ON THE EQUIVALENCE OF DIFFERENT LAX PAIRS FOR THE KAC–VAN MOERBEKE HIERARCHY

JOHANNA MICHOR AND GERALD TESCHL

Abstract. We give a simple algebraic proof that the two different Lax pairs for the Kac–van Moerbeke hierarchy, constructed from Jacobi respectively super-symmetric Dirac-type difference operators, give rise to the same hierarchy of evolution equations. As a byproduct we obtain some new recursions for computing these equations.

1. Introduction

There are two different Lax equations for the Kac–van Moerbeke equation: The original one, found independently by Kac and van Moerbeke [6] and Manakov [7], based on a Jacobi matrix with zero diagonal elements and its skew-symmetrized square and the second one based on super-symmetric Dirac-type matrices. Both approaches can be generalized to give corresponding hierarchies of evolution equations in the usual way and both reveal a close connection to the Toda hierarchy. In fact, the first approach shows that the Kac–van Moerbeke hierarchy (KM hierarchy) is contained in the Toda hierarchy by setting b = 0 in the odd equations. The second one relates both hierarchies via a Bäcklund transformation since the Dirac-type difference operator gives rise to two Jacobi operators by taking squares (respectively factorizing positive Jacobi operators to obtain the other direction). Both ways of introducing the KM hierarchy have their merits, however, tough it is obvious that both produce the same hierarchy by looking at the first few equations, we could not find a formal proof in the literature. The purpose of this short note is to give a simple algebraic proof for this fact. As a byproduct we will also obtain some new recursions for computing the equations in the KM hierarchy.

In Section 2 we review the recursive construction of the Toda hierarchy via Lax pairs involving Jacobi operators and obtain the KM hierarchy by setting b = 0 in the odd equations. In Section 3 we introduce the KM hierarchy via Lax pairs involving Dirac-type difference operators. In Section 4 we show that both constructions produce the same equations. Finally, we recall how to identify Jacobi operators with b = 0 in Section 5.

2. The Toda hierarchy

In this section we introduce the Toda hierarchy using the standard Lax formalism following [2] (see also [5], [10]).

2000 Mathematics Subject Classification. Primary 47B36, 37K15; Secondary 81U40, 39A10.

Key words and phrases. Kac–van Moerbeke hierarchy, Lax pair, Toda hierarchy.

Work supported by the Austrian Science Fund (FWF) under Grants No. Y330 and J2655.

in Modern Analysis and Applications, V. Adamyan (ed.) et al., 437–445, Oper. Theory Adv.

Appl. 191, Birkhäuser, Basel, 2009.

We will only consider bounded solutions and hence require

Hypothesis H.2.1. Suppose a(t), b(t) satisfy

a(t) ∈ ℓ^{∞}(ℤ, ℝ),  b(t) ∈ ℓ^{∞}(ℤ, ℝ),  a(n, t) ≠ 0,  (n, t) ∈ ℤ × ℝ,

and let t ↦ (a(t), b(t)) be differentiable in ℓ^{∞}(ℤ) ⊕ ℓ^{∞}(ℤ).

Associated with a(t), b(t) is a Jacobi operator

(2.1)  H(t) = a(t)S^{+} + a^{−}(t)S^{−} + b(t)

in ℓ^{2}(ℤ), where S^{±}f(n) = f^{±}(n) = f(n ± 1) are the usual shift operators and ℓ^{2}(ℤ) denotes the Hilbert space of square summable (complex-valued) sequences over ℤ. Moreover, choose constants c_{0} = 1, c_{j}, 1 ≤ j ≤ r, c_{r+1} = 0, and set

\begin{aligned}g_{j}(n, t) & = ∑_{ℓ=0}^{j} c_{j−ℓ}⟨δ_{n}, H(t)^{ℓ}δ_{n}⟩, \\ h_{j}(n, t) & = 2a(n, t) ∑_{ℓ=0}^{j} c_{j−ℓ}⟨δ_{n+1}, H(t)^{ℓ}δ_{n}⟩ + c_{j+1}. \end{aligned} (2.2)

