\Bigl{(}v(x+\eta)+v(x)\Bigr{)}+\partial_{t_{1}}\Bigl{(}v^{2}(x+\eta)-v^{2}(x)\Bigr{)}.$$

$$\left\{\begin{array}[]{l}\partial_{t_{1}}\log c(x)=v(x)-v(x-\eta)\\ \\ \partial_{\bar{t}_{1}}v(x)=c(x)-c(x+\eta).\end{array}\right.$$

$${\partial }_{t_1}{\partial }_{{\bar{t}}_1}\log c(x)=2c(x)-c(x+\eta )-c(x-\eta )$$

$${\partial }_{t_1}{\partial }_{{\bar{t}}_1}\phi (x)=e^{\phi (x)-\phi (x-\eta )}-e^{\phi (x+\eta )-\phi (x)}.$$

$${\partial }_{t_m}\psi ={\mathcal{A}}_m(x)\psi ,{\partial }_{{\bar{t}}_m}\psi ={\bar{\mathcal{A}}}_m(x)\psi ,$$

$$\psi =z^{x/\eta }{e}^{\xi (\mathbf{t},z)}\left(1+\frac{{\xi }_1(x,\mathbf{t},\bar{\mathbf{t}})}{z}+\frac{{\xi }_2(x,\mathbf{t},\bar{\mathbf{t}})}{z^2}+\dots \right),$$

$$\xi (\mathbf{t},z)=\sum _{k\ge 1}{t}_k{z}^k.$$

$$\mathcal{W}(x)=1+{\xi }_1(x)e^{-\eta {\partial }_x}+{\xi }_2(x)e^{-2\eta {\partial }_x}+\dots$$

$$\psi =\mathcal{W}(x)z^{x/\eta }{e}^{\xi (\mathbf{t},z)}.$$

$${\psi }^{†}=({\mathcal{W}}^{†}(x-\eta ){)}^{-1}{z}^{-x/\eta }{e}^{-\xi (\mathbf{t},z)}$$

$$-{\partial }_{t_m}{\psi }^{†}={\mathcal{A}}_m^{†}(x-\eta ){\psi }^{†}.$$

$$\begin{array}[]{l}\phantom{-}\partial_{t_{1}}\psi(x)=\psi(x+\eta)+v(x)\psi(x),\\ \\ -\partial_{t_{1}}\psi^{{\dagger}}(x)=\psi^{{\dagger}}(x-\eta)+v(x-\eta)\psi^{{\dagger}}(x),\end{array}$$

$${\partial }_{{\bar{t}}_1}\psi (x)=c(x)\psi (x-\eta ).$$

$$\begin{array}[]{c}\displaystyle{\oint_{\infty}z^{\frac{x-x^{\prime}}{\eta}-1}e^{\xi({\bf t},z)-\xi({\bf t}^{\prime},z)}\tau\Bigl{(}x,{\bf t}-[z^{-1}],\bar{\bf t}\Bigr{)}\tau\Bigl{(}x^{\prime}+\eta,{\bf t}^{\prime}+[z^{-1}],\bar{\bf t}^{\prime}\Bigr{)}dz}\\ \\ \displaystyle{=\,\oint_{0}z^{\frac{x-x^{\prime}}{\eta}-1}e^{\xi(\bar{\bf t},z^{-1})-\xi(\bar{\bf t}^{\prime},z^{-1})}\tau\Bigl{(}x+\eta,{\bf t},\bar{\bf t}-[z]\Bigr{)}\tau\Bigl{(}x^{\prime},{\bf t}^{\prime},\bar{\bf t}^{\prime}+[z]\Bigr{)}dz}\end{array}$$

$${\bf t}\pm[z]=\Bigl{\{}t_{1}\pm z,t_{2}\pm\frac{1}{2}z^{2},t_{3}\pm\frac{1}{3}z^{3},\ldots\Bigl{\}}.$$

$$\frac{1}{2\pi i}\oint_{\infty}z^{-1}e^{\xi({\bf t},z)-\xi({\bf t}^{\prime},z)}\tau\Bigl{(}x,{\bf t}-[z^{-1}]\Bigr{)}\tau\Bigl{(}x+\eta,{\bf t}^{\prime}+[z^{-1}]\Bigr{)}dz=\tau(x+\eta,{\bf t})\tau(x,{\bf t}^{\prime}).$$

