1 Introduction

Dynamics of poles of singular solutions to nonlinear integrable equations is a well-known subject in the theory of integrable systems. Investigations in this direction were initiated in the seminal paper [1]. In [2, 3], it was shown that poles \(x_i\) of rational solutions to the Kadomtsev–Petviashvili (KP) equation move as particles of the many-body Calogero–Moser system [4,5,6] with the pairwise potential \(1/(x_i-x_j)^2\). This remarkable connection was further generalized to elliptic (double-periodic) solutions in [7]: Poles \(x_i\) of the elliptic solutions were shown to move according to the equations of motion of Calogero–Moser particles with the elliptic interaction potential \(\wp (x_i-x_j)\), where \(\wp \) is the elliptic Weierstrass \(\wp \)-function. This many-body system of classical mechanics is known to be integrable. For a review of the models of the Calogero–Moser type, see [8].

The correspondence between singular solutions of nonlinear integrable equations and integrable many-body systems allows for generalizations in various directions. The extension to the matrix KP equation was discussed in [9]; in this case the poles and matrix residues at the poles move as particles of the spin generalization of the Calogero–Moser model known also as the Gibbons–Hermsen model [10]. In the paper [11] by the authors, the dynamics of poles of elliptic solutions to the (matrix) two-dimensional Toda lattice was shown to be isomorphic to the relativistic extension of the Calogero–Moser system known also as the Ruijsenaars–Schneider system [12, 13] and its version with spin degrees of freedom [11]. Another generalizations concern B- and C-versions of the KP equation (BKP and CKP). Equations of motion for poles of elliptic solutions to the BKP and CKP equations were recently obtained in [14] and [15], respectively. For a review, see [16].

The method suggested in [2, 7] and used in all subsequent works on the subject is based on the well-known fact that nonlinear integrable equations such as KP, BKP, CKP and the 2D Toda lattice can be represented as compatibility conditions for systems of certain overdetermined linear problems which are partial differential or difference equations in two variables (space and time). The suggested scheme of finding the dynamics of poles consists in substituting the elliptic solution not in the nonlinear equation itself but in the linear problems for it, using a suitable pole ansatz for the wave function depending on a spectral parameter.

In this paper, we systematically use another method suggested by one of the authors in [17] on the example of the KP/Calogero–Moser correspondence and further discussed in [18, 19]. The key point of this method is existence of meromorphic solutions to the linear partial differential or difference equations. We call them monodromy free linear equations. It turns out that the conditions of existence of meromorphic solutions in the space variable are equivalent to equations of motion for the poles which are equations of motion for integrable many-body systems of Calogero–Moser and Ruijsenaars–Schneider type.

We also prove that existence of at least one meromorphic solution implies existence of a whole family of meromorphic wave solutions depending on a spectral parameter.

For completeness, we include in this paper the analysis of meromorphic solutions to the linear problems for the KP and Toda lattice equations leading to the Calogero–Moser and Ruijsenaars–Schneider systems, respectively. (This is contained in the earlier works [17, 18].) Conditions of existence of meromorphic solutions to the linear problems for the BKP and CKP equations as well as for the Toda lattice of type C were not discussed in the literature; in this paper we find them and show that they are equivalent to the equations of motion for poles of these equations obtained in [14, 15, 22].

The main new result of this paper is the equations of motion (3.59) for poles of elliptic solutions to the Toda lattice of type B recently introduced in [20]. They have the form

$$\begin{aligned} \ddot{x}_i +\sum _{k=1, \, \ne i}^N \dot{x}_i \dot{x}_k \Bigl (\zeta (x_{ik}+\eta )+ \zeta (x_{ik}-\eta )-2\zeta (x_{ik})\Bigr ) -U(x_{i1}, \ldots \, x_{iN})=0,\nonumber \\ \end{aligned}$$
(1.1)

where dot means the time derivative, \(x_{ik}\equiv x_i-x_k\),

$$\begin{aligned}{} & {} U(x_{i1}, \ldots \, x_{iN}) \nonumber \\{} & {} \quad = \sigma (2\eta )\left[ \prod _{j\ne i}\frac{\sigma (x_{ij}+2\eta )\sigma (x_{ij}-\eta )}{\sigma (x_{ij}+\eta )\sigma (x_{ij})}- \prod _{j\ne i}\frac{\sigma (x_{ij}-2\eta )\sigma (x_{ij}+\eta )}{\sigma (x_{ij}-\eta )\sigma (x_{ij})}\right] \nonumber \\ \end{aligned}$$
(1.2)

and \(\sigma (x)\), \(\zeta (x)\) are the standard Weierstrass functions (see Appendix A). These equations are obtained from the condition of existence of meromorphic solutions to the differential–difference auxiliary linear problem for the Toda lattice of type B. We also show that the same equations can be obtained by restriction of the Ruijsenaars–Schneider dynamics with respect to the time flow \(\partial _{t_1}-\partial _{\bar{t}_1}\) of the system containing 2N particles to the half-dimensional subspace of the 4N-dimensional phase space corresponding to the configuration in which the particles stick together joining in pairs such that the distance between particles in each pair is equal to \(\eta \). This configuration is destroyed by the flow \(\partial _{t_1}+\partial _{\bar{t}_1}\) but is preserved by the flow \(\partial _{t_1}-\partial _{\bar{t}_1}\) (and, hypothetically, by all higher flows \(\partial _{t_k}-\partial _{\bar{t}_k}\)), and the time evolution in \(t=t_1-\bar{t}_1\) of the pairs with coordinates \(x_i\) is given by equations (1.1), (1.2). This fact is not so surprising if we recall that the tau-function \(\tau ^\textrm{Toda}(x)\) of the Toda lattice (whose zeros move as Ruijsenaars–Schneider particles) is connected with the tau-function \(\tau (x)\) of the Toda lattice of type B (whose zeros move according to equations (1.1)) by the relation

$$\begin{aligned} \tau ^\textrm{Toda}(x)=\tau (x)\, \tau (x-\eta ) \end{aligned}$$
(1.3)

(see [20]), so zeros of \(\tau ^\textrm{Toda}(x)\) stick together in pairs.

To avoid a confusion, we should stress that what we mean by the Toda lattice of type B or C is very different from the systems introduced in [23] under similar names. Our equations are natural integrable discretizations of the BKP and CKP equations, that is why we found it appropriate to call them “Toda lattices of type B and C.”

The reader should be aware that the notation for coefficients of Laurent expansions below is valid only throughout each section and the same notation may mean different things in different sections. We hope that this will not lead to a misunderstanding since each section is devoted to its own linear equation and the contents do not intersect.

2 Differential equations

2.1 The KP case

We start with a worm-up exercise following [17].

Consider the linear equation

$$\begin{aligned} (\partial _t -\partial _x^2 -2u)\psi =0, \end{aligned}$$
(2.1)

which is one of the auxiliary linear problems for the KP equation. It is easy to see that if u(x) has a pole a in the complex x-plane, it must be a second-order pole. Expanding the left-hand side in a neighborhood of the pole, one can find a necessary condition of existence of a meromorphic solution in this neighborhood.

Proposition 2.1

If equation (2.1) with

$$\begin{aligned} u(x)=-\frac{1}{(x-a)^2} +u_0 +u_1 (x-a) + \ldots , \end{aligned}$$
(2.2)

has a meromorphic in x solution, then the condition

$$\begin{aligned} \ddot{a} +4u_1=0 \end{aligned}$$
(2.3)

holds, where dot means the time derivative.

Proof

Let the expansion of \(\psi (x)\) around the point a be of the form

$$\begin{aligned} \psi (x) = \frac{\alpha }{x-a} +\beta +\gamma (x-a) +\delta (x-a)^2 +\ldots . \end{aligned}$$
(2.4)

Substituting expansions (2.2), (2.4) in the left-hand side of (2.1), we see that the highest (third-order)-order poles cancel identically. Equating the coefficients in front of \((x-a)^{-2}\), \((x-a)^{-1}\) and \((x-a)^0\) to zero, we get the conditions

$$\begin{aligned} \left\{ \begin{array}{l} \dot{a} \alpha +2\beta =0, \\ \\ \dot{\alpha }+2\gamma -2u_0 \alpha =0, \\ \\ \dot{\beta }-\dot{a} \gamma -2u_0 \beta -2u_1 \alpha =0, \end{array}\right. \end{aligned}$$
(2.5)

Taking t-derivative of the first equation, plugging \(\dot{\alpha }\) and \(\dot{\beta }\) from the second and the third ones and using the first one again, we obtain the necessary condition (2.3). \(\square \)

One can see that this condition encodes equations of motion for the (elliptic in general) Calogero–Moser system. Indeed, let u(x) be the doubly-periodic meromorphic function

$$\begin{aligned} u(x)=-\sum _i \wp (x-x_i), \end{aligned}$$
(2.6)

where \(\wp (x)\) is the Weierstrass \(\wp \)-function, then the expansion (2.2) near the pole at \(a=x_i\) holds true with

$$\begin{aligned} u_0=-\sum _{j\ne i}\wp (x_i-x_j), \qquad u_1 = -\sum _{j\ne i}\wp ' (x_i-x_j), \end{aligned}$$

so the conditions (2.3) for each \(x_i\) read

$$\begin{aligned} \ddot{x}_i =4\sum _{j\ne i}\wp '(x_i-x_j), \end{aligned}$$
(2.7)

which are the equations of motion for the elliptic Calogero–Moser system.

Next, we show that (2.3) is simultaneously a sufficient condition for local existence of a meromorphic wave solution to equation (2.1), i.e., a solution depending on a spectral parameter k with the expansion of the form

$$\begin{aligned} \psi (x) = e^{kx +k^2t}\Bigl (1+\sum _{s\ge 1} \xi _s k^{-s}\Bigr ), \quad k\rightarrow \infty . \end{aligned}$$
(2.8)

Proposition 2.2

Suppose that condition (2.3) for the pole a of u(x) holds. Then, all wave solutions of equation (2.1) of the form (2.8) are meromorphic in a neighborhood of the point a with a simple pole at \(x=a\) and regular elsewhere in this neighborhood.