The sequences g_{j}, h_{j} satisfy the recursion relations

g_{0} = 1, h_{0} = c_{1},
2g_{j+1} − h_{j} − h_{j}^{−} − 2bg_{j} = 0, 0 ≤ j ≤ r,
(2.3)  h_{j+1} − h_{j+1}^{−} − 2(a^{2}g_{j}^{+} − (a^{−})^{2}g_{j}^{−}) − b(h_{j} − h_{j}^{−}) = 0, 0 ≤ j < r.

Introducing

(2.4)  P_{2r+2}(t) = −H(t)^{r+1} + ∑_{j=0}^{r}(2a(t)g_{j}(t)S^{+} − h_{j}(t))H(t)^{r−j} + g_{r+1}(t),

a straightforward computation shows that the Lax equation

(2.5)  \frac{d}{dt}H(t) − [P_{2r+2}(t), H(t)] = 0,  t ∈ ℝ,

is equivalent to

(2.6)  TL_{r}(a(t), b(t)) = (\begin{matrix}ȧ(t) − a(t)(g_{r+1}^{+}(t) − g_{r+1}(t)) \\ ḃ(t) − (h_{r+1}(t) − h_{r+1}^{−}(t)) \end{matrix}) = 0,

where the dot denotes a derivative with respect to t. Varying r ∈ ℕ_{0} yields the Toda hierarchy TL_{r}(a, b) = 0. The corresponding homogeneous quantities obtained by taking all summation constants equal to zero, c_{ℓ} ≡ 0, ℓ ∈ ℕ, are denoted by ĝ_{j}, ĥ_{j}, etc., resp.

(2.7)  T̂L_{r}(a, b) = TL_{r}(a, b)|_{c_{ℓ}≡0,1≤ℓ≤r}.

Next we show that we can set b ≡ 0 in the odd equations of the Toda hierarchy.

Lemma 2.2. Let b ≡ 0. Then the homogeneous coefficients satisfy

ĝ_{2j+1} = ĥ_{2j} = 0, j ∈ ℕ_{0}.

Proof. We use induction on the recursion relations (2.3). The claim is true for j = 0. If ĥ_{2j} = 0 then ĝ_{2j+1} = 0, and ĥ_{2j} = 0 follows from the last equation in (2.3). □

In particular, if we choose c_{2ℓ} = 0 in TL_{2r+1}, then we can set b ≡ 0 to obtain a hierarchy of evolution equations for a alone. In fact, set

(2.8)  G_{j} = ĝ_{2j}, K_{j} = ĥ_{2j+1},

in this case. Then they satisfy the recursion

G_{0} = 1, K_{0} = 2a^{2},
2G_{j+1} − K_{j} − K_{j}^{−} = 0, 0 ≤ j ≤ r,
(2.9)  K_{j+1} − K_{j+1}^{−} − 2(a^{2}G_{j}^{+} − (a^{−})^{2}G_{j}^{−}) = 0, 0 ≤ j < r,

and TL_{2r+1}(a, 0) = 0 is equivalent to the KM hierarchy defined as

(2.10)  KM_{r}(a) = ȧ − a(G_{r+1}^{+} − G_{r+1}),  r ∈ ℕ_{0}.

3. The Kac–van Moerbeke hierarchy as a modified Toda hierarchy

In this section we review the construction of the KM hierarchy as a modified Toda hierarchy. We refer to [2], [10] for further details.

Suppose ρ(t) satisfies

Hypothesis H.3.1. Let

(3.1)  ρ(t) ∈ ℓ^{∞}(ℤ, ℝ), ρ(n, t) ≠ 0, (n, t) ∈ ℤ × ℝ

and let t ↦ ρ(t) be differentiable in ℓ^{∞}(ℤ).