$$\begin{array}[]{c}\mu\tau(x+\eta,{\bf t}+[\lambda^{-1}]-[\mu^{-1}],\bar{\bf t})\tau(x,{\bf t},\bar{\bf t})-\lambda\tau(x+\eta,{\bf t},\bar{\bf t})\tau(x,{\bf t}+[\lambda^{-1}]-[\mu^{-1}],\bar{\bf t})\\ \\ +\,(\lambda-\mu)\tau(x+\eta,{\bf t}+[\lambda^{-1}],\bar{\bf t})\tau(x,{\bf t}-[\mu^{-1}],\bar{\bf t})=0,\end{array}$$

$$\begin{array}[]{c}\tau(x,{\bf t}-[\lambda^{-1}],\bar{\bf t})\tau(x,{\bf t},\bar{\bf t}-[\nu])-\tau(x,{\bf t},\bar{\bf t})\tau(x,{\bf t}-[\lambda^{-1}],\bar{\bf t}-[\nu])\\ \\ =\,\nu\lambda^{-1}\tau(x+\eta,{\bf t},\bar{\bf t}-[\nu])\tau(x-\eta,{\bf t}-[\lambda^{-1}],\bar{\bf t}).\end{array}$$

Full text

1. 1 Introduction

2. 2 The 2D Toda latttice hierarchy

3. 3 Trigonometric solutions to the mKP equation

4. 4 The higher Hamiltonian equations

5. 4.1 Positive times

6. 4.2 Negative times

7. 5 The generating form of equations of motion in higher times

8. 6 The tau-function for trigonometric solutions

9. 7 Conclusion

10. References

1 Introduction

The 2D Toda lattice (2DTL) hierarchy is an infinite set of compatible nonlinear differential-difference equations involving infinitely many time variables ${\bf t}=\{t_{1},t_{2},t_{3},\ldots\}$ (“positive” times), $\bar{\bf t}=\{\bar{t}_{1},\bar{t}_{2},\bar{t}_{3},\ldots\}$ (“negative” times) in which the equations are differential and the “zeroth” time $t_{0}=x$ in which the equations are difference. When the negative times are frozen, the equations involving $x$ and ${\bf t}$ variables form the modified Kadomtsev-Petviashvili (mKP) hierarchy which is a subhierarchy of the 2DTL one. Among all solutions to these equations, of special interest are solutions which have a finite number of poles in the variable $x$ in a fundamental domain of the complex plane. In particular, one can consider solutions which are trigonometric or hyperbolic functions of $x$ with poles depending on the times.

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. It was shown that the poles move as particles of the integrable Calogero-Moser many-body system [3, 4, 5, 6] with some restrictions in the phase space. As it was proved in [7, 8], this connection becomes most natural for the more general Kadomtsev-Petviashvili (KP) equation, in which case there are no restrictions in the phase space for the Calogero-Moser dynamics of poles. The method suggested by Krichever [9] 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 fuction). This method allows one to obtain the equations of motion together with their Lax representation.

The further progress was achieved in Shiota’s work [10]. Shiota has shown that the correspondence between rational solutions to the KP equation and the Calogero-Moser system with rational potential can be extended to the level of hierarchies. The evolution of poles with respect to the higher times $t_{k}$ of the KP hierarchy was shown to be governed by the higher Hamiltonians $H_{k}=\mbox{tr}\,L^{k}$ of the integrable Calogero-Moser system, where $L$ is the Lax matrix. Later this correspondence was generalized to trigonometric solutions of the KP hierarchy (see [11, 12]).

Dynamics of poles of elliptic solutions to the 2DTL and mKP hierarchies was studied in [13]. It was proved that the poles move as particles of the integrable Ruijsenaars-Schneider many-body system [14] which is a relativistic generalization of the Calogero-Moser system. The extension to the level of hierarchies for rational solutions to the mKP equation has been made in [15] (see also [16]): again, the evolution of poles with respect to the higher times $t_{k}$ of the mKP hierarchy is governed by the higher Hamiltonians $\mbox{tr}\,L^{k}$ of the Ruijsenaars-Schneider system.