Proof

Substitution of the series (2.8) into the equation (2.1) gives the recurrence relation

$$\begin{aligned} 2\xi _{s+1}' =\dot{\xi }_s -2u\xi _s -\xi _s'', \qquad s\ge 0, \qquad \xi _0\equiv 1. \end{aligned}$$
(2.9)

In particular, at \(s=0\) we have

$$\begin{aligned} \xi _1'=-u. \end{aligned}$$
(2.10)

Let the Laurent expansion of \(\xi _s\) near the pole at \(x=a\) be of the form

$$\begin{aligned} \xi _s = \frac{r_s}{x-a} +r_{s,0} +r_{s,1}(x-a) +\ldots , \end{aligned}$$
(2.11)

and the expansion of u(x) be as in (2.2). The solution is meromorphic if the residue of the right-hand side of (2.9) vanishes:

$$\begin{aligned} 2\mathop {\hbox {res}}\limits _{x=a} \xi _{s+1}' =\dot{r}_s +2r_{s,1} -2u_0 r_s =0. \end{aligned}$$
(2.12)

At \(s=0\), we have \(\hbox {res}_{x=a}\xi _1'=0\) from (2.10). We are going to prove (2.12) by induction in s. Assume that (2.12) holds for some s, then it is easy to see that the condition (2.3) implies that it holds for \(s+1\). Indeed, substituting the expansion (2.11) into the equation and equating the coefficients of \((x-a)^{-2}\), \((x-a)^{-1}\) and \((x-a)^0\) to zero, we get the conditions

$$\begin{aligned} \left\{ \begin{array}{l} 2r_{s+1}=-r_s\dot{a} -2r_{s,0}, \\ \\ \dot{r}_s +2r_{s,1} -2u_0 r_s =0, \\ \\ 2r_{s+1, 1}=\dot{r}_{s,0} -r_{s,1}\dot{a} -2u_0 r_{s,0}-2u_1 r_s. \end{array} \right. \end{aligned}$$
(2.13)

Substituting them into the right-hand side of (2.12) at \(s\rightarrow s+1\), we have:

$$\begin{aligned}{} & {} 2\mathop {\hbox {res}}\limits _{x=a} \xi _{s+2}' =\dot{r}_{s+1} +2r_{s+1,1} -2u_0 r_{s+1} \\{} & {} \begin{array}{c} \quad =(-\frac{1}{2}\dot{r}_s \dot{a} -\frac{1}{2} r_s \ddot{a} -\dot{r}_{s,0}) +(\dot{r}_{s,0} -r_{s,1}\dot{a} -2u_0 r_{s,0} -2u_1 r_s) +(u_0r_s\dot{a} +2u_0 r_{s,0}), \end{array} \end{aligned}$$

where we have used the first and the third equations in (2.13). After cancellations, we get:

$$\begin{aligned} 2\mathop {\hbox {res}}\limits _{x=a} \xi _{s+2}' =-\frac{1}{2} r_s (\ddot{a} +4u_1) -\frac{1}{2} \dot{a} (\dot{r}_s +2r_{s,1} -2u_0 r_s)=0. \end{aligned}$$

(The second term vanishes by the induction assumption, and the first one is zero due to the condition (2.3).) \(\square \)

2.2 The CKP case

Consider the linear equation

$$\begin{aligned} (\partial _t -\partial _x^3 -6u\partial _x -3u')\psi =0, \end{aligned}$$
(2.14)

which is one of the auxiliary linear problems for the CKP equation.

Proposition 2.3

Suppose that u(x) in (2.14) has a pole at \(x=a\) and equation (2.14) has a meromorphic solution, then the condition

$$\begin{aligned} \dot{a} + 6u_0=0 \end{aligned}$$
(2.15)

holds, where \(u_0\) is the coefficient in the Laurent expansion

$$\begin{aligned} u(x)=-\frac{1}{2(x-a)^2} +u_0 +u_1 (x-a) + \ldots .\nonumber \\ \end{aligned}$$
(2.16)

Proof

We have the expansion near the pole at \(x=a\):

$$\begin{aligned} \psi (x)= \frac{\alpha }{x-a} +\beta +\gamma (x-a) +\delta (x-a)^2 + \ldots .\nonumber \\ \end{aligned}$$
(2.17)

Substituting the expansions in the left-hand side of (2.14), we see that the highest (fourth-order)-order poles cancel identically. The necessary condition of cancellation of the third-order poles is \(\beta =0\). Equating the coefficients in front of \((x-a)^{-2}\), \((x-a)^{-1}\) and \((x-a)^0\) to zero, we get the conditions

$$\begin{aligned} \left\{ \begin{array}{l} \alpha \dot{a} +6\alpha u_0 =0, \\ \\ \dot{\alpha }+3\alpha u_1 +3\delta =0, \\ \\ \gamma \dot{a} +6\gamma u_0 =0. \end{array}\right. \end{aligned}$$
(2.18)

The first and the third equations are equivalent and lead to the necessary condition (2.15). \(\square \)

Let u(x) be the elliptic function

$$\begin{aligned} u(x)=-\frac{1}{2}\sum _i \wp (x-x_i), \end{aligned}$$
(2.19)

then the expansion (2.16) near the pole at \(x=x_i\) holds true with \(u_0=-\frac{1}{2}\sum _{j\ne i}\wp (x_i-x_j),\) so the conditions (2.15) for each \(x_i\) read

$$\begin{aligned} \dot{x}_i =3\sum _{j\ne i}\wp (x_i-x_j), \end{aligned}$$
(2.20)

which are the equations of motion for poles of elliptic solutions to the CKP equation derived in [15]. As shown in [15], they are obtained by restriction of the third flow of the Calogero–Moser system to the submanifold of turning points.

Next, we show that (2.15) is simultaneously a sufficient condition for local existence of meromorphic wave solutions to equation (2.14) with the expansion of the form

$$\begin{aligned} \psi = e^{kx +k^3t}\Bigl (1+\sum _{s\ge 1} \xi _s k^{-s}\Bigr ), \quad k\rightarrow \infty . \end{aligned}$$
(2.21)

Proposition 2.4

Suppose that condition (2.15) for the pole a of u(x) holds. Then, all wave solutions of equation (2.14) of the form (2.21) are meromorphic in a neighborhood of the point a with a simple pole at \(x=a\) and regular elsewhere in this neighborhood.

Proof

Substitution of the series into equation (2.14) gives the recurrence relation

$$\begin{aligned}{} & {} \dot{\xi }_s -3\xi _{s+2}' -3\xi _{s+1}'' -\xi _s''' -6u\xi _{s+1} -6u\xi _s' -3u'\xi _s =0,\nonumber \\{} & {} \quad s\ge -1, \quad \xi _{-1}\equiv 0, \; \xi _0\equiv 1. \end{aligned}$$
(2.22)

In particular, at \(s=-1\) we have

$$\begin{aligned} \xi _1'=-2u. \end{aligned}$$
(2.23)

It is convenient to represent equation (2.22) in the form

$$\begin{aligned} f_s'=\dot{\xi }_s -6u\xi _{s+1} -3u\xi _s', \quad f_s=3\xi _{s+2}+3\xi _{s+1}' +\xi _s'' +3u\xi _s. \end{aligned}$$
(2.24)

Let the Laurent expansion of \(\xi _s\) near the pole at \(x=a\) be

$$\begin{aligned} \xi _s = \frac{r_s}{x-a} +r_{s,0} +r_{s,1}(x-a) +r_{s,2}(x-a)^2 +\ldots , \end{aligned}$$
(2.25)

and the expansion of u(x) be as in (2.16). The solution is meromorphic if the residue of the right-hand side of (2.24) vanishes:

$$\begin{aligned} \mathop {\hbox {res}}\limits _{x=a} f_{s}' =\dot{r}_s +3r_{s+1,1} +3r_{s,2}-6u_0 r_{s+1}+3u_1r_s =0. \end{aligned}$$
(2.26)

At \(s=-1\), we have \(\begin{array}{l}\textrm{res}_{x=a}f_{-1}'=0\end{array}\) from (2.23). We are going to prove (2.26) by induction in s. Assume that (2.26) holds for some s, then it is easy to see that the condition (2.15) implies that it holds for \(s+1\). Indeed, substituting the expansion (2.25) into the equation and equating the coefficients of \((x-a)^{-3}\), \((x-a)^{-2}\), \((x-a)^{-1}\) and \((x-a)^0\) to zero, we get the conditions

$$\begin{aligned} \left\{ \begin{array}{l} r_{s+1}+r_{s,0}=0, \\ \\ 3r_{s+2}+r_s\dot{a} +3r_{s+1,0}+6u_0r_s=0, \\ \\ \dot{r}_s +3r_{s+1,1} +3r_{s,2}-6u_0 r_{s+1}+3u_1r_s =0, \\ \\ \dot{r}_{s,0} -r_{s,1}\dot{a} -3r_{s+2, 1} -3r_{s+1,2} -6u_0 r_{s+1,0}-6u_0r_{s,1}-3u_1 r_{s+1}=0. \end{array} \right. \nonumber \\ \end{aligned}$$
(2.27)

Substituting them into the right-hand side of (2.26) at \(s\rightarrow s+1\), we have:

$$\begin{aligned} \mathop {\hbox {res}}\limits _{x=a} f_{s+1}'= & {} \dot{r}_{s+1} +3r_{s+2,1} + 3r_{s+1,2}-6u_0 r_{s+2}+3u_1r_{s+1}\\= & {} (2u_0r_s -r_{s,1})(\dot{a}+6u_0)-2u_0(3r_{s+2}\\{} & {} + r_s\dot{a} +3r_{s+1,0}+6u_0r_s)=0. \end{aligned}$$

(The second term vanishes because of the second equation in (2.27), and the first one is zero due to the condition (2.15).) \(\square \)

2.3 The BKP case

Consider the linear equation

$$\begin{aligned} (\partial _t -\partial _x^3 -6u\partial _x)\psi =0, \end{aligned}$$
(2.28)

which is one of the auxiliary linear problems for the BKP equation.