Define the “even” and “odd” parts of ρ(t) by

(3.2)  ρ_{e}(n, t) = ρ(2n, t), ρ_{o}(n, t) = ρ(2n + 1, t), (n, t) ∈ ℤ × ℝ,

and consider the bounded operators (in ℓ^{2}(ℤ))

(3.3)  A(t) = ρ_{o}(t)S^{+} + ρ_{e}(t), A(t)^{∗} = ρ_{o}^{−}(t)S^{−} + ρ_{e}(t).

In addition, we set

(3.4)  H_{1}(t) = A(t)^{∗}A(t), H_{2}(t) = A(t)A(t)^{∗},

with

(3.5)  H_{k}(t) = a_{k}(t)S^{+} + a_{k}^{−}(t)S^{−} + b_{k}(t),  k = 1, 2,

and

(3.6)  a_{1}(t) = ρ_{e}(t)ρ_{o}(t),  b_{1}(t) = ρ_{e}(t)^{2} + ρ_{o}^{−}(t)^{2},
(3.7)  a_{2}(t) = ρ_{e}^{+}(t)ρ_{o}(t),  b_{2}(t) = ρ_{e}(t)^{2} + ρ_{o}(t)^{2}.

Now we define operators D(t), Q_{2r+2}(t) in ℓ^{2}(ℤ, ℂ^{2}) as follows,

(3.8)  D(t) = ( \begin{matrix}0 & A(t)^{∗} \\ A(t) & 0 \end{matrix} ) ,
(3.9)  Q_{2r+2}(t) = ( \begin{matrix}P_{1,2r+2}(t) & 0 \\ 0 & P_{2,2r+2}(t) \end{matrix} ) , r ∈ ℕ_{0}.

Here P_{k,2r+2}(t), k = 1, 2 are defined as in (2.4), that is,

(3.10) P_{k,2r+2}(t) = −H_{k}(t)^{r+1} + ∑_{j=0}^{r}(2a_{k}(t)g_{k,j}(t)S^{+} − h_{k,j}(t))H_{k}(t)^{j} + g_{k,r+1},

{g_{k,j}(n, t)}_{0≤j≤r}, {h_{k,j}(n, t)}_{0≤j≤r+1} are defined as in (2.2). Moreover, we choose the same integration constants in P_{1,2r+2}(t) and P_{2,2r+2}(t) (i.e., c_{1,ℓ} = c_{2,ℓ} ≡ c_{ℓ}, 1 ≤ ℓ ≤ r).

Analogous to equation (2.5) one obtains that

(3.11)  \frac{d}{dt}D(t) − [Q_{2r+2}(t), D(t)] = 0

is equivalent to

KM_{r}(ρ) = (KM_{r}(ρ)_{e}, KM_{r}(ρ)_{o})
(3.12)  = ( \begin{matrix}ρ̇_{e} − ρ_{e}(g_{2,r+1} − g_{1,r+1}) \\ ρ̇_{o} + ρ_{o}(g_{2,r+1} − g_{1,r+1}^{+}) \end{matrix} ) = 0.

As in the Toda context (2.6), varying r ∈ ℕ_{0} yields the KM hierarchy which we denote by

(3.13)  KM_{r}(ρ) = 0, r ∈ ℕ_{0}.

The homogeneous KM hierarchy is denoted by

(3.14)  K̂M_{r}(ρ) = KM_{r}(ρ)|_{c_{ℓ}≡0,1≤ℓ≤r}.

One look at the transformations (3.6), (3.7) verifies that the equations for ρ_{o}, ρ_{e} are in fact one equation for ρ. More explicitly, combining g_{k,j}, resp. h_{k,j}, into one sequence

(3.15)  \begin{aligned}G_{j}(2n) & = & g_{1,j}(n) \\ G_{j}(2n + 1) & = & g_{2,j}(n) \end{aligned} , resp. \begin{aligned}H_{j}(2n) & = & h_{1,j}(n) \\ H_{j}(2n + 1) & = & h_{2,j}(n) \end{aligned} ,

we can rewrite (3.12) as

(3.16)  KM_{r}(ρ) = ρ̇ − ρ(G_{r+1}^{+} − G_{r+1}).

