Abstract
We further develop the approach to many-body systems based on finding conditions of existence of meromorphic solutions to certain linear partial differential and difference equations which serve as auxiliary linear problems for nonlinear integrable equations such as KP, BKP, CKP and different versions of the Toda lattice. These conditions imply equations of the time evolution for poles of singular solutions to the nonlinear equations which are equations of motion for integrable many-body systems of Calogero–Moser and Ruijsenaars–Schneider type. A new many-body system is introduced, which governs dynamics of poles of elliptic solutions to the Toda lattice of type B.
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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
where dot means the time derivative, \(x_{ik}\equiv x_i-x_k\),
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
(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
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
has a meromorphic in x solution, then the condition
holds, where dot means the time derivative.
Proof
Let the expansion of \(\psi (x)\) around the point a be of the form
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
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
where \(\wp (x)\) is the Weierstrass \(\wp \)-function, then the expansion (2.2) near the pole at \(a=x_i\) holds true with
so the conditions (2.3) for each \(x_i\) read
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
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
In particular, at \(s=0\) we have
Let the Laurent expansion of \(\xi _s\) near the pole at \(x=a\) be of the form
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:
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
Substituting them into the right-hand side of (2.12) at \(s\rightarrow s+1\), we have:
where we have used the first and the third equations in (2.13). After cancellations, we get:
(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
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
holds, where \(u_0\) is the coefficient in the Laurent expansion
Proof
We have the expansion near the pole at \(x=a\):
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
The first and the third equations are equivalent and lead to the necessary condition (2.15). \(\square \)
Let u(x) be the elliptic function
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
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
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
In particular, at \(s=-1\) we have
It is convenient to represent equation (2.22) in the form
Let the Laurent expansion of \(\xi _s\) near the pole at \(x=a\) be
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:
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
Substituting them into the right-hand side of (2.26) at \(s\rightarrow s+1\), we have:
(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
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
holds, where \(u_0, u_1, u_3\) are coefficients in the Laurent expansion
Proof
We have the expansion near the pole at \(x=a\):
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
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
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
then the expansion (2.30) near the pole at \(x=x_i\) holds true with
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]:
(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
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
In particular, at \(s=-1\) we have
It is convenient to represent equation (2.36) in the form
Let the Laurent expansion of \(\xi _s\) near the pole at \(x=a\) be
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:
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
Substituting them into the right-hand side of (2.40) at \(s\rightarrow s+1\), we have:
(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
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):
Let b(x) have first-order poles at \(x=a\) and \(x=a-\eta \):
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
holds.
Proof
Let the expansion of \(\psi (x)\) around the point a be of the form
then
Substituting the expansions around the point a in the equation, we write:
Equating the coefficients in front of the poles, we obtain the conditions
Around the point \(a-\eta \), we have:
Equating the coefficients in front of the terms or order \((x-a+\eta )^{-1}\) and \((x-a+\eta )^{0}\), we obtain the conditions
Taking the time derivative of the first equation in (3.8) and using (3.7), we get
where
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
Let \(a=x_i\) for some i, then the coefficients \(\mu _0\), \(\mu _1\) in (3.4) are
and (3.5) is equivalent to the equation of motion
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
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 \(\xi _s\):
. (It is convenient to put \(\xi _0=1\).) In particular,
Let the Laurent expansion of \(\xi _s\) near the point a be
Substituting this into (3.15) with \(x\rightarrow a\) and equating the coefficients at the poles, we get the recurrence relation
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
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
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:
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):
Suppose that the function v(x) has a second-order pole at \(x=a\) with the expansion
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
holds.