Proposition 2.5

Suppose that u(x) in (2.28) has a pole at \(x=a\) and equation (2.28) has a meromorphic solution, then the condition

$$\begin{aligned} \ddot{a} +6\dot{u}_0 -12u_1(\dot{a} +6u_0) -36 u_3=0 \end{aligned}$$
(2.29)

holds, where \(u_0, u_1, u_3\) are coefficients in the Laurent expansion

$$\begin{aligned} u(x)=-\frac{1}{(x-a)^2} +u_0 +u_1 (x-a) + u_2 (x-a)^2 +u_3 (x-a)^3 +\ldots .\nonumber \\ \end{aligned}$$
(2.30)

Proof

We have the expansion near the pole at \(x=a\):

$$\begin{aligned} \psi (x)= \frac{\alpha }{x-a} +\beta +\gamma (x-a) +\delta (x-a)^2 + \varepsilon (x-a)^3+ \mu (x-a)^4 +\ldots .\nonumber \\ \end{aligned}$$
(2.31)

Substituting the expansions in the left-hand side of (2.28), we see that possible fourth- and third-order poles cancel identically. Equating the coefficients in front of \((x-a)^{-2}\), \((x-a)^{-1}\), \((x-a)^0\) and \((x-a)\) to zero, we get the conditions

$$\begin{aligned} \left\{ \begin{array}{l} \alpha \dot{a} +6\alpha u_0 +6\gamma =0, \\ \\ \dot{\alpha }+6\alpha u_1 +12\delta =0, \\ \\ \dot{\beta }-\gamma \dot{a} -6\gamma u_0 +6\alpha u_2 +12\varepsilon =0, \\ \\ \dot{\gamma }-2\delta \dot{a} -12\delta u_0 -6\gamma u_1 +6\alpha u_3 =0. \end{array}\right. \end{aligned}$$
(2.32)

Note that the terms with the coefficient \(\mu \) in (2.31) which might enter the last condition actually cancel. Taking t-derivative of the first equation, we have

$$\begin{aligned} \ddot{a} +6\dot{u}_0 +6\frac{\dot{\gamma }}{\alpha }-6\frac{\gamma \dot{\alpha }}{\alpha ^2}=0. \end{aligned}$$

Plugging here \(\dot{\alpha }\) from the second equation and \(\dot{\gamma }\) from the fourth one, we obtain the necessary condition (2.29). \(\square \)

One can see that this condition encodes equations of motion for the many-body system obtained in [14] as a dynamical system for motion of poles of elliptic solutions to the BKP equation. Indeed, let u(x) be the doubly-periodic meromorphic function

$$\begin{aligned} u(x)=-\sum _i \wp (x-x_i), \end{aligned}$$
(2.33)

then the expansion (2.30) near the pole at \(x=x_i\) holds true with

$$\begin{aligned} u_0=-\sum _{j\ne i}\wp (x_i-x_j), \quad u_1=-\sum _{j\ne i}\wp ' (x_i-x_j), \quad u_3=-\frac{1}{6}\sum _{j\ne i}\wp ''' (x_i-x_j) \end{aligned}$$

and \({ \dot{u}_0 =-\sum _{j\ne i}(\dot{x}_i-\dot{x}_j)\wp '(x_i-x_j).} \) Therefore, the conditions (2.29) for each \(x_i\) give the equations of motion derived in [14]:

$$\begin{aligned} \ddot{x}_i+6 \sum _{j\ne i}(\dot{x}_i+\dot{x}_j)\wp '(x_i-x_j)- 72 \sum _{j\ne k\ne i} \wp (x_i-x_j)\wp '(x_i-x_k)=0. \nonumber \\ \end{aligned}$$
(2.34)

(The identity \(\wp '''(x)=12\wp (x)\wp '(x)\) is used.) As shown in [21], the same equations can be obtained by restriction of the third flow of the elliptic Calogero–Moser system to the subspace of the phase space in which 2N particles stick together in pairs in such a way that the two particles in each pair are in one and the same point. This configuration is immediately destroyed by the second flow of the Calogero–Moser system but is preserved by the third one, and coordinates of the pairs are subject to equations (2.34).

Next, we will show that (2.29) is simultaneously a sufficient condition for local existence of meromorphic wave solutions to equation (2.28) of the form

$$\begin{aligned} \psi (x) = e^{kx +k^3t}\Bigl (1+\sum _{s\ge 1} \xi _s k^{-s}\Bigr ), \quad k\rightarrow \infty . \end{aligned}$$
(2.35)

Proposition 2.6

Suppose that condition (2.29) for the pole of u(x) holds. Then, all wave solutions of equation (2.28) of the form (2.35) are meromorphic in a neighborhood of the point a with a simple pole at \(x=a\) and regular elsewhere in this neighborhood.

Proof

Substitution of the series into the equation (2.28) gives the recurrence relation

$$\begin{aligned} \dot{\xi }_s -3\xi _{s+2}' -3\xi _{s+1}'' -\xi _s''' -6u\xi _{s+1} -6u\xi _s' =0, \quad s\ge -1, \quad \xi _{-1}\equiv 0, \; \xi _0\equiv 1.\nonumber \\ \end{aligned}$$
(2.36)

In particular, at \(s=-1\) we have

$$\begin{aligned} \xi _1'=-2u. \end{aligned}$$
(2.37)

It is convenient to represent equation (2.36) in the form

$$\begin{aligned} g_s'=\dot{\xi }_s -6u\xi _{s+1} -6u\xi _s', \quad g_s=3\xi _{s+2}+3\xi _{s+1}' +\xi _s''. \end{aligned}$$
(2.38)

Let the Laurent expansion of \(\xi _s\) near the pole at \(x=a\) be

$$\begin{aligned} \xi _s = \frac{r_s}{x-a} +r_{s,0} +r_{s,1}(x-a) + r_{s,2}(x-a)^2 +r_{s,3}(x-a)^3 +\ldots , \end{aligned}$$
(2.39)

and the expansion of u(x) be as in (2.30). The solution is meromorphic if the residue of the right-hand side of (2.38) vanishes:

$$\begin{aligned} \mathop {\hbox {res}}\limits _{x=a} g_{s}' =\dot{r}_s +6r_{s+1,1} +12r_{s,2}-6u_0 r_{s+1}+6u_1r_s =0. \end{aligned}$$
(2.40)

At \(s=-1\), we have \(\displaystyle {\mathop {\hbox {res}}\limits _{x=a}g_{-1}'=0}\) from (2.37). As before, we will prove (2.40) by induction in s. Assume that (2.40) holds for some s, then it is easy to see that the condition (2.29) implies that it holds for \(s+1\). Indeed, substituting the expansion (2.39) into the equation and equating the coefficients of \((x-a)^{-2}\), \((x-a)^{-1}\) \((x-a)^{0}\) and \((x-a)\) to zero, we get the conditions

$$\begin{aligned} \left\{ \begin{array}{l} 3r_{s+2}+r_s\dot{a} +6r_{s+1,0}+6r_{s,1}+6u_0r_s=0, \\ \\ \dot{r}_s +6r_{s+1,1} +12r_{s,2}-6u_0 r_{s+1}+6u_1r_s =0, \\ \\ \dot{r}_{s,0} -r_{s,1}\dot{a} -3r_{s+2, 1} +12r_{s,3} -6u_0 r_{s+1,0}-6u_0r_{s,1}-6u_1 r_{s+1}+6u_2r_s=0, \\ \\ \begin{array}{l} \dot{r}_{s,1} -2r_{s,2}\dot{a} -6r_{s+2, 2} -12r_{s+1,3} -6u_0 r_{s+1,1}-12u_0r_{s,2} \\ \\ -6u_1 r_{s+1,0}-6u_1r_{s,1}-6u_2r_{s+1}+6u_3r_s=0.\end{array} \end{array} \right. \nonumber \\ \end{aligned}$$
(2.41)

Substituting them into the right-hand side of (2.40) at \(s\rightarrow s+1\), we have:

$$\begin{aligned}{} & {} 3\mathop {\hbox {res}}\limits _{x=a} g_{s+1}' = 3\dot{r}_{s+1} +18r_{s+2,1} +36r_{s+1,2}-18u_0 r_{s+2}+18u_1r_{s+1} \\{} & {} =-r_{s-1}(\ddot{a} +6\dot{u}_0 -12u_1(\dot{a}+6u_0)-36u_3)\\{} & {} \quad - \dot{a}(\dot{r}_{s-1}+12r_{s-1,2}+6r_{s,1}-6u_0r_s +6u_1r_{s-1}) \\{} & {} -6u_0(r_s\dot{a} +3r_{s+2}+6r_{s+1,0}+6r_{s,1}+6u_0r_s ) -6u_1(r_{s-1}\dot{a} +3r_{s+1}\\{} & {} +6r_{s,0}+6r_{s-1,1}+6u_0r_{s-1} )=0. \end{aligned}$$

(The last two terms vanish because of the first equation in (2.41), the second term vanishes due to the induction assumption, and the first one is zero due to the condition (2.29).) \(\square \)

3 Differential–difference equations

3.1 The Toda lattice case

The 2D Toda lattice equation is the compatibility condition for the linear differential–difference equations

$$\begin{aligned} \partial _{t_1} \psi (x)= & {} \psi (x+\eta ) +b(x)\psi (x), \end{aligned}$$
(3.1)
$$\begin{aligned} \partial _{\bar{t}_1}\psi (x)= & {} v(x)\psi (x-\eta ), \end{aligned}$$
(3.2)

where \(\eta \) is a parameter (the lattice spacing). Suppose that b(x) and v(x) are meromorphic functions; let us investigate when these equations have meromorphic solutions in x.

3.1.1 The linear problem with respect to \(t_1\)

First we consider the linear problem (3.1):

$$\begin{aligned} \partial _{t} \psi (x)=\psi (x+\eta ) +b(x)\psi (x). \end{aligned}$$
(3.3)

Let b(x) have first-order poles at \(x=a\) and \(x=a-\eta \):

$$\begin{aligned} b(x)=\left\{ \begin{array}{l} \displaystyle { \frac{\nu }{x-a}\, +\, \mu _0 \, + \, O(x-a), \quad x\rightarrow a} \\ \\ \displaystyle {-\frac{\nu }{x-a+\eta }+\mu _1 +O(x-a+\eta ), \quad x\rightarrow a-\eta .} \end{array} \right. \end{aligned}$$
(3.4)

Proposition 3.1

Suppose that b(x) in (3.3) has poles at \(x=a\) and \(x=a-\eta \) with expansions near the poles of the form (3.4). If equation (3.3) has a meromorphic solution with a pole at the point a regular at \(a\pm \eta \), then the condition

$$\begin{aligned} \ddot{a} -\dot{a} (\mu _0 +\mu _1)=0 \end{aligned}$$
(3.5)

holds.