In this paper we study the correspondence of the 2DTL hierarchy and the Ruijsenaars-Schneider hierarchy for trigonometric solutions of the former. Our method consists in a direct solution of the auxiliary linear problems for the wave function and its adjoint using a suitable pole ansatz. The tau-function of the 2DTL hierarchy for trigonometric solutions has the form

[TABLE]

where $\gamma$ is a complex parameter. (The zeros $x_{i}$ of the tau-function are poles of the solution.) When $\gamma$ is purely imaginary (respectively, real), one deals with trigonometric (respectively, hyperbolic) solutions. The limit $\gamma\to 0$ corresponds to rational solutions. We show that the evolution of the $x_{i}$’s with respect to the time $t_{m}$ is governed by the Hamiltonian

[TABLE]

where the parameter $\eta$ has the meaning of the inverse velocity of light and

[TABLE]

is the Lax matrix of the trigonometric Ruijsenaars-Schneider system. In particular,

[TABLE]

is the standard first Hamiltonian of the Ruijsenaars-Schneider system. In a similar way, the evolution of the $x_{i}$’s with respect to the time $\bar{t}_{m}$ is governed by the Hamiltonian

[TABLE]

The paper is organized as follows. In section 2 we remind the reader the main facts about the 2DTL hierarchy. Section 3 is devoted to solutions which are trigonometric functions of $x=t_{0}$. We derive equations of motion for their poles as functions of the time $t_{1}$. In section 4 we consider the dynamics of poles with respect to the higher times and derive the corresponding Hamiltonian equations. In section 5 we derive the self-dual form of equations of motion and show that it encodes all higher equations of motion in the generating form. In section 6 the determinant formula for the tau-function of trigonometric solutions is proved.

2 The 2D Toda latttice hierarchy

Here we very briefly review the 2DTL hierarchy (see [17]). Let us consider the pseudo-difference operators

[TABLE]

where $e^{\eta\partial_{x}}$ is the shift operator ($e^{\pm\eta\partial_{x}}f(x)=f(x\pm\eta)$) and the coefficient functions $U_{k}$, $\bar{U}_{k}$ are functions of $x$, ${\bf t}$ and $\bar{\bf t}$. They are the Lax operators of the 2DTL hierarchy. The equations of the hierarchy are differential-difference equations for the functions $U_{k}$, $\bar{U}_{k}$. They are encoded in the Lax equations

[TABLE]

[TABLE]

where \displaystyle{\Bigl{(}\sum_{\raise-1.0pt\hbox{\mbox{\Bbbb Z}}}U_{k}e^{k\eta\partial_{x}}\Bigr{)}_{\geq 0}=\sum_{k\geq 0}U_{k}e^{k\eta\partial_{x}}}, \displaystyle{\Bigl{(}\sum_{k\in\raise-1.0pt\hbox{\mbox{\Bbbb Z}}}U_{k}e^{k\eta\partial_{x}}\Bigr{)}_{<0}=\sum_{k<0}U_{k}e^{k\eta\partial_{x}}} For example, ${\cal A}_{1}=e^{\eta\partial_{x}}+U_{0}(x)$, $\bar{\cal A}_{1}=c(x)e^{-\eta\partial_{x}}$.

An equivalent formulation is through the zero curvature (Zakharov-Shabat) equations

[TABLE]

[TABLE]

[TABLE]

In particular, at $n=1$, $m=2$ we obtain from (2.4)

[TABLE]

Excluding $U_{1}$ from this system, one gets the mKP equation for $v(x)=U_{0}(x)$:

[TABLE]

From (2.5) at $m=n=1$ we have

[TABLE]

Excluding $v(x)$, we get the second order differential-difference equation for $c(x)$:

[TABLE]

which is one of the forms of the 2D Toda equation. After the change of variables $c(x)=e^{\varphi(x)-\varphi(x-\eta)}$ it acquires the most familiar form

[TABLE]

The zero curvature equations are compatibility conditions for the auxiliary linear problems

[TABLE]

where the wave function $\psi$ depends on a spectral parameter $z$: $\psi=\psi(z;{\bf t})$. The wave function has the following expansion in powers of $z$:

[TABLE]

where

[TABLE]

The wave operator is the pseudo-difference operator of the form

[TABLE]

with the same coefficient functions $\xi_{k}$ as in (2.11), then the wave function can be written as

[TABLE]

The adjoint wave function $\psi^{{\dagger}}$ is defined by the formula

[TABLE]