From (2.3) we see that G_{j}, H_{j} satisfy the recursions

G_{0} = 1, H_{0} = c_{1},
2G_{j+1} − H_{j} − H_{j}^{−−} − 2(ρ^{2} + (ρ^{−})^{2})G_{j} = 0, 0 ≤ j ≤ r,
H_{j+1} − H_{j+1}^{−−} − 2((ρρ^{+})^{2}G_{j}^{+} − (ρ^{−}ρ)^{2}G_{j}^{−−})
(3.17)  −(ρ^{2} + (ρ^{−})^{2})(H_{j} − H_{j}^{−−}) = 0, 0 ≤ j < r.

The homogeneous quantities are denoted by Ĝ_{j}, Ĥ_{j}, etc., as before.

As a simple consequence of (3.11) we have

(3.18)  \frac{d}{dt}D(t)^{2} − [Q_{2r+2}(t), D(t)^{2}] = 0

and observing

(3.19)  D(t)^{2} = ( \begin{matrix}H_{1}(t) & 0 \\ 0 & H_{2}(t) \end{matrix} )

yields the implication

(3.20)  KM_{r}(ρ) = 0 ⇒ TL_{r}(a_{k}, b_{k}) = 0, k = 1, 2,

that is, given a solution ρ of the KM_{r} equation (3.13), one obtains two solutions, (a_{1}, b_{1}) and (a_{2}, b_{2}), of the TL_{r} equations (2.6) related to each other by the Miuratype transformations (3.6), (3.7). For more information we refer to [4], [5], [9], [10], [11], and [12].

4. Equivalence of both constructions

In this section we want to show that the constructions of the KM hierarchy outlined in the previous two sections yield in fact the same set of evolution equations. This will follow once we show that G_{j} defined in (2.8) is the same as G_{j} defined in (3.15). It will be sufficient to consider the homogeneous quantities, however, we will omit the additional hats for notational simplicity. Moreover, we will denote the sequence G_{j} defined in (2.8) by G̃_{j} to distinguish it from the one defined in (3.15). Since both are defined recursively via the recursions (2.9) for G̃_{j}, K_{j} respectively (3.17) for G_{j}, H_{j} our first aim is to eliminate the additional sequences K_{j} respectively H_{j} and to get a recursion for G̃_{j} respectively G_{j} alone.

Lemma 4.1. The coefficients g_{j}(n) satisfy the following linear recursion

\begin{aligned}g_{j+3}^{+} − g_{j+3} & = (b + 2b^{+})g_{j+2}^{+} − (2b + b^{+})g_{j+2} \\ & − (2b + b^{+})b^{+}g_{j+1}^{+} + b(2b^{+} + b)g_{j+1} + k_{j+1}^{+} + k_{j+1} \\ & + b(b^{+})^{2}g_{j}^{+} − b^{+}b^{2}g_{j} − bk_{j}^{+} − b^{+}k_{j}, \end{aligned} (4.1)

where

(4.2)  k_{j} = a^{2}g_{j}^{+} − (a^{−})^{2}g_{j}^{−}, j ∈ ℕ.