Proof

Let the expansion of \(\psi (x)\) around the point a be of the form

$$\begin{aligned} \psi (x)=\frac{\alpha }{x-a} +\beta +O(x-a), \end{aligned}$$
(3.6)

then

$$\begin{aligned} \partial _t \psi (x)=\frac{\alpha \dot{a}}{(x-a)^2}+\frac{\dot{\alpha }}{x-a}+ O(1). \end{aligned}$$

Substituting the expansions around the point a in the equation, we write:

$$\begin{aligned}{} & {} \frac{\alpha \dot{a}}{(x-a)^2}+\frac{\dot{\alpha }}{x-a}+ O(1) =\left( \frac{\nu }{x-a}\, +\, \mu _0 +\ldots \right) \\{} & {} \quad \left( \frac{\alpha }{x-a} +\beta +\ldots \right) . \end{aligned}$$

Equating the coefficients in front of the poles, we obtain the conditions

$$\begin{aligned} \left\{ \begin{array}{l} \nu =\dot{a}, \\ \\ \dot{\alpha }=\nu \beta +\mu _0 \alpha . \end{array} \right. \end{aligned}$$
(3.7)

Around the point \(a-\eta \), we have:

$$\begin{aligned} \begin{array}{c} \partial _t \psi (a-\eta ) +O(x-a+\eta ) \\ \displaystyle {=\frac{\alpha }{x-a+\eta } +\beta -\Bigl (\frac{\nu }{x-a+\eta } +\mu _1 +\ldots \Bigr ) \Bigl (\psi (a-\eta ) +(x-a+\eta )\psi '(a-\eta )+\ldots \Bigr ).} \end{array} \end{aligned}$$

Equating the coefficients in front of the terms or order \((x-a+\eta )^{-1}\) and \((x-a+\eta )^{0}\), we obtain the conditions

$$\begin{aligned} \left\{ \begin{array}{l} \alpha =\nu \psi (a-\eta ), \\ \\ \partial _t \psi (a-\eta )=\beta -\mu _1 \psi (a-\eta )-\nu \psi '(a-\eta ). \end{array} \right. \end{aligned}$$
(3.8)

Taking the time derivative of the first equation in (3.8) and using (3.7), we get

$$\begin{aligned} \dot{\alpha }= \ddot{a} \psi (a-\eta )+\dot{a} \dot{\psi }(a-\eta ), \end{aligned}$$
(3.9)

where

$$\begin{aligned} \dot{\psi }(a-\eta )=\partial _t \psi (a-\eta )+\dot{a} \psi '(a-\eta ) \end{aligned}$$
(3.10)

is the full time derivative of \(\psi (a-\eta )\). Now, combining equations (3.7)–(3.10), we arrive at the condition (3.5). \(\square \)

Suppose now that b(x) is an elliptic function of x having 2N first-order poles in the fundamental domain at some points \(x_j\) and at the points \(x_j-\eta \), then it must have the form

$$\begin{aligned} b(x)=\sum _j \dot{x}_j \Bigl (\zeta (x-x_j)-\zeta (x-x_j+\eta )\Bigr ). \end{aligned}$$
(3.11)

Let \(a=x_i\) for some i, then the coefficients \(\mu _0\), \(\mu _1\) in (3.4) are

$$\begin{aligned} \begin{array}{l} \displaystyle { \mu _0 =\sum _{k\ne i} \dot{x}_k \zeta (x_i-x_k)-\sum _k \dot{x}_k \zeta (x_i-x_k +\eta ),} \\ \\ \displaystyle { \mu _1 =\sum _{k\ne i} \dot{x}_k \zeta (x_i-x_k)-\sum _k \dot{x}_k \zeta (x_i-x_k -\eta )} \end{array} \end{aligned}$$
(3.12)

and (3.5) is equivalent to the equation of motion

$$\begin{aligned} \ddot{x}_i +\sum _{k\ne i}\dot{x}_i \dot{x}_k \Bigl ( \zeta (x_i-x_k+\eta )+\zeta (x_i-x_k-\eta )-2\zeta (x_i-x_k)\Bigr )=0 \end{aligned}$$
(3.13)

of the Ruijsenaars–Schneider model.

Let us show that (3.5) is a sufficient condition for existence of a meromorphic wave solution to equation (3.3) depending on a spectral parameter k with a pole at \(x=a\) and regular at the points \(x=a\pm \eta \). This solution can be found in the form

$$\begin{aligned} \psi (x)= k^{x/\eta }e^{kt} \Bigl (1+ \sum _{s\ge 1} \xi _s(x)k^{-s}\Bigr ), \quad k\rightarrow \infty . \end{aligned}$$
(3.14)

Proposition 3.2

Suppose that b(x) in (3.3) has poles at \(x=a\) and \(x=a-\eta \) with expansions near the poles of the form (3.4) and condition (3.5) for the pole of b(x) holds. Then, all wave solutions of equation (3.3) of the form (3.14) are meromorphic in a neighborhood of the point a with a simple pole at \(x=a\) and regular at \(x=a\pm \eta \).

Proof

Substituting the expansions into the equation, we obtain the recurrence relation for the coefficients \(\xi _s\):

$$\begin{aligned} \xi _{s+1}(x)-\xi _{s+1}(x+\eta )=b(x)\xi _s(x)-\dot{\xi }_s (x), \quad s\ge 0 \end{aligned}$$
(3.15)

. (It is convenient to put \(\xi _0=1\).) In particular,

$$\begin{aligned} \xi _{1}(x)-\xi _{1}(x+\eta )=b(x). \end{aligned}$$
(3.16)

Let the Laurent expansion of \(\xi _s\) near the point a be

$$\begin{aligned} \xi _s(x)=\frac{r_s}{x-a} +r_{s,0}+r_{s,1}(x-a) +\ldots \,, \quad x\rightarrow a. \end{aligned}$$
(3.17)

Substituting this into (3.15) with \(x\rightarrow a\) and equating the coefficients at the poles, we get the recurrence relation

$$\begin{aligned} r_{s+1}=\dot{a} r_{s,0} +\mu _0 r_s -\dot{r}_s. \end{aligned}$$
(3.18)

Expanding (3.15) near the point \(x\rightarrow a-\eta \) and equating the coefficients in front of \((x-a+\eta )^{-1}\) and \((x-a+\eta )^{0}\), we get

$$\begin{aligned} \left\{ \begin{array}{l} r_{s+1}=\dot{a} \xi _s(a-\eta ), \\ \\ r_{s+1, 0}-\xi _{s+1}(a-\eta )=\mu _1 \xi _s(a-\eta )+\dot{a} \xi _s'(a-\eta ) +\partial _t \xi _s (a-\eta ). \end{array} \right. \end{aligned}$$
(3.19)

The meromorphic wave solution with the required properties exists if the sum of residues of the right-hand side of (3.15) at the points \(x=a\) and \(x=a-\eta \) is equal to zero for all \(s\ge 0\). This sum of residues is given by

$$\begin{aligned} R_s= \dot{a} r_{s,0} +\mu _0 r_s -\dot{r}_s -\dot{a} \xi _s(a-\eta ). \end{aligned}$$

We are going to prove that \(R_s=0\) for all \(s\ge 0\) by induction. This is obviously true for \(s=0\) due to equation (3.16). Assume that \(R_s=0\) for some s, this is our induction assumption. Using (3.19), we find:

$$\begin{aligned} R_{s+1}= & {} \dot{a} \Bigl (r_{s+1,0}-\xi _{s+1}(a-\eta )\Bigr ) +\mu _0 r_{s+1} -\dot{r}_{s+1} \\= & {} \Bigl (\dot{a}(\mu _0+\mu _1)-\ddot{a}\Bigr )\xi _s(a-\eta )-\partial _t R_s=0 \end{aligned}$$

because the first term vanishes due to the condition (3.5) and the second one vanishes due to the induction assumption. \(\square \)

3.1.2 The linear problem with respect to \(\bar{t}_1\)

Consider now the linear problem (3.2):

$$\begin{aligned} \partial _{t}\psi (x) =v(x)\psi (x-\eta ). \end{aligned}$$
(3.20)

Suppose that the function v(x) has a second-order pole at \(x=a\) with the expansion

$$\begin{aligned} v(x)=\frac{\nu }{(x-a)^2} +\frac{\mu }{x-a}+O(1), \quad x\rightarrow a. \end{aligned}$$
(3.21)

We also assume that v(x) has a zero at \(x=a-\eta \): \( v(a-\eta )=0. \)

Proposition 3.3

Suppose that v(x) in (3.20) has a second-order pole at \(x=a\) with the expansion (3.21) and \(v(a-\eta )=0.\) If equation (3.20) has a meromorphic solution with a simple pole at the point a regular at \(a\pm \eta \), then the condition

$$\begin{aligned} \nu \ddot{a}+\mu \dot{a}^2 -\dot{\nu }\dot{a}=0 \end{aligned}$$
(3.22)

holds.

Proof

For \(\psi (x)\) near the point a, we have

$$\begin{aligned} \psi (x)=\frac{\alpha }{x-a}+O(1), \quad x\rightarrow a. \end{aligned}$$
(3.23)

Substituting the expansions around the point a in the equation (3.20), we write:

$$\begin{aligned} \frac{\alpha \dot{a}}{(x-a)^2}+\frac{\dot{\alpha }}{x-a} =\Bigl (\frac{\nu }{(x-a)^2} +\frac{\mu }{x-a}+O(1)\Bigr ) \Bigl (\psi (a-\eta )+(x-a)\psi '(a-\eta )+\ldots \Bigr ). \end{aligned}$$

Equating the coefficients in front of the poles, we obtain the conditions

$$\begin{aligned} \left\{ \begin{array}{l} \alpha \dot{a} =\nu \psi (a-\eta ), \\ \\ \dot{\alpha }=\nu \psi '(a-\eta ) +\mu \psi (a-\eta ). \end{array} \right. \end{aligned}$$
(3.24)

At \(x=a-\eta \), equation (3.20) gives \( \partial _t \psi (a-\eta )=0 \) due to the fact that \(v(a-\eta )=0\), so the full time derivative of \(\psi (a-\eta )\) is

$$\begin{aligned} \dot{\psi }(a-\eta )=\dot{a} \psi '(a-\eta ). \end{aligned}$$
(3.25)

Combining the time derivative of the first equation in (3.24) with the second one and using (3.25), we obtain the condition (3.22). \(\square \)

Suppose that v(x) is an elliptic function of the form

$$\begin{aligned} v(x)=\prod _{j=1}^N \frac{\sigma (x-x_j+\eta ) \sigma (x-x_j-\eta )}{\sigma ^2 (x-x_j)}, \end{aligned}$$
(3.26)

then, setting \(a=x_i\), we have:

$$\begin{aligned} \nu= & {} -\sigma ^2(\eta )\prod _{j\ne i} \frac{\sigma (x_i-x_j+\eta )\sigma (x_i-x_j-\eta )}{\sigma ^2 (x_i-x_j)}, \end{aligned}$$
(3.27)
$$\begin{aligned} \mu= & {} \nu \sum _{k\ne i}\Bigl ( \zeta (x_i-x_k+\eta )+\zeta (x_i-x_k-\eta )-2\zeta (x_i-x_k)\Bigr ), \end{aligned}$$
(3.28)

and the condition (3.22) is equivalent to the equation of motion which is the same as (3.13).