(see, e.g., [18]), where the adjoint difference operator is defined according to the rule $(f(x)\circ e^{n\eta\partial_{x}})^{{\dagger}}=e^{-n\eta\partial_{x}}\circ f(x)$. The auxiliary linear problems for the adjoint wave function have the form

[TABLE]

In particular, we have:

[TABLE]

[TABLE]

A common solution to the 2DTL hierarchy is provided by the tau-function $\tau=\tau(x,{\bf t},\bar{\bf t})$ [19, 20]. The hierarchy is encoded in the bilinear relation

[TABLE]

valid for all $x,x^{\prime}$, ${\bf t},{\bf t}^{\prime}$, $\bar{\bf t},\bar{\bf t}^{\prime}$, where

[TABLE]

The integration contour in the left hand side is a big circle around infinity separating the singularities coming from the exponential factor from those coming from the tau-functions. The integration contour in the right hand side is a small circle around zero separating the singularities coming from the exponential factor from those coming from the tau-functions. In particular, setting $x=x^{\prime}$, $\bar{\bf t}=\bar{\bf t}^{\prime}$, one obtains from (2.19) the bilinear relation for the mKP hierarchy

[TABLE]

Consequences of the bilinear relations (which are in fact equivalent to the whole hierarchy, see [21]) are the equations

[TABLE]

[TABLE]

There is also an equation similar to (2.22) with shifts of the negative times. Together with the tau-function $\tau$ it is convenient to introduce another tau-function, $\tau^{\prime}$, which differs from $\tau$ by a simple factor:

[TABLE]

The coefficient functions of the Lax operators can be expressed through the tau-function. In particular,

[TABLE]

After this substitution the mKP equation (2.8) becomes

[TABLE]

which can be also represented in the bilinear form

[TABLE]

The Toda equation (2.9) becomes

[TABLE]

The wave function and its adjoint are expressed through the tau-function according to the formulas

[TABLE]

[TABLE]

One may also introduce the complimentary wave functions $\bar{\psi}$, $\bar{\psi}^{{\dagger}}$ by the formulas

[TABLE]

[TABLE]

They satisfy the same auxiliary linear problems as the wave functions $\psi$, $\psi^{{\dagger}}$. It will be more convenient for us to work with the renormalized wave functions

[TABLE]

[TABLE]

It is easy to check that they satisfy the linear equations

[TABLE]

where $\displaystyle{\bar{v}(x)=\partial_{\bar{t}_{1}}\log\frac{\tau(x+\eta)}{\tau(x}}$.

Finally, let us point out useful corollaries of the bilinear relation (2.21). Differentiating it with respect to $t_{m}$ and putting ${\bf t}={\bf t}^{\prime}$ after that, we obtain:

[TABLE]

or

[TABLE]

In a similar way, differentiating the bilinear relation (2.19) with respect to $\bar{t}_{m}$ and putting $x=x^{\prime}$, ${\bf t}={\bf t}^{\prime}$, $\bar{\bf t}=\bar{\bf t}^{\prime}$ after that, we obtain the relation

[TABLE]

Here $\displaystyle{\mathop{\hbox{res}}\limits_{\infty}}$, $\displaystyle{\mathop{\hbox{res}}\limits_{0}}$ are defined according to the convention $\displaystyle{\mathop{\hbox{res}}\limits_{\infty}}(z^{-n})=\delta_{n1}$, $\displaystyle{\mathop{\hbox{res}}\limits_{0}}(z^{-n})=\delta_{n1}$.

3 Trigonometric solutions to the mKP equation

We are going to consider solutions which are trigonometric (i.e. single-periodic) functions of the variable $x$. For trigonometric solutions the tau-function has the form

[TABLE]

(In this section we ignore the dependence on the negative times keeping them fixed to zero.) This function has a single period $\pi i/\gamma$ in the complex plane. As in [12], we pass to the exponentiated variables

[TABLE]

In these variables, the tau-function becomes a polynomial of degree $N$ with roots $w_{i}$ which are supposed to be distinct: $\displaystyle{\tau=\prod_{i}(w-w_{i})}$. The function v(x)=\partial_{t_{1}}\log\Bigl{(}\tau(x+\eta)/\tau(x)\Bigr{)} is

[TABLE]

where

[TABLE]

and here and below dot means the $t_{1}$-derivative.