Proof. It suffices to consider the homogeneous case g_{j}(n) = ⟨δ_{n}, H^{j}δ_{n}⟩. Then (compare [10, Sect 6.1])

g(z, n) = ⟨δ_{n}, (H − z)^{−1}δ_{n}⟩ = − ∑_{j=0}^{∞} \frac{g_{j}(n)}{z^{j+1}}

satisfies [10, (1.109)]

\frac{(a^{+})^{2}g^{++} − a^{2}g}{z − b^{+}} + \frac{a^{2}g^{+} − (a^{−})^{2}g^{−}}{z − b} = (z − b^{+})g^{+} − (z − b)g,

and the claim follows after comparing coefficients. □

Corollary 4.2. For j ∈ ℕ_{0}, the sequences G̃_{j}, defined by (2.8) and corresponding to the TL hierarchy with b ≡ 0, satisfy

G̃_{j+1}^{+} − G̃_{j+1} = (a^{+})^{2}G̃_{j}^{++} + a^{2}(G̃_{j}^{+} − G̃_{j}) − (a^{−})^{2}G̃_{j}^{−}. (4.3)

The corresponding sequences G_{j} for the KM hierarchy defined in (3.15) satisfy

\begin{aligned}G_{j+3} − G_{j+3}^{++} & = ((a^{−})^{2} + a^{2})^{2}((a^{+})^{2} + (a^{++})^{2})G_{j} \\ & + (a^{−−})^{2}(a^{−})^{2}G_{j+1}^{−−} + a^{2}(a^{+})^{2}G_{j+1} \\ & + ((a^{+})^{2} + (a^{++})^{2})(2(a^{−})^{2} + 2a^{2} + (a^{+})^{2} + (a^{++})^{2})G_{j+1}^{++} \\ & + (2(a^{−})^{2} + 2a^{2} + (a^{+})^{2} + (a^{++})^{2})G_{j+2} \\ & − ((a^{−})^{2} + a^{2})((a^{+})^{2} + (a^{++})^{2})^{2}G_{j}^{++} \\ & − ((a^{+})^{2} + (a^{++})^{2})((a^{−−})^{2}(a^{−})^{2}G_{j}^{−−} − a^{2}(a^{+})^{2}G_{j}^{++}) \\ & − ((a^{−})^{2} + a^{2})(a^{2}(a^{+})^{2}G_{j} − (a^{++})^{2}(a^{+++})^{2}G_{j}^{++++}) \\ & − ((a^{−})^{2} + a^{2})((a^{−})^{2} + a^{2} + 2(a^{+})^{2} + 2(a^{++})^{2})G_{j+1} \\ & − a^{2}(a^{+})^{2}G_{j+1}^{++} − (a^{++})^{2}(a^{+++})^{2}G_{j+1}^{++++} \\ & − ((a^{−})^{2} + a^{2} + 2(a^{+})^{2} + 2(a^{++})^{2})G_{j+2}^{++}. \end{aligned} (4.4)

Proof. Use (4.1) with b ≡ 0 for (4.3) resp. (3.6), (3.7) with a = ρ for (4.4). □ Lemma 4.3. For all n ∈ ℤ,

(4.5) G̃_{j}(n) = G_{j}(n), j ∈ ℕ_{0}.

Proof. Our aim is to show that G̃_{j} satisfy the linear recursion relation (4.4) for Ĝ_{j}. We start with (4.3),

G̃_{j+3} − G̃_{j+3}^{+} + G̃_{j+3}^{+} − G̃_{j+3}^{++} = −(a^{+})^{2}G̃_{j+2}^{++} + a^{2}(G̃_{j+2} − G̃_{j+2}^{+}) + (a^{−})^{2}G̃_{j+2}^{−}

− (a^{++})^{2}G̃_{j+2}^{+++} + (a^{+})^{2}(G̃_{j+2}^{+} − G̃_{j+2}^{++}) + a^{2}G̃_{j+2}, (4.6) and observe that the right hand side of (4.4) only involves even shifts of G_{j}. Hence we systematically replace in (4.6) odd shifts of G̃_{j} by (4.3),

G̃_{j} = \{ \begin{matrix}G_{1,j} := G̃_{j}^{+} − (a^{+})^{2}G̃_{j−1}^{++} + a^{2}(G̃_{j−1} − G̃_{j−1}^{+}) + (a^{−})^{2}G̃_{j−1}^{−} \\ G_{2,j} := G̃_{j}^{−} + a^{2}G̃_{j−1}^{+} + (a^{−})^{2}(G̃_{j−1} − G̃_{j−1}^{−}) − (a^{−−})^{2}G̃_{j−1}^{−−} \end{matrix} ,

as follows:

G̃_{j+2}^{+++} → G_{2,j+2}^{+++}, G̃_{j+2}^{+} → xG_{1,j+2}^{+} + (1 − x)G_{2,j+2}^{+}, G̃_{j+2}^{−} → G_{1,j+2}^{−}, with

x = \frac{(a^{−})^{2} + a^{2} + (a^{++})^{2}}{a^{2} − (a^{+})^{2}} .

In the resulting equation we replace

G̃_{j+1}^{+++} → G_{2,j+1}^{+++}, G̃_{j+1}^{+} → yG_{1,j+1}^{+} + (1 − y)G_{2,j+1}^{+}, G̃_{j+1}^{−} → G_{1,j+1}^{−}, where

y = \frac{(a^{−})^{2}(a^{++})^{2} + a^{2}(a^{++})^{2}}{a^{2}(a^{++})^{2} − (a^{−})^{2}(a^{+})^{2}} .

This gives (4.4) for G̃_{j}. □

Hence both constructions for the KM hierarchy are equivalent and we have Theorem 4.4. Let r ∈ ℕ_{0}. Then

(4.7)  TL_{2r+1}(a, 0) = KM_{r}(a).

provided c_{2j+1}^{TL} = c_{j}^{KM} and c_{2j}^{TL} = 0 for j = 0, …, r.

Remark 4.5. As pointed out by M. Gekhtman to us, an alternate way of proving equivalence is by showing that (in the semi-infinite case, n ∈ ℕ) both constructions give rise to the same set of evolutions for the moments of the underlying spectral measure (compare [1]). Our purely algebraic approach has the advantage that it does neither require the semi-infinite case nor self-adjointness.

5. Appendix: Jacobi operators with b ≡ 0

In order to get solutions for the Kac–van Moerbeke hierarchy out of solutions of the Toda hierarchy one clearly needs to identify those cases which lead to Jacobi operators with b ≡ 0. For the sake of completeness we recall some folklore results here.

Let H be a Jacobi operator associated with the sequences a, b as in (2.1). Recall that under the unitary operator Uf(n) = (−1)^{n}f(n) our Jacobi operator transforms according to U^{−1}H(a, b)U = H(−a, b), where we write H(a, b) in order to display the dependence of H on the sequences a and b. Hence, in the special case b ≡ 0 we infer that H and −H are unitarily equivalent, U^{−1}HU = −H. In particular, the spectrum is symmetric with respect to the reflection z → −z and it is not surprising, that this symmetry plays an important role.

Denote the diagonal and first off-diagonal of the Green’s function of a Jacobi operator H by

\begin{aligned}g(z, n) & = ⟨δ_{n}, (H − z)^{−1}δ_{n}⟩, \\ h(z, n) & = 2a(n)⟨δ_{n+1}, (H − z)^{−1}δ_{n}⟩ − 1. \end{aligned} (5.1)

Then we have

Theorem 5.1. For a given Jacobi operator, b ≡ 0 is equivalent to g(z, n) = −g(−z, n) and h(z, n) = h(−z, n).

Proof. Set H̃ = −U^{−1}HU, then the corresponding diagonal and first off-diagonal elements are related via g̃(z, n) = −g(−z, n) and h̃(z, n) = h(−z, n). Hence the claim follows since g(z, n) and h(z, n) uniquely determine H (see [10, Sect. 2.7] respectively [8] for the unbounded case). □

Note that one could alternatively use recursions: Since g_{j}(n) and h_{j}(n) are just the coefficients in the asymptotic expansions of g(z, n) respectively h(z, n) around z = ∞ (see [10, Chap. 6]), our claim is equivalent to g_{2j+1}(n) = 0 and h_{2j}(n) = 0.