Similarly to the previous case, equation (3.22) is a sufficient condition for local existence of a meromorphic wave solution to equation (3.20) depending on a spectral parameter k with a pole at \(x=a\) and regular at the points \(x=a\pm \eta \). However, the arguments need some modifications.

Proposition 3.4

Suppose that v(x) in (3.20) has a second-order pole at \(x=a\) with expansion near the pole of the form (3.21) and \(v(a\pm \eta )=0\). Let v(x) be of the form

$$\begin{aligned} v(x)=e^{\varphi (x)-\varphi (x-\eta )}, \end{aligned}$$
(3.29)

where \(\varphi (x)\) has logarithmic singularities at \(x=a\) and \(x=a-\eta \), so that

$$\begin{aligned} \dot{\varphi }(x)=\left\{ \begin{array}{l} \displaystyle {\frac{\dot{a}}{x-a} +\varphi _0 +O(x-a)}, \quad x\rightarrow a, \\ \\ \displaystyle {-\frac{\dot{a}}{x-a+\eta } +\varphi _1 +O(x-a+\eta )}, \quad x\rightarrow a-\eta . \end{array} \right. \end{aligned}$$
(3.30)

Suppose that condition (3.22) for the pole of v(x) holds. Then, all wave solutions of equation (3.20) of the form

$$\begin{aligned} \psi (x) = k^{-x/\eta }e^{kt +\varphi (x)} \Bigl (1+ \sum _{s\ge 1} \xi _s(x)k^{-s}\Bigr ), \quad k\rightarrow \infty \end{aligned}$$
(3.31)

are meromorphic in a neighborhood of the point a with a simple pole at \(x=a\) and regular at \(x=a\pm \eta \).

Proof

Expanding the equality \(\partial _t \log v(x)=\dot{\varphi }(x)-\dot{\varphi }(x-\eta )\) near the point \(x=a\) with the help of (3.21) and comparing the coefficients, we get:

$$\begin{aligned} \varphi _0-\varphi _1 =\frac{\dot{\nu }}{\nu }-\frac{\mu }{\nu }\, \dot{a}. \end{aligned}$$
(3.32)

Substituting the expansions into the equation, we obtain the recurrence relation for the coefficients \(\xi _s\):

$$\begin{aligned} \xi _{s+1}(x-\eta )-\xi _{s+1}(x)= \dot{\varphi }(x)\xi _s(x)+\dot{\xi }_s (x), \quad s\ge 0. \end{aligned}$$
(3.33)

In particular,

$$\begin{aligned} \xi _{1}(x-\eta )-\xi _{1}(x)=\dot{\varphi }(x). \end{aligned}$$
(3.34)

The factor \(e^{\varphi (x)}\) in (3.31) has a pole at \(x=a\) and a zero at \(x=a-\eta \). This means that the coefficients \(\xi _s(x)\) are regular at \(x=a\) and may have a pole at \(x=a-\eta \). Let the Laurent expansion of \(\xi _s\) near the point \(a-\eta \) be of the form

$$\begin{aligned} \xi _s(x)=\frac{r_s}{x-a+\eta } +r_{s,0}+r_{s,1}(x-a+\eta ) +\ldots \,, \quad x\rightarrow a-\eta . \end{aligned}$$
(3.35)

The meromorphic wave solution with the required properties exists if the sum of residues of the right-hand side of (3.33) at the points \(x=a\) and \(x=a-\eta \) is equal to zero for all \(s\ge 0\). This can be proved by induction in the same way as for the linear problem (3.3). \(\square \)

3.2 The case of the Toda lattice of type C

The Toda lattice of type C was introduced in the paper [22] by the authors. The auxiliary linear problem has the form

$$\begin{aligned} \partial _t \psi (x)=\psi (x+\eta )+\frac{1}{2}\, \dot{\varphi }(x)\psi (x)+ v(x)\psi (x-\eta ), \end{aligned}$$
(3.36)

where \( v(x)=e^{\varphi (x)-\varphi (x-\eta )}. \) We assume that \(\dot{\varphi }(x)\) is expanded near the points \(x=a\) and \(x=a-\eta \) as in (3.30) and

$$\begin{aligned} v(x)=\frac{\nu }{(x-a)^2}+\frac{\mu }{x-a}+O(1), \quad x\rightarrow a. \end{aligned}$$
(3.37)

We also note that the relation (3.32) holds.

Proposition 3.5

Suppose that \(v(x)=e^{\varphi (x)-\varphi (x-\eta )}\) in (3.36) has a second-order pole at \(x=a\) with the expansion (3.37), \(v(a-\eta )=0\) and \(\dot{\varphi }(x)\) is expanded near the points \(x=a\) and \(x=a-\eta \) as in (3.30). If equation (3.36) has a meromorphic solution with a simple pole at the point a regular at \(a\pm \eta \), then the condition

$$\begin{aligned} \dot{a}^2=-4\nu \end{aligned}$$
(3.38)

holds.

Proof

Let the function \(\psi (x)\) have a pole at \(x=a\) with the expansion near this point of the form (3.6). Substituting the expansions near \(x=a\) into the equation and equating the coefficients in front of the highest poles, we get the condition

$$\begin{aligned} \alpha \dot{a} + 2\nu \psi (a-\eta )=0. \end{aligned}$$
(3.39)

The same procedure at the pole at \(x=a-\eta \) leads to the condition

$$\begin{aligned} 2\alpha =\dot{a} \psi (a-\eta ). \end{aligned}$$
(3.40)

Combining (3.39) and (3.40), we obtain the condition (3.38). \(\square \)

Suppose now that v(x) is the elliptic function (3.26), then, expanding it near the point \(a=x_i\), we see that \(\nu \) is given by (3.27). Equations (3.38) for any i are then equivalent to the equations of motion for poles of elliptic solutions to the Toda lattice of type C:

$$\begin{aligned} \dot{x}_i = 2\sigma (\eta )\prod _{j\ne i} \frac{(\sigma (x_i-x_j+\eta )\sigma (x_i-x_j-\eta ))^{1/2}}{\sigma (x_i-x_j)}. \end{aligned}$$
(3.41)

These equations were obtained in [22]. The properly taken limit \(\eta \rightarrow 0\) leads to equations (2.20).

Let us show that (3.38) is a sufficient condition for existence of a meromorphic wave solution to equation (3.36) depending on a spectral parameter k with a pole at \(x=a\) and regular at the points \(x=a\pm \eta \).

Proposition 3.6

Suppose that \(v(x)=e^{\varphi (x)-\varphi (x-\eta )}\) in (3.36) has a second-order pole at \(x=a\) with expansion near the pole of the form (3.37) and \(v(a-\eta )=0\). Assume also that the function \(\dot{\varphi }(x)\) has expansion of the form (3.30). If condition (3.22) for the pole of v(x) holds, then all wave solutions of equation (3.36) of the form

$$\begin{aligned} \psi (x)= k^{x/\eta }e^{kt} \Bigl (1+ \sum _{s\ge 1} \xi _s(x)k^{-s}\Bigr ), \quad k\rightarrow \infty \end{aligned}$$
(3.42)

are meromorphic in a neighborhood of the point a with a simple pole at \(x=a\) and regular at \(x=a\pm \eta \).

Proof

Substituting the expansions into the equation, we obtain the recurrence relation for the coefficients \(\xi _s\) in (3.42):

$$\begin{aligned} \xi _{s+1}(x)-\xi _{s+1}(x+\eta )= \frac{1}{2}\, \dot{\varphi }(x)\xi _s(x)-\dot{\xi }_s (x) -v(x)\xi _{s-1}(x-\eta ), \quad s\ge 0,\nonumber \\ \end{aligned}$$
(3.43)

where we set \(\xi _{-1}=0\), \(\xi _0=1\). In particular,

$$\begin{aligned} \xi _{1}(x)-\xi _{1}(x+\eta )=\frac{1}{2}\, \dot{\varphi }(x). \end{aligned}$$
(3.44)

Assuming the expansion of \(\xi _s(x)\) near the point a of the form (3.17), we get from (3.43) expanded near the this point:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{1}{2}\, r_s \dot{a} +\nu \xi _{s-1}(a-\eta )=0, \\ \\ r_{s+1}=\frac{1}{2}\, \dot{a} r_{s,0} +\frac{1}{2}\, \varphi _0 r_s - \dot{r}_s -\nu \xi _{s-1}'(a-\eta )-\mu \xi _{s-1}(a-\eta ). \end{array} \right. \end{aligned}$$
(3.45)

The expansion near the point \(x=a-\eta \) gives:

$$\begin{aligned} \left\{ \begin{array}{l} r_{s+1}=\frac{1}{2}\, \dot{a} \xi _s(a-\eta ), \\ \\ \xi _{s+1}(a-\eta )-r_{s+1,0}=\frac{1}{2}\, \varphi _1 \xi _s(a-\eta )- \frac{1}{2}\, \dot{a} \xi _{s}'(a-\eta )-\partial _t\xi _{s}(a-\eta ). \end{array} \right. \end{aligned}$$
(3.46)

Let

$$\begin{aligned} R_s=\frac{1}{2}\, \dot{a} (r_{s,0}-\xi _s(a-\eta ))+\frac{1}{2}\, \varphi _0 r_s -\dot{r}_s -\nu \xi _{s-1}'(a-\eta )-\mu \xi _{s-1}(a-\eta ) \end{aligned}$$

be sum of the residues at the points \(x=a\) and \(x=a-\eta \) which must be zero. At \(s=0\), this is true due to (3.44). Our induction assumption is that \(R_s=0\) for some s; let us show that this implies that \(R_{s+1}=0\). We have:

$$\begin{aligned} R_{s+1}=\frac{1}{2}\, \dot{a} (r_{s+1,0}-\xi _{s+1} (a-\eta ))+\frac{1}{2}\, \varphi _0 r_{s+1} -\dot{r}_{s+1} -\nu \xi _{s}'(a-\eta )-\mu \xi _{s}(a-\eta ). \end{aligned}$$