We begin with the investigation of the $t_{1}$-dynamics of the poles. The ansatz for the $\psi$-function depending on the spectral parameter $z$ suggested by equation (2.28) is

[TABLE]

where we have put $t_{k}=0$ for $k\geq 2$. The coefficients $c_{i}$ depend on ${\bf t}$ and on $z$. Substituting $\psi$ and $v$ into the first auxiliary linear problem in (2.17) $-\partial_{t_{1}}\psi(x)+\psi(x+\eta)+v(x)\psi(x)=0$, we get:

[TABLE]

[TABLE]

The left hand side is a rational function of $w$ vanishing at infinity with simple poles at $w=w_{i}$ and $w=q^{-1}w_{i}$ (the second order poles cancel identically). We should equate residues at the poles to zero. This gives the following system of linear equations for the coefficients $c_{i}$:

[TABLE]

In a similar way, the adjoint linear problem $\partial_{t_{1}}\psi^{\dagger}(x)+\psi^{\dagger}(x-\eta)+v(x-\eta)\psi^{{\dagger}}(x)=0$ with the ansatz for the $\psi^{{\dagger}}$-function

[TABLE]

leads to the linear equations for the coefficients $c_{i}^{*}$:

[TABLE]

After the gauge transformation $\tilde{c}_{i}=c_{i}w_{i}^{-1/2}$, $\tilde{c}^{*}_{i}=c^{*}_{i}w_{i}^{-1/2}$ the above conditions can be written in the matrix form

[TABLE]

[TABLE]

where $\tilde{\bf c}=(\tilde{c}_{1},\ldots,\tilde{c}_{N})^{T}$ is a column vector, $\tilde{\bf c}^{*}=(\tilde{c}^{*}_{1},\ldots,\tilde{c}^{*}_{N})$ is a row vector, ${\bf e}=(1,1,\ldots,1)^{T}$, $I$ is the identity matrix and the matrices $X$, $W$, $L$, $M$, $\tilde{M}$ are

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The following commutation relation can be checked directly:

[TABLE]

Here $E={\bf e}\otimes{\bf e}^{T}$ is the rank 1 matrix with matrix elements $E_{ij}=1$. This commutation relation will be important in what follows.

The system of linear equations (3.9) is overdetermined. Taking the $t_{1}$-derivative of the first equation in (3.9) and substituting the second equation, one obtains the compatibility condition of the system in the form

[TABLE]

A straightforward calculation shows that

[TABLE]

[TABLE]

where

[TABLE]

and the diagonal matrices $D^{\pm}$, $D_{0}$ are

[TABLE]

Therefore, the compatibility condition takes the form $R\tilde{\bf c}=0$ which means that $R_{ii}=0$ for all $i$. This leads to the equations of motion of the trigonometric Ruijsenaars-Schneider model

[TABLE]

where $x_{ik}=x_{i}-x_{k}$. The matrix equation $\dot{L}+[L,M^{\prime}]=0$ with

[TABLE]

[TABLE]

provides the Lax representation for them.

Equations (3.17) are Hamiltonian with the Hamiltonian

[TABLE]

and the canonical Poisson brackets between $p_{i}$ and $x_{i}$. The velocity is given by

[TABLE]

The Lax representation implies that the higher conserved quantities are $\mbox{tr}\,L^{m}$. It is proved in [22] that they are in involution, i.e., the system is integrable.

Let us consider the transformation of the phase space coordinates $(p_{i},x_{i})\to(u_{i},x_{i}^{\prime})$, where $x_{i}^{\prime}=x_{i}$ and

[TABLE]

Then the derivatives transform as follows:

[TABLE]

[TABLE]

At this point we finish the discussion of the $t_{1}$-dynamics of poles and pass to the higher times in the next section.