Proof
For \(\psi (x)\) near the point a, we have
Substituting the expansions around the point a in the equation (3.20), we write:
Equating the coefficients in front of the poles, we obtain the conditions
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
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
then, setting \(a=x_i\), we have:
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
where \(\varphi (x)\) has logarithmic singularities at \(x=a\) and \(x=a-\eta \), so that
Suppose that condition (3.22) for the pole of v(x) holds. Then, all wave solutions of equation (3.20) of the form
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:
Substituting the expansions into the equation, we obtain the recurrence relation for the coefficients \(\xi _s\):
In particular,
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
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
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
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
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
The same procedure at the pole at \(x=a-\eta \) leads to the condition
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:
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
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):
where we set \(\xi _{-1}=0\), \(\xi _0=1\). In particular,
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:
The expansion near the point \(x=a-\eta \) gives:
Let
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:
Substituting here the recurrence relations (3.45), (3.46), we obtain, after some calculations and cancellations:
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:
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
Assume that the function v(x) has a second-order pole at \(x=a\) with the expansion
We also assume that v(x) has zeros at \(x=a-\eta \) and \(x=a+\eta \):
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
holds.
Proof
The principal part of the Laurent expansion of \(\psi (x)\) near the point \(x=a\) is
As \(x\rightarrow a\), we have from equation (3.47):
Equating the coefficients in front of the poles, we obtain the conditions
At \(x=a\pm \eta \), equation (3.47) gives:
Therefore,
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
then, setting \(a=x_i\), we have:
and the condition (3.50) is equivalent to the equation of motion
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
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
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
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:
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:
where \(X_{ik}\equiv X_i-X_k\) and
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
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 \(a\pm \eta \):
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
It is also convenient to introduce the function
The function \(\dot{\varphi }_+ (x)\) has simple poles at the points \(x=a\) and \(x=a+\eta \) with the expansions
where
It is easy to check that
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
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 \(\xi _s\):
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
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:
whence
where
Expanding equation (3.71) near the point \(x=a+\eta \), we get, equating the coefficients in front of the poles:
Similarly, expanding equation (3.71) near the point \(x=a\), we get, equating the coefficients in front of \((x-a)^{-1}\) and \((x-a)^0\):
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:
. (The last term in (3.71) vanishes at this point.)
Let
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:
A straightforward calculation which uses recurrence relations (3.76), (3.77) and (3.78) yields:
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
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
Proof
Tending x to \(a_{t+1}\), \(a_t-\eta \) and \(a_{t+1}-\eta \) in equation (4.1), we obtain the relations
Combining them, we arrive at the condition (4.2). \(\square \)
Let \(\tau _t(x)\) be an “elliptic polynomial” of the form
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
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
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
Let the expansion of the function \(\psi _t(x)\) near the pole be of the form
Tending x to \(a_{t+1}\), \(a_t-\eta \) and \(a_{t+1}-\eta \) in equation (4.7), we obtain the equalities
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
which follows from the fact that the left-hand side is sum of the residues at the 2N poles of the elliptic function
at the points \(x=x_i\), \(x=x_i-\eta \), \(i=1, \ldots \,, N\).
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Appendices
Appendix A: The Weierstrass functions
Throughout the main text, we use the standard Weierstrass functions: the \(\sigma \)-function, the \(\zeta \)-function and the \(\wp \)-function.
The Weierstrass \(\sigma \)-function with quasi-periods \(2\omega \), \(2\omega '\) such that \(\textrm{Im} (\omega '/ \omega )>0\) is defined by the infinite product
It is an odd entire quasiperiodic function in the complex plane. The Weierstrass \(\zeta \)-function is defined as
It is an odd function with first-order poles at the points of the lattice \(s=2\omega m+2\omega ' m'\) with integer \(m, m'\). The definition of the Weierstrass \(\wp \)-function is
It is an even double-periodic function with periods \(2\omega , 2\omega '\) and with second-order poles at the points of the lattice \(s=2\omega m+2\omega ' m'\) with integer \(m, m'\).
The monodromy properties of the \(\sigma \)-function under shifts by the quasi-periods are as follows:
The \(\zeta \)-function acquires an additive constant when the argument is shifted by any quasi-period:
These constants are related by the identity \(2\omega ' \zeta (\omega )-2\omega \zeta (\omega ')=\pi i\).