Substituting here the recurrence relations (3.45), (3.46), we obtain, after some calculations and cancellations:

$$\begin{aligned} R_{s+1}= \frac{1}{4}\Bigl ( (\varphi _0\! -\! \varphi _1)\dot{a} -4\mu -2\ddot{a}\Bigr ) \xi _s(a-\eta ) -\frac{1}{4}\, (\dot{a}^2+4\nu )\xi _s'(a-\eta )-\partial _t R_s. \end{aligned}$$

The last two terms are equal to zero by virtue of the induction assumption and the condition (3.38). As for the first term, we have:

$$\begin{aligned}{} & {} (\varphi _0\! -\! \varphi _1)\dot{a} -4\mu -2\ddot{a}= \Bigl (\frac{\dot{\nu }}{\nu }-\frac{\mu }{\nu }\, \dot{a} \Bigr )\dot{a} -4\mu - 2\ddot{a} \\{} & {} =(\dot{a}^2 +4\nu )\frac{\dot{\nu }-\mu \dot{a}}{\nu \dot{a}}- \frac{1}{\dot{a}}\, \partial _t(\dot{a}^2 +4\nu )=0 \end{aligned}$$

by virtue of the condition (3.38). Therefore, \(R_{s+1}=0\) and we have proved the existence of a meromorphic wave solution. \(\square \)

3.3 The case of the Toda lattice of type B

3.3.1 Existence of a meromorphic solution

The Toda lattice of type B was recently introduced by the authors in [20]. The linear equation for the first time flow has the form

$$\begin{aligned} \partial _t \psi (x)=v(x)(\psi (x+\eta )-\psi (x-\eta )). \end{aligned}$$
(3.47)

Assume that the function v(x) has a second-order pole at \(x=a\) with the expansion

$$\begin{aligned} v(x)=\frac{\nu }{(x-a)^2} +\frac{\mu }{x-a}+O(1), \quad x\rightarrow a. \end{aligned}$$
(3.48)

We also assume that v(x) has zeros at \(x=a-\eta \) and \(x=a+\eta \):

$$\begin{aligned} v(x)=\left\{ \begin{array}{l} (x-a-\eta )V^+(a)+O((x\! -\! a\! -\! \eta )^2), \quad x\rightarrow a+\eta , \\ \\ (x-a+\eta )V^-(a)+O((x\! -\! a\! +\! \eta )^2), \quad x\rightarrow a-\eta . \end{array}\right. \end{aligned}$$
(3.49)

Proposition 3.7

Suppose that v(x) in (3.47) has a second-order pole at \(x=a\) and zeros at \(x=a\pm \eta \) with the expansions (3.48), (3.49). If equation (3.47) has a meromorphic solution with a simple pole at the point a regular at \(a\pm \eta \), then the condition

$$\begin{aligned} \nu \ddot{a} +\mu \dot{a}^2 -\dot{\nu }\dot{a} +\nu ^2 (V^+(a)+V^-(a))=0 \end{aligned}$$
(3.50)

holds.

Proof

The principal part of the Laurent expansion of \(\psi (x)\) near the point \(x=a\) is

$$\begin{aligned} \psi (x)=\frac{\alpha }{x-a}+O(1), \quad x\rightarrow a. \end{aligned}$$
(3.51)

As \(x\rightarrow a\), we have from equation (3.47):

$$\begin{aligned}{} & {} \frac{\alpha \dot{a}}{(x-a)^2}+\frac{\dot{\alpha }}{x-a} =\Bigl (\frac{\nu }{(x-a)^2} \! +\! \frac{\mu }{x-a}\! +\! O(1)\Bigr ) \\{} & {} \Bigl (\psi (a\! +\! \eta )\! -\! \psi (a\! -\! \eta )\! + \!(x-a) (\psi '(a\! +\! \eta )\! -\! \psi '(a\! -\! \eta ))\! +\ldots \Bigr ). \end{aligned}$$

Equating the coefficients in front of the poles, we obtain the conditions

$$\begin{aligned} \left\{ \begin{array}{l} \alpha \dot{a} =\nu (\psi (a+\eta )-\psi (a-\eta )), \\ \\ \dot{\alpha }=\mu (\psi (a+\eta ) - \psi (a-\eta ))+\nu (\psi '(a+\eta ) -\psi '(a-\eta ) ). \end{array} \right. \end{aligned}$$
(3.52)

At \(x=a\pm \eta \), equation (3.47) gives:

$$\begin{aligned} \partial _t \psi (a\pm \eta )=\mp \alpha \, V^{\pm }(a). \end{aligned}$$
(3.53)

Therefore,

$$\begin{aligned} \dot{\psi }(a\pm \eta )=\mp \alpha \, V^{\pm }(a)+\dot{a} \psi '(a\pm \eta ). \end{aligned}$$
(3.54)

Taking the time derivative of the first equation in (3.52) and combining it with the other equations, we obtain the condition (3.50). \(\square \)

3.3.2 Dynamics of poles of elliptic solutions

Suppose that v(x) is an elliptic function of the form

$$\begin{aligned} v(x)=\prod _{j=1}^N \frac{\sigma (x-x_j+\eta )\sigma (x-x_j-\eta )}{\sigma ^2 (x-x_j)}, \end{aligned}$$
(3.55)

then, setting \(a=x_i\), we have:

$$\begin{aligned} \nu= & {} -\sigma ^2(\eta )\prod _{j\ne i} \frac{\sigma (x_i-x_j+\eta )\sigma (x_i-x_j-\eta )}{\sigma ^2 (x_i-x_j)}, \end{aligned}$$
(3.56)
$$\begin{aligned} \mu= & {} \nu \sum _{k\ne i}\Bigl ( \zeta (x_i-x_k+\eta )+\zeta (x_i-x_k-\eta )-2\zeta (x_i-x_k)\Bigr ), \end{aligned}$$
(3.57)
$$\begin{aligned} V^{\pm }(x_i)= & {} \pm \frac{\sigma (2\eta )}{\sigma ^2(\eta )} \prod _{j\ne i}\frac{\sigma (x_i-x_j\pm 2\eta )\sigma (x_i-x_j)}{\sigma ^2 (x_i-x_j\pm \eta )}, \end{aligned}$$
(3.58)

and the condition (3.50) is equivalent to the equation of motion

$$\begin{aligned} \begin{array}{c} \displaystyle { \ddot{x}_i +\sum _{k\ne i}\dot{x}_i \dot{x}_k \Bigl ( \zeta (x_i\! -\! x_k\! +\! \eta )+\zeta (x_i\! -\! x_k\! -\! \eta ) -2\zeta (x_i-x_k)\Bigr )} \\ \\ \displaystyle { -\sigma (2\eta )\left[ \prod _{j\ne i} \frac{\sigma (x_i\! -\! x_j\! + \! 2\eta ) \sigma (x_i\! -\! x_j-\eta )}{\sigma (x_i-x_j \! +\! \eta ) \sigma (x_i \! -\! x_j)}- \prod _{j\ne i} \frac{\sigma (x_i\! -\! x_j\! -\! 2\eta ) \sigma (x_i\! -\! x_j+\eta )}{\sigma (x_i\! -\! x_j\! -\! \eta ) \sigma (x_i\! -\! x_j)}\right] }=0. \end{array}\nonumber \\ \end{aligned}$$
(3.59)

As shown below, the properly taken limit \(\eta \rightarrow 0\) of this equation coincides with equation (2.34). In the rational limit, one should substitute \(\zeta (x)\rightarrow 1/x\), \(\sigma (x)\rightarrow x\).

In Appendix C, we show that the same equations can be obtained by restriction of the Ruijsenaars–Schneider dynamics with respect to the time flow \(\partial _{t_1}-\partial _{\bar{t}_1}\) of the system containing 2N particles to the half-dimensional subspace of the 4N-dimensional phase space corresponding to the configuration in which the particles stick together in pairs such that the distance between particles in each pair is equal to \(\eta \). This configuration is destroyed by the flow \(\partial _{t_1}+\partial _{\bar{t}_1}\) but is preserved by the flow \(\partial _{t_1}-\partial _{\bar{t}_1}\). We conjecture that it is preserved also by all higher flows \(\partial _{t_k}-\partial _{\bar{t}_k}\). The time evolution in \(t=t_1-\bar{t}_1\) of the pairs with coordinates \(x_i\) is given by equations (3.59).

3.3.3 The limit \(\eta \rightarrow 0\)

The “non-relativistic limit” \(\eta \rightarrow 0\) in (3.55) yields

$$\begin{aligned} v(x)=1+\eta ^2 u(x) + O(\eta ^4), \end{aligned}$$
(3.60)

where u(x) is given by (2.33). Then, the limit of the difference operator \(v(x)(e^{\eta \partial _x}-e^{-\eta \partial _x})\) is

$$\begin{aligned} v(x)(e^{\eta \partial _x}-e^{-\eta \partial _x})=2\eta \partial _x + \frac{\eta ^3}{3}\Bigl (\partial _x^3 +6u(x)\partial _x \Bigr ) +O(\eta ^5), \end{aligned}$$
(3.61)

i.e., in the next-to-leading order the differential operator \(\partial _x^3 +6u\partial _x\) participating in the linear problem for the BKP equation arises (see equation (2.28)). Let us pass to the variables

$$\begin{aligned} X=x+2\eta t, \quad T=\frac{\eta ^3}{3}\, t, \end{aligned}$$
(3.62)

then \(\partial _X =\partial _x\), \(\partial _T = \displaystyle {\frac{3}{\eta ^3}\, (\partial _t -2\eta \partial _x)}\) and in the limit \(\eta \rightarrow 0\) the linear problem (3.47) becomes \(\partial _T \psi =(\partial _X^3 -6u\partial _X)\psi \) which is the linear problem (2.28) for the BKP equation.