4 The higher Hamiltonian equations

4.1 Positive times

In order to study the dynamics of poles in the higher positive times ${\bf t}$, we use the relation (2.36), which, after the substitution of the wave functions for the trigonometric solutions, takes the form

[TABLE]

The both sides are rational functions of $w$ with simple poles at $w=w_{i}$ and $w=q^{-1}w_{i}$ vanishing at infinity. Identifying the residues at the poles at $w=w_{i}$ in the both sides, we obtain:

[TABLE]

From (3.9), (3.10) we have:

[TABLE]

Substituting this into (4.1), we get:

[TABLE]

[TABLE]

where $E_{i}$ is the diagonal matrix with the matrix elements $(E_{i})_{jk}=\delta_{ij}\delta_{ik}$. Using the commutation relation (3.15), we have:

[TABLE]

[TABLE]

Next, we use the easily proved identity

[TABLE]

(see (3.22)) to continue the chain of equalities:

[TABLE]

[TABLE]

In this way we have obtained one half of the Hamiltonian equations for the higher flows

[TABLE]

where the Lax matrix $L$ is given by (1.2). In particular, the Hamiltonian $H_{1}$ coincides with (3.20).

The derivation of the second half of the Hamiltonian equations is more involved. The idea of the derivation is the same as in [15]. First of all, we note that (4.3) can be written in the form

[TABLE]

Differentiating this equality with respect to $t_{1}$ and using the Lax equation, we get:

[TABLE]

Now we apply $\partial_{t_{m}}$ to equation (3.21):

[TABLE]

[TABLE]

[TABLE]

where the matrix $A^{(i)}$ is

[TABLE]

Note that the diagonal part of the matrix $M^{\prime}$ (3.19) does not contribute, so instead of the matrix $M^{\prime}$ here we can substitute its off-diagonal part

[TABLE]

Let us calculate the matrix elements:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Combining everything together, we obtain the matrix elements of the matrix $A^{(i)}$:

[TABLE]

Our next goal is to prove that

[TABLE]

where the matrix $C^{(i)}$ is given by

[TABLE]

From this one concludes that

[TABLE]

This yields the second half of the Hamiltonian equations for the higher flows:

[TABLE]

In order to prove the identity (4.6), we calculate matrix elements of the right hand side and compare them with (4.5). Indeed, we have:

[TABLE]

[TABLE]

[TABLE]

and one can check that $A^{(i)}+\partial L/\partial x_{i}+[C^{(i)},L]=0$.

4.2 Negative times

In order to investigate the dynamics of zeros of the tau-function in the negative times, we first consider the $\bar{t}_{1}$-evolution. We will work with the complimentary wave functions $\phi$, $\phi^{{\dagger}}$ given by (2.32), (2.33) for which we use the ansatz

[TABLE]

[TABLE]

where $b_{i}$, $b_{i}^{*}$ are some unknown coefficients depending on the times and on $z$ but not on $x$. Substituting them into the linear equations (2.34), we can write down the conditions of cancellation of the poles in the way similar to that of section 3. In fact the equations for $b_{i}$ are the same as for $c_{i}^{*}$ up to changing $z$ to $-z^{-1}$, $w$ to $qw$ and $\partial_{t_{1}}$ to $\partial_{\bar{t}_{1}}$. The equations for $b_{i}^{*}$ and $c_{i}$ are connected in a similar way. Passing to $\tilde{b}_{i}=w_{i}^{-1/2}b_{i}$, $\tilde{b}_{i}^{*}=w_{i}^{-1/2}b_{i}^{*}$, we have, after some algebra, in the notation of section 3:

[TABLE]

[TABLE]

where $\tilde{\bf b}^{T}=(\tilde{b}_{1},\ldots,\tilde{b}_{N})$, $\tilde{\bf b}^{*}=(\tilde{b}_{1}^{*},\ldots,\tilde{b}_{N}^{*})$ and the matrix $\bar{L}$ reads

[TABLE]

This matrix satisfies the commutation relation (3.15) with $\partial_{\bar{t}_{1}}W$ instead of $\dot{W}=\partial_{t_{1}}W$:

[TABLE]

Using the relation (2.37), we find, similarly to (4.1):

[TABLE]

Substituting here the solutions of linear systems (4.11), (4.12) and using (4.14), one can repeat the calculation from section 4.1 with the result

[TABLE]

Our next goal is to derive a relation between the Lax matrices $L$ and $\bar{L}$. For this, we need a relation between the velocities $\dot{x}_{i}=\partial_{t_{1}}x_{i}$ and $\partial_{\bar{t}_{1}}x_{i}$. The latter can be derived from the Toda equation (2.27). Substituting the tau-function in the form $\tau=\prod_{i}(w-w_{i})$, we get

[TABLE]

Identifying the second order poles in the both sides, we obtain the relation

[TABLE]

or

[TABLE]

where the diagonal matrices $U_{\pm}$ are

[TABLE]