Appendix B: The Ruijsenaars–Schneider model
Here, we collect some facts on the elliptic Ruijsenaars–Schneider system [12] following the paper [13]. The N-particle elliptic Ruijsenaars–Schneider system is a completely integrable model. The canonical Poisson brackets between coordinates and momenta are \(\{x_i, p_j\}=\delta _{ij}\). The integrals of motion in involution have the form
It is convenient to put \(I_0=1\). Important particular cases of (B1) are
which is the Hamiltonian \(H_1\) of the chiral Ruijsenaars–Schneider model and
Let us denote the time variable of the Hamiltonian flow with the Hamiltonian \(H_1\) by \(t_1\). The velocities of the particles are
where dot means the \(t_1\)-derivative. The Hamiltonian equations \(\dot{x}_i=\partial H_1/\partial p_i\), \(\dot{p}_i=-\partial H_1/ \partial x_i\) are equivalent to the following equations of motion:
The properly taken limit \(\eta \rightarrow 0\) (the “non-relativistic limit”) gives equations of motion of the elliptic Calogero–Moser system.
One can also introduce integrals of motion \(I_{-k}\) as
In particular,
It can be easily verified that equations of motion in the time \(\bar{t}_1\) corresponding to the Hamiltonian \(\bar{H}_1=\sigma ^2(\eta )I_{-1}\) are the same as (B5).
The “physical” Hamiltonian of the Ruijsenaars–Schneider model is \(H_+=H_1 +\bar{H}_1\). Below in Appendix C we consider the Hamiltonian flow corresponding to the Hamiltonian \(H_-=H_1 -\bar{H}_1\).
Appendix C: How Ruijsenaars–Schneider particles stick together
In this appendix, we show how to restrict the Ruijsenaars–Schneider dynamics of the \(N=2n\)-particle system to the subspace in which the particles stick together in n pairs such that
We introduce the variables
which are coordinates of the pairs. Such configuration is destroyed by the \(H_+\)-Hamiltonian flow \(\partial _{t_1}+\partial _{\bar{t}_1}\) but is preserved by the \(H_-\)-flow \(\partial _t=\partial _{t_1}-\partial _{\bar{t}_1}\), as we shall see below.
The Hamiltonian \(H_-\) reads
For the velocities \(\dot{x}_i =\partial H_-/\partial p_i\), we have:
where \(x_{ik}\equiv x_i-x_k\). Suppose that the momenta remain finite under the restriction to (C1). Then in terms of coordinates \(X_i\) of the pairs, we have from these formulas:
The dynamics preserves the configuration (C1) if \(\dot{x}_{2i-1}= \dot{x}_{2i}\), whence we should require
Resolving this constraint, we can put
where
and \(P_i\) are arbitrary. We have thus restricted the original 4n-dimensional phase space \(\mathcal{F}\) with coordinates \(\{p_j, x_j\}\), \(j=1, \ldots , 2n\) to the 2n-dimensional subspace \(\mathcal{P}\) with coordinates \(\{P_i, X_i\}\), \(i=1, \ldots , n\) corresponding to joining the particles into pairs of the form (C1). Equations (C3) are then equivalent to
Let us now pass to the second set of the Hamiltonian equations, \(\dot{p}_i=-\partial H_-/\partial x_i\):
Restricting to the subspace \(\mathcal{P}\), we have:
When passing from (C7) to (C8) with the constraint (C1), one encounters expressions like \(\sigma (x_{2i}\! -\! x_{2i-1}\! -\! \eta ) \zeta (x_{2i}\! -\! x_{2i-1}\! -\! \eta )\) which is an indeterminacy of the form 0/0. To resolve it, one should put \(x_{2i}\! -\! x_{2i-1}=\eta +\varepsilon \) and tend \(\varepsilon \rightarrow 0\).
Taking the time derivative of (C6), we obtain:
where we should substitute \(\dot{P}_i=-\dot{\alpha }_i +\dot{p}_{2i-1}\) from (C8) taking into account (C6):
Substituting here
we obtain, after cancellations:
These are equations (1.1), (1.2).
A similar calculation for \(\dot{p}_{2i}\) leads to the same result. This means that the restriction to the subspace \(\mathcal{P}\) is consistent.
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Krichever, I., Zabrodin, A. Monodromy free linear equations and many-body systems. Lett Math Phys 113, 75 (2023). https://doi.org/10.1007/s11005-023-01699-3
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DOI: https://doi.org/10.1007/s11005-023-01699-3