Taking the change of variables (3.62) into account, let us find the \(\eta \rightarrow 0\) limit of equation (3.59). We have:

$$\begin{aligned} \sigma (x-x_i)=\sigma \Bigl (X-\frac{6}{\eta ^2}\, T -x_i \Bigr )= \sigma (X-X_i), \end{aligned}$$

whence \(X_i= \displaystyle {\frac{6}{\eta ^2}T +x_i}\) and \(\displaystyle { \dot{x}_i =\partial _t x_i =-2\eta + \frac{\eta ^3}{3}\, \partial _T \! X_i} \). Expanding equation (3.59) in powers of \(\eta \), we obtain:

$$\begin{aligned}{} & {} \frac{\eta ^6}{9}\, \partial _T^2 \! X_i -4\eta ^2 \sum _{k\ne i} \Bigl (1\! -\! \frac{\eta ^2}{6}\, \partial _T \! X_i\Bigr )\\{} & {} \quad \Bigl (1\! -\! \frac{\eta ^2}{6}\, \partial _T \! X_k\Bigr ) \Bigr ( \eta ^2 \wp '(X_{ik})\! +\! \frac{\eta ^4}{12}\, \wp '''(X_{ik}) \! +\! O(\eta ^6)\Bigr ) \! -\! U_i=0, \end{aligned}$$

where \(X_{ik}\equiv X_i-X_k\) and

$$\begin{aligned}{} & {} U_i=\sigma (2\eta ) \left[ \prod _{j\ne i}\frac{\sigma (X_{ij}+2\eta )\sigma (X_{ij}-\eta )}{\sigma (X_{ij}+\eta )\sigma (X_{ij})}- \prod _{j\ne i}\frac{\sigma (X_{ij}-2\eta )\sigma (X_{ij}+\eta )}{\sigma (X_{ij}-\eta )\sigma (X_{ij})}\right] \\{} & {} =-4\eta ^4 \sum _{j\ne i}\wp '(X_{ij})+8\eta ^6 \sum _{j\ne i}\wp (X_{ij}) \sum _{l\ne i}\wp '(X_{il}) -\eta ^6 \sum _{j\ne i}\wp '''(X_{ij}) +O(\eta ^8). \end{aligned}$$

It is easy to see that the terms of order \(\eta ^4\) cancel in the equation and in the leading-order \(\eta ^6\) equation (2.34) arises.

3.3.4 Existence of meromorphic wave solutions

Similarly to the previous cases, equation (3.50) is a sufficient condition for local existence of a meromorphic wave solution to equation (3.47) depending on a spectral parameter k with a pole at \(x=a\) and regular at the points \(x=a\pm \eta \). However, the proof requires more sophisticated calculations than in the Toda lattice case. To proceed, we represent v(x) in the form

$$\begin{aligned} v(x)=\frac{\tau (x+\eta )\tau (x-\eta )}{\tau ^2(x)} \end{aligned}$$
(3.63)

which is motivated by the result of [20]. (\(\tau (x)\) is the tau-function of the Toda lattice of type B.) We assume that \(\tau (x)\) has a simple zero at the point a and that it is regular and nonzero in some neighborhood of this point including the points \(a\pm \eta \):

$$\begin{aligned} \tau (x)=(x-a)\rho (x-a), \end{aligned}$$
(3.64)

where the function \(\rho (x)\) is regular and nonzero at \(x=0\). It depends also on the time t. Then, the coefficients in (3.48), (3.49) are expressed as

$$\begin{aligned}{} & {} \nu =-\eta ^2 \frac{\rho (\eta )\rho (-\eta )}{\rho ^2(0)}, \quad \mu =\nu \left( \frac{\rho '(\eta )}{\rho (\eta )}+ \frac{\rho '(-\eta )}{\rho (-\eta )}-2 \frac{\rho '(0)}{\rho (0)}\right) , \end{aligned}$$
(3.65)
$$\begin{aligned}{} & {} V^{\pm }(a)=\pm \frac{2}{\eta }\, \frac{\rho (0)\rho (\pm 2\eta )}{\rho ^2(\pm \eta )}. \end{aligned}$$
(3.66)

It is also convenient to introduce the function

$$\begin{aligned} \varphi _+(x)=\log \frac{\tau (x-\eta )}{\tau (x)}. \end{aligned}$$
(3.67)

The function \(\dot{\varphi }_+ (x)\) has simple poles at the points \(x=a\) and \(x=a+\eta \) with the expansions

$$\begin{aligned} \dot{\varphi }_+ (x)=\left\{ \begin{array}{l} \displaystyle { \frac{\dot{a}}{x-a} +\varphi _0 +\ldots \,, \quad x\rightarrow a,} \\ \\ \displaystyle {-\frac{\dot{a}}{x\! -\! a\! -\! \eta } +\varphi _1 + \ldots \,, \quad x\rightarrow a+\eta ,} \end{array} \right. \end{aligned}$$
(3.68)

where

$$\begin{aligned} \begin{array}{l} \displaystyle { \varphi _0 = \dot{a} \left( \frac{1}{\eta }- \frac{\rho '(-\eta )}{\rho (-\eta )}+\frac{\rho '(0)}{\rho (0)}\right) +\frac{\dot{\rho }(-\eta )}{\rho (-\eta )}-\frac{\dot{\rho }(0)}{\rho (0)}}, \\ \\ \displaystyle { \varphi _1 = \dot{a} \left( \frac{1}{\eta }\, +\, \frac{\rho '(\eta )}{\rho (\eta )}\, -\, \frac{\rho '(0)}{\rho (0)}\right) -\frac{\dot{\rho }(\eta )}{\rho (\eta )}+\frac{\dot{\rho }(0)}{\rho (0)}}. \end{array} \end{aligned}$$
(3.69)

It is easy to check that

$$\begin{aligned} \varphi _0-\varphi _1=\frac{\dot{\nu }}{\nu }-\frac{\mu }{\nu }\, \dot{a}. \end{aligned}$$

Proposition 3.8

Assume that v(x) in (3.47) is represented in the form (3.63), \(\tau (x)\) has a simple zero at a point a and \(\tau (a+\eta )\tau (a-\eta )\ne 0\). If condition (3.50) holds, then all wave solutions of equation (3.47) of the form

$$\begin{aligned} \psi (x)= k^{x/\eta }e^{kt+\varphi _+(x)} \Bigl (1+ \sum _{s\ge 1} \xi _s(x)k^{-s}\Bigr ), \quad k\rightarrow \infty , \end{aligned}$$
(3.70)

where \(\varphi _+(x)\) is given by (3.67), are meromorphic in a neighborhood of the point a with a simple pole at \(x=a\) and regular at \(x=a\pm \eta \).

Proof

Substituting the series (3.70) into the equation (3.47), we obtain the recurrence relation for the coefficients \(\xi _s\):

$$\begin{aligned} \xi _{s+1}(x+\eta )-\xi _{s+1}(x)= \dot{\varphi }_+ (x)\xi _s(x)+\dot{\xi }_s (x) +v(x)v(x-\eta )\xi _{s-1}(x-\eta ), \quad s\ge 0,\nonumber \\ \end{aligned}$$
(3.71)

where we set \(\xi _{-1}=0\), \(\xi _0=1\). The factor \(e^{\varphi _+ (x)}\) in (3.70) has a pole at \(x=a\) and a zero at \(x=a+\eta \). This means that the coefficients \(\xi _s(x)\) are regular at \(x=a\) and may have a pole at \(x=a+\eta \). Let the Laurent expansion of \(\xi _s\) near the point \(a+\eta \) be

$$\begin{aligned} \xi _s(x)=\frac{r_s}{x-a-\eta } +r_{s,0}+O(x-a-\eta )\,, \quad x\rightarrow a+\eta . \end{aligned}$$
(3.72)

The meromorphic wave solution with the required properties exists if the sum of residues of the right-hand side of (3.71) at the points \(x=a\) and \(x=a+\eta \) is equal to zero for all \(s\ge 0\). This can be proved by induction in a similar way as before but the calculations are more involved.

To proceed, we need some properties of the coefficient function \(v(x)v(x-\eta )\) in (3.71). We have:

$$\begin{aligned} v(x)v(x-\eta )=\frac{\tau (x-2\eta )\tau (x+\eta )}{\tau (x-\eta ) \tau (x)}, \end{aligned}$$

whence

$$\begin{aligned}{} & {} v(x)v(x-\eta )=\frac{\nu V^+(a)}{x-a-\eta }+O(1), \quad x\rightarrow a+\eta , \qquad v(x)v(x-\eta )\Bigr |_{x=a-\eta }=0, \end{aligned}$$
(3.73)
$$\begin{aligned}{} & {} v(x)v(x-\eta )=\frac{\nu V^-(a)}{x-a}+\Omega \nu V^-(a)+O(x-a), \quad x\rightarrow a, \end{aligned}$$
(3.74)

where

$$\begin{aligned} \Omega = \frac{3}{2\eta }+\frac{\rho '(-2\eta )}{\rho (-2\eta )}- \frac{\rho '(-\eta )}{\rho (-\eta )}+\frac{\rho '(\eta )}{\rho (\eta )} -\frac{\rho '(0)}{\rho (0)}. \end{aligned}$$
(3.75)

Expanding equation (3.71) near the point \(x=a+\eta \), we get, equating the coefficients in front of the poles:

$$\begin{aligned} r_{s+1}=\dot{a} r_{s,0} -\varphi _1 r_s -\dot{r}_s -\nu V^+(a) \xi _{s-1}(a). \end{aligned}$$
(3.76)

Similarly, expanding equation (3.71) near the point \(x=a\), we get, equating the coefficients in front of \((x-a)^{-1}\) and \((x-a)^0\):

$$\begin{aligned} \left\{ \begin{array}{l} r_{s+1}=\dot{a} \xi _s(a)+\nu V^-(a)\xi _{s-1}(a-\eta ), \\ \\ r_{s+1, 0}-\xi _{s+1}(a)= \dot{\xi }_s(a) + \varphi _0 \xi _s(a) +\nu V^-(a)\xi _{s-1}'(a-\eta )+ \nu V^-(a)\Omega \xi _{s-1}(a-\eta ), \end{array} \right. \nonumber \\ \end{aligned}$$
(3.77)

where \( \dot{\xi }_s(a)= \dot{a} \xi _s'(a)+\partial _t \xi _s(a) \) is the full time derivative. At the point \(x=a-\eta \), all terms in equation (3.71) are regular and the equation gives:

$$\begin{aligned} \xi _{s+1}(a)-\xi _{s+1}(a-\eta )=\partial _t \varphi _+(a-\eta )\xi _s (a-\eta ) +\partial _t \xi _s (a-\eta ) \end{aligned}$$
(3.78)

. (The last term in (3.71) vanishes at this point.)