Next, we need the formula for the inverse of the Cauchy matrix

[TABLE]

We have:

[TABLE]

or

[TABLE]

Now we write $L=2\gamma q^{1/2}\partial_{t_{1}}X\,W^{1/2}CW^{1/2}$ and find, using (4.22), (4.18):

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We see that the matrix $-\bar{L}^{T}$ is connected with $L^{-1}$ by a similarity transformation with a diagonal matrix. Using the fact that $E_{i}\bar{L}=-\eta^{-1}\partial\bar{L}/\partial p_{i}$, we can therefore rewrite equation (4.16) as

[TABLE]

which is one half of the Hamiltonian equations for the negative time flows.

The derivation of the second half is straightforward. We note that

[TABLE]

where we use the notation of section 4.1. In the complete analogy with the calculation in the previous subsection, we have

[TABLE]

with the same matrix $A^{(i)}$ (4.4). By virtue of (4.6) we obtain:

[TABLE]

which are the Hamiltonian equations

[TABLE]

with $\bar{H}_{m}$ given by (1.4). In particular,

[TABLE]

5 The generating form of equations of motion in higher times

In the above analysis we parametrized the wave function by residues at its poles. Another possible parametrization is by zeros and poles. It leads to the so-called self-dual form of equations of motion. In this section we derive these equations and show that they provide a generating form of equations of motion for the Ruijsenaars-Schneider model in the higher times.

In this section we keep the negative times $\bar{\bf t}$ fixed and consider only the dependence on ${\bf t}$. In accordance with (2.28) we have $\psi(\mu,x,{\bf t})=\mu^{x/\eta}e^{\xi({\bf t},\mu)}\hat{\tau}(x,{\bf t})/\tau(x,{\bf t})$ (here $\hat{\tau}(x,{\bf t})=\tau(x,{\bf t}-[\mu^{-1}])$), then the auxiliary linear problem (2.17) acquires the form

[TABLE]

For trigonometric solutions $\tau$ is of the form (3.1) and for $\hat{\tau}$ we write

[TABLE]

parametrizing it by its zeros $y_{i}$. Substituting this into (5.1), we have:

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Identifying residues at the simple poles at $w=q^{-1}w_{i}$ and $w=v_{i}$, we get the system of equations

[TABLE]

or

[TABLE]

This is the Ruijsenaars-Schneider analog of the Bäcklund transformation for the Calogero-Moser system [23, 24]. These equations appeared in [25] in the context of the integrable time discretization of the Ruijsenaars-Schneider model (see also [26, 27]). One can show that the equations of motion of the Ruijsenaars-Schneider model for $x_{i}$’s follow from (5.4) and $y_{i}$’s obey the same equations (for the proof see [26]).

At the same time these equations contain all the higher equations of motion in an encoded form. To see this, we introduce the differential operator

[TABLE]

then $\hat{\tau}=e^{-D(\mu)}\tau$ and $y_{i}=e^{-D(\mu)}x_{i}$. Performing an overall time shift in the second equation in (5.4), we can rewrite them in the form

[TABLE]

Dividing one equation by the other, we obtain the equations

[TABLE]

which are, on one hand, equations of motion for the Ruijsenaars-Schheider system in discrete time (see [28]) and, on the other, provide the generating form of the higher equations of motion in continuous hierarchical times. Indeed, expanding (5.7) in (inverse) powers of $\mu$, one gets the set of the higher equations of motion. In particular, equations (3.17) are obtained by expansion of (5.7) up to $\mu^{-1}$.

6 The tau-function for trigonometric solutions

In this section we prove the determinant formula for the tau-function of trigonometric solutions

[TABLE]

where $L_{0}=L(0)$, $W_{0}=W(0)$. We recall that the tau-function $\tau^{\prime}$ is connected with $\tau$ by formula (2.24).