Similarly, b ≡ 0 is equivalent to m_{±}(z, n) = −m_{±}(−z, n), where

(5.2)  m_{±}(z, n) = ⟨δ_{n±1}, (H_{±,n} − z)^{−1}δ_{n±1}⟩

are the Weyl m-functions. Here H_{±,n} are the two half-line operators obtained from H by imposing an additional Dirichlet boundary condition at n. The corresponding spectral measures are of course symmetric in this case.

For a quasi-periodic algebro-geometric solution (see e.g. [10, Chap. 9]), this implies b ≡ 0 if and only if both the spectrum and the Dirichlet divisor are symmetric with respect to the reflection z → −z (cf. [3, Chap. 3]). For an N soliton solution this implies b ≡ 0 if and only if the eigenvalues come in pairs, E and −E, and the norming constants associated with each eigenvalue pair are equal.

Acknowledgments

We thank Michael Gekhtman and Fritz Gesztesy for valuable discussions on this topic and hints with respect to the literature. G.T. would like to thank all organizers of the international conference on Modern Analysis and Applications in honor of Mark Krein, Odessa, April 2008, for their kind invitation and the stimulating atmosphere during the meeting.

References

[1] Y. Berezansky and M. Shmoish, Nonisospectral flows on semi-infinite Jacobi matrices, J. Nonlinear Math. Phys. 1, no. 2, 116–146 (1994).

[2] W. Bulla, F. Gesztesy, H. Holden, and G. Teschl, Algebro-Geometric Quasi-Periodic Finite-Gap Solutions of the Toda and Kac-van Moerbeke Hierarchies, Mem. Amer. Math. Soc. 135-641, (1998).

[3] B. A. Dubrovin and V. B. Matveev and S. P. Novikov, Non-linear equations of the Korteweg–de Vries type, finite-zone linear operators and Abelian varieties, Russ. Math. Surv. 31, 59–146, (1976).

[4] F. Gesztesy, H. Holden, B. Simon, and Z. Zhao, On the Toda and Kac-van Moerbeke systems, Trans. Amer. Math. Soc. 339, 849–868 (1993).

[5] F. Gesztesy, H. Holden, J. Michor, and G. Teschl, Soliton Equations and Their Algebro-Geometric Solutions. Volume II: (1+1)-Dimensional Discrete Models, Cambridge Studies in Advanced Mathematics 114, Cambridge University Press, Cambridge, 2008.

[6] M. Kac and P. van Moerbeke, On an explicitly soluble system of nonlinear differential equations, related to certain Toda lattices, Adv. Math. 16, 160–169 (1975).

[7] S. V. Manakov, Complete integrability and stochastization of discrete dynamical systems, Soviet Physics JETP 40, 269–274 (1975).

[8] G. Teschl, Trace formulas and inverse spectral theory for Jacobi operators, Comm. Math. Phys. 196, 175–202 (1998).

[9] G. Teschl, On the Toda and Kac–van Moerbeke hierarchies, Math. Z. 231, 325–344 (1999).

[10] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surv. and Mon. 72, Amer. Math. Soc., Rhode Island, 2000.

[11] M. Toda, Theory of Nonlinear Lattices, 2^{nd} enl. edition, Springer, Berlin, 1989.

[12] M. Toda and M. Wadati, A canonical transformation for the exponential lattice, J. Phys. Soc. Jpn. 39, 1204–1211 (1975).

Imperial College, 180 Queen’s Gate, London SW7 2BZ, and International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria

E-mail address: Johanna.Michor@esi.ac.at
URL: http://www.mat.univie.ac.at/^{∼}jmichor/

Faculty of Mathematics, Nordbergstrasse 15, 1090 Wien, Austria, and International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Wien, Austria

E-mail address: Gerald.Teschl@univie.ac.at
URL: http://www.mat.univie.ac.at/^{∼}gerald/