Let

$$\begin{aligned} R_s=\dot{a} (r_{s,0}-\xi _s(a))-\varphi _1 r_s -\dot{r}_s -\nu V^+(a)\xi _{s-1}(a)-\nu V^-(a)\xi _{s-1}(a-\eta ) \end{aligned}$$

be sum of the residues at the left-hand side of (3.71) at the points \(x=a\) and \(x=a+\eta \) which must be zero. It is seen from (3.71) that \(R_1=0\). Our induction assumption is that \(R_s=0\) for some s; let us show that this implies that \(R_{s+1}=0\). We have:

$$\begin{aligned} R_{s+1}=\dot{a} (r_{s+1,0}-\xi _{s+1}(a))-\varphi _1 r_{s+1} -\dot{r}_{s+1} -\nu V^+(a)\xi _{s}(a)-\nu V^-(a)\xi _{s}(a-\eta ). \end{aligned}$$

A straightforward calculation which uses recurrence relations (3.76), (3.77) and (3.78) yields:

$$\begin{aligned}{} & {} R_{s+1}=\left[ \dot{a}\Bigl (\frac{\dot{\nu }}{\nu }-\frac{\mu }{\nu }\, \dot{a} \Bigr )-\ddot{a} -\nu (V^+(a)+V^-(a))\right] -\partial _t R_s \\{} & {} +\nu V^-(a)\xi _{s-1}(a-\eta )\left[ \dot{a}\Omega -\varphi _1 - \partial _t \log (\nu V^-(a)) +\partial _t \varphi _+(a-\eta )\right] . \end{aligned}$$

The first two terms vanish by virtue of the condition (3.50) and the induction assumption. Using equations (3.65), (3.66), (3.69) and (3.75), one can show that the third term also vanishes. Therefore, we have proved that from \(R_s=0\) it follows that \(R_{s+1}=0\) which implies the existence of a meromorphic wave solution. \(\square \)

4 Fully difference equation

The case of fully difference equation was considered by one of the authors in [18] but we find it appropriate to include it here for completeness.

Let us consider the difference equation

$$\begin{aligned} \psi _{t+1}(x)=\psi _t(x+\eta )+ u_t(x)\psi _t (x), \qquad u_t(x)=\frac{\tau _t(x) \tau _{t+1}(x+\eta )}{\tau _t(x+\eta )\tau _{t+1}(x)} \end{aligned}$$
(4.1)

which serves as the auxiliary linear problem for the Hirota bilinear difference equation for the tau-function \(\tau _t(x)\) [24, 25].

Proposition 4.1

Let \(\tau _t(x)\) have a simple zero at some point \(a_t\): \(\tau _t(a_t)=0\), \(\tau _t '(a_t)\ne 0\) and \(\tau _t(a_t-\eta ) \tau _{t+1}(a_t)\ne 0\). Then, the necessary condition that equation (4.1) has a meromorphic solution with a simple pole at \(x=a_t\) regular at \(x=a_t \pm \eta \), \(x=a_{t+1}\) is

$$\begin{aligned} \frac{\tau _{t+1}(a_t)\tau _t(a_t-\eta ) \tau _{t-1}(a_t+\eta )}{\tau _{t+1}(a_t-\eta )\tau _t(a_t+\eta ) \tau _{t-1}(a_t)}=-1. \end{aligned}$$
(4.2)

Proof

Tending x to \(a_{t+1}\), \(a_t-\eta \) and \(a_{t+1}-\eta \) in equation (4.1), we obtain the relations

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle { \alpha _{t+1}=\frac{\tau _t(a_{t+1}) \tau _{t+1}(a_{t+1}+\eta )}{\tau _{t+1}'(a_{t+1})\tau _t(a_{t+1}+\eta )}\, \psi _t(a_{t+1}),} \\ \\ \displaystyle { \alpha _{t}=-\frac{\tau _t(a_{t}-\eta ) \tau _{t+1}(a_{t})}{\tau _{t}'(a_{t})\tau _{t+1}(a_{t}-\eta )}\, \psi _t(a_{t}-\eta ),} \\ \\ \psi _{t+1}(a_{t+1}-\eta )=\psi _t(a_{t+1}). \end{array} \right. \end{aligned}$$
(4.3)

Combining them, we arrive at the condition (4.2). \(\square \)

Let \(\tau _t(x)\) be an “elliptic polynomial” of the form

$$\begin{aligned} \tau _t(x)=\prod _{j=1}^N \sigma (x-x_j^t) \end{aligned}$$
(4.4)

with N simple roots \(x_j^t\) in the fundamental domain, then the conditions (4.2) with \(a_t=x_i^t\) for each \(i=1, \ldots , N\) yield the equations

$$\begin{aligned} \prod _{j=1}^N \frac{\sigma (x_i^t-x_j^{t+1})\sigma (x_i^t -x_j^t -\eta ) \sigma (x_i^t-x_j^{t-1}+\eta )}{\sigma (x_i^t-x_j^{t+1}-\eta ) \sigma (x_i^t -x_j^t +\eta ) \sigma (x_i^t-x_j^{t-1})}=-1. \end{aligned}$$
(4.5)

These are equations of motion for the discrete time version of the Ruijsenaars–Schneider system first obtained in [26].

Finally, we will show that (4.2) is a sufficient condition for local existence of a meromorphic wave solution to equation (4.1) of the form

$$\begin{aligned} \psi _t(x)=k^{x/\eta }k^t \Bigl (1+\sum _{s\ge 1}\xi _s^t(x)k^{-s}\Bigr ) \end{aligned}$$
(4.6)

having a simple pole at \(x=a_t\) and regular at \(x=a_t\pm \eta \), \(x=a_{t+1}\).

Proposition 4.2

Assume that \(\tau _t (x)\) in (4.1) has a zero at some point \(a_t\) such that \(\tau _t '(a_t)\ne 0\), \(\tau _t(a_t-\eta ) \tau _{t+1}(a_t)\ne 0\) and condition (4.2) holds. Then, all wave solutions to equation (4.1) of the form (4.6) are meromorphic in a neighborhood of the point a with a simple pole at \(x=a\) and regular at \(x=a\pm \eta \).

Proof

Substituting the series (4.6) into the equation, we obtain the recurrence relation

$$\begin{aligned} \xi _{s+1}^{t+1}(x)-\xi _{s+1}^t(x+\eta )=u_t(x)\xi _s^t(x). \end{aligned}$$
(4.7)

Let the expansion of the function \(\psi _t(x)\) near the pole be of the form

$$\begin{aligned} \psi _t(x)=\frac{r_s^t}{x-a_t}+r_{s,0}^t +O(x-a_t). \end{aligned}$$
(4.8)

Tending x to \(a_{t+1}\), \(a_t-\eta \) and \(a_{t+1}-\eta \) in equation (4.7), we obtain the equalities

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle { r_{s+1}^{t+1}=\frac{\tau _t(a_{t+1}) \tau _{t+1}(a_{t+1}+\eta )}{\tau _{t+1}'(a_{t+1})\tau _t(a_{t+1}+\eta )}\, \xi _s^t(a_{t+1}),} \\ \\ \displaystyle { r_{s+1}^t=-\frac{\tau _t(a_{t}-\eta ) \tau _{t+1}(a_{t})}{\tau _{t}'(a_{t})\tau _{t+1}(a_{t}-\eta )}\, \xi _s^t(a_{t}-\eta ),} \\ \\ \xi _{s}^{t+1}(a_{t+1}-\eta )=\xi _{s}^t(a_{t+1}), \end{array} \right. \end{aligned}$$
(4.9)

where we shifted \(s\rightarrow s-1\) in the last equation. Combining them and taking into account the condition (4.2), we see that the two expressions for \(r_{s+1}^t\) that follow from (4.7) actually coincide whence we conclude that the solution of the form (4.6) exists. \(\square \)

5 Concluding remarks

In this paper, we have further developed the approach to integrable many-body systems based on monodromy free linear equations, i.e., on finding conditions of existence of meromorphic solutions to linear partial differential and difference equations for all cases which arise as linear problems for integrable nonlinear equations. Some of them were previously discussed by one of the authors in [17,18,19]. We have also shown that these conditions are simultaneously sufficient conditions for local existence of meromorphic wave solutions depending on a spectral parameter. These conditions straightforwardly lead to equations of motion for poles of elliptic solutions to nonlinear integrable equations. It turns out that the dynamics of poles is isomorphic to many-body systems of Calogero–Moser type which include the Calogero–Moser system itself, its relativistic extension (the Ruijsenaars–Schneider system) and a rather exotic system with three-body interaction found in [14]. We note that the approach of this paper is the shortest way to obtain the equations of motion. Presumably, all these systems are integrable as they arise as certain finite-dimensional reductions of integrable nonlinear systems with infinitely many degrees of freedom. Independent proofs of integrability exist for the Calogero–Moser and Ruijsenaars–Schneider systems, while for the system introduced in [14] this is still a hypothesis.

The main new result of this paper is the many-body system with equations of motion (1.1), (1.2) representing dynamics of poles of elliptic solutions to the Toda lattice of type B recently introduced in [20]. This system can be regarded as a kind of relativistic extension of the system found in [14] with equations of motion (2.34) in the sense that the former is related to the latter in the same way as the Ruijsenaars–Schneider system is related to the Calogero–Moser system.

There are some interesting open problems. First, it is not clear whether the system (1.1), (1.2) is Hamiltonian. Second, a commutation representation for it is not yet known. Presumably, such commutation representation is of the form of the Manakov’s triple as it is the case for the system (2.34) which arises from (1.1), (1.2) in the \(\eta \rightarrow 0\) limit. Last but not least, there is the problem of proving integrability of the system (1.1), (1.2). At the moment, we know only one integral of motion which is \(\displaystyle {I=\sum _i \dot{x}_i}\). Its conservation (\(\ddot{I}=0\)) can be seen directly from the equations of motion taking into account that

$$\begin{aligned} \sum _{i=1}^N U(x_{i1}, \ldots , x_{iN})=0 \end{aligned}$$

which follows from the fact that the left-hand side is sum of the residues at the 2N poles of the elliptic function

$$\begin{aligned} f(x)=\prod _{j=1}^N \frac{\sigma (x-x_j+2\eta ) \sigma (x-x_j-\eta )}{\sigma (x-x_j+\eta )\sigma (x-x_j)} \end{aligned}$$

at the points \(x=x_i\), \(x=x_i-\eta \), \(i=1, \ldots \,, N\).