The matrix \displaystyle{\exp\Bigl{(}\sum_{k\geq 1}(q^{-k/2}\!-\!q^{k/2})(t_{k}L_{0}^{k}-\bar{t}_{k}L_{0}^{-k})\Bigr{)}W_{0}} can be diagonalized with the help of a diagonalizing matrix $V$:

[TABLE]

There is a freedom in the definition of $V$: it can be multiplied by a diagonal matrix from the left. We fix this freedom by the condition

[TABLE]

The matrices $W_{0}$, $L_{0}$ satisfy the commutation relation (3.15) which we write here in the form

[TABLE]

Let us prove, following [25], that the matrices $W$ and $L=VL_{0}V^{-1}$ satisfy the same commutation relation. We have:

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[TABLE]

[TABLE]

[TABLE]

[TABLE]

(The last equality follows from the condition (6.2).) Denoting $W^{3/2}VW_{0}^{-3/2}\dot{W}_{0}\,{\bf e}=\dot{W}{\bf e}$, we arrive at the desired commutation relation.

We are going to prove that the function (6.1) satisfies the bilinear equations (2.22), (2.23) of the 2DTL hierarchy. We begin with equation (2.22):

[TABLE]

(Here and below in the proof we put $\bar{\bf t}=0$ and identify $\tau$ and $\tau^{\prime}$.) The similarity transformation with the matrix $V$ under the determinant in (6.1) allows one to write the following formulas:

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[TABLE]

[TABLE]

where the matrix $\tilde{E}=W^{-1/2}\dot{W}EW^{1/2}$ has rank 1 and we have used the commutation relation (3.15) and the formula $\det(I+A)=1+\mbox{tr}\,A$ valid for any rank 1 matrix $A$. Similar calculations yield

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[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Substituting everything into the left hand side of (6.4), we obtain:

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[TABLE]

[TABLE]

[TABLE]

This expression is a rational function of $w$ with simple poles at $w=w_{i}$ and $w=q^{-1}w_{i}$ vanishing at $\infty$. To prove that it actually vanishes identically it is enough to prove that the residues at the poles are zero. The residue at the pole at $w=w_{i}$ is equal to

[TABLE]

[TABLE]

Recalling that $\displaystyle{L_{ij}=q^{1/2}\frac{\dot{w}_{i}w_{i}^{-1/2}w_{j}^{1/2}}{w_{i}-qw_{j}}}$, we can rewrite the last line (the triple sum) in the form

[TABLE]

from which it is seen that the residue is zero. The calculation for the residue at $w=q^{-1}w_{i}$ is similar.

The fact that the function (6.1) is a KP tau-function with respect to the times ${\bf t}$ follows also from the result of Kasman and Gekhtman [29]: for any matrices $X$, $Y$, $Z$ such that the matrix $XZ-YX$ has rank $1$ the function

[TABLE]

is a tau-function of the KP hierarchy. In our case $X=-W_{0}$, $Z=q^{-1/2}L_{0}$, $Y=q^{1/2}L_{0}$ and the condition that $XZ-YX$ has rank $1$ is equivalent to the commutation relation (3.15).

Let us pass to the proof of equation (2.23) which we write here in the equivalent form

[TABLE]

The calculations similar to the ones done above yield:

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[TABLE]

[TABLE]

[TABLE]

Using the results of the above calculation, it is not difficult to see that the substitution into (6.7) gives the identity, so the equation (2.23) is proved.

7 Conclusion

The main result of this paper is establishing the precise correspondence between trigonometric solutions of the 2D Toda lattice hierarchy and the hierarchy of the Hamiltonian equations for the integrable Ruijsenaars-Schneider model with higher Hamiltonians. The zeros of the tau-function move as particles of the Ruijsenaars-Schneider model. We have shown that the $m$th time flow $t_{m}$ of the 2DTL hierarchy gives rise to the flow with the Hamiltonian $H_{m}$ of the Ruijsenars-Schneider model proportional to $\mbox{tr}\,L^{m}$, where $L$ is the Lax matrix, while the time flow $\bar{t}_{m}$ gives rise to the Hamiltonian flow with the Hamiltonian $\bar{H}_{m}$ proportional to $\mbox{tr}\,L^{-m}$. In some sense this correspondence is simpler and more natural then a similar correspondence between the KP hierarchy and trigonometric Calogero-Moser hierarchy [12], which in principle can be obtained from our results in the limit $\eta\to 0$.

Acknowledgments

The work of V.P. was supported in part by RFBR grant 18-01-00273. The research of A.Z. was carried out within the HSE University Basic Research Program and funded (jointly) by the Russian Academic Excellence Project ’5-100’. The work of A.Z. was also supported in part by RFBR grant 18-01-00461.