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5Russian Journal of Nonlinear Dynamics, 2020, vol. 16, no. 3, pp. 397-419. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd200301
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 35Q58, 37J35, 47B39, 39A70
Inverse Spectral Problems for Second-Order Difference Operators and Their Application to the Study of Volterra Type Systems
In this paper, some links between inverse problem methods for the second-order difference operators and nonlinear dynamical systems are studied. In particular, the systems of Volterra type are considered. It is shown that the classical inverse problem method for semi-infinite Jacobi matrices can be applied to obtain a hierarchy of Volterra lattices, and this approach is compared with the one based on Magri's bi-Hamiltonian formalism. Then, using the inverse problem method for nonsymmetric difference operators (which amounts to reconstruction of the operator from the moments of its Weyl function), the hierarchies of Volterra and Toda lattices are studied. It is found that the equations of Volterra hierarchy can be transformed into their Toda counterparts, and this transformation can be easily described in terms of the above-mentioned moments.
Keywords: inverse spectral problems, difference operators, Jacobi matrices, Volterra lattices, Toda lattices
1. Introduction
Since the classical works of Kac, van Moerbeke and Moser [1, 2] the inverse spectral problems for difference operators have been applied for integration of nonlinear dynamical systems. A typical example of such application is the work of Berezanski [7], where the semi-infinite Toda lattice was integrated by means of the classical inverse spectral problem for Jacobi operators. This work inspired the development of the inverse spectral problem methods with an aim to cover wider classes of nonlinear systems. One of the advantages of these methods is that they impose loose restrictions on the operator's coefficients and the inverse problem data. Also, since the inverse spectral problems for difference operators are connected with the theory of continued fractions, Hermite-Pade approximations and orthogonal polynomials, the development of integration methods for nonlinear systems had an influence on these "neighboring" areas, see [12? ].
Received December 12, 2019 Accepted July 23, 2020
Andrey S. Osipov osipa68@yahoo.com
Scientific Research Institute for System Analysis of the Russian Academy of Sciences Nakhimovskii pr. 36-1, Moscow, 117218 Russia
A. S. Osipov
Our aim here is to look at some issues arising in the study of integrable systems from the inverse spectral theory point of view. Although most of these issues are not new, the proposed approach allows generalization (namely, to the case of systems with matrix (or operator) coefficients).
In this paper, we mainly consider the Volterra lattice (also referred to, due to its physical applications, as Langmuir lattice), which is a typical object of study in the theory of integrable systems. This system has a number of interesting properties, for example, it can be considered as a discrete analog of the Korteweg-de Vries equation, see [9, 12, 16]. It also appears in the discretization of conformal field theory; the Poisson bracket for this system, found by Faddeev and Takhtajan, provides a lattice generalization of a certain Virasoro algebra [5, 20].
The paper is organized as follows. The next section contains a brief summary of the inverse spectral problem method for the Jacobi operators (we skipped some details, e.g., those related to J-continued fractions, which are not relevant to our further discussion). In Section 3 it is shown how, using this method, one can construct a hierarchy of Volterra lattices, and the proposed approach is compared with the construction method based on Magri's bi-Hamiltonian formalism [15, 16]. In Section 4, using an inverse spectral method for nonsymmetric difference operators of the second order, we build up the hierarchies of Volterra and Toda systems and show how the systems of Volterra hierarchy can be transformed into Toda-type systems. Here we mention that a similar transformation between the finite Volterra and Toda lattices was established in [10], which allows one to transform the Poisson brackets, master symmetries,first integrals of a given Volterra lattice to the similar objects of the corresponding Toda lattice. We describe this transformation in terms of the inverse spectral data of Lax operators, corresponding to both types of systems. Also we outline the application of our approach to the finite systems of Volterra and Toda type.
2. Inverse problems for Jacobi operators
First we recall some basic facts and definitions from the theory of Jacobi operators, essential for further consideration [3, 7]. Namely, consider the semi-infinite Jacobi matrix
/be ao 0 0
J = ao bi ai 0
0 'J
such that
ai > 0, bi G R, i G Z+.
Denote by l2 the Hilbert space of complex quadratic summable sequences and by (en)00=o its standard orthonormal basis. The matrix J generates the (minimal) closed symmetric operator in l2 and in the basis (en) this operator (also denoted by J) has the matrix representation J. For the operator J, its spectral function is defined as: p(A) = (E\e0,e0), where E\ is the spectral measure of a certain self-adjoint extension Jext of J (i. e., according to the spectral theorem for
oo
self-adjoint operators, J C Jext = f XdE\; more details and definitions can be found in [3, 6]). The Weyl function of J is defined as the Stieltjes transform of p(A): m{z) = f in terms
of the inverse spectral problems, m(z) is the object equivalent to p(A).
The inverse spectral problem for the operator J can be formulated as follows: given p(A) or m(z), find ai and bi (restore the matrix J).
oo
m,
For the moments sn := f Andp(A), n G Z+ we have
Sn = / And(Exea, eo) by the spe=al theorem (Jneo, eo) = (Jn)o,o;
—oo
(the latter notation is for the matrix J). In fact, these moments are related to the asymptotic expansion of the resolvent Rz := (zE — J)-1 of the operator J at infinity, because for sufficiently
large z we have (zE — J)~l ~ ^ + + +____ By the spectral theorem, m(z) = (R*e o, eo).
z z2 z
Thus, m(z) is a resolvent function of the operator J, and we can call the sequence S = {sj }°=o the moment sequence corresponding to the Weyl function m(z). Also note that although there may be different self-adjoint operator extensions corresponding to the same matrix J (and therefore different p(A) and m(z)), the sequence S is unique for the matrix J.
As follows from the Hamburger theorem (its detailed proof is contained, e. g., in [3]), S is a positive sequence, which means that for all n
An(J) > 0; (2.1)
where An(J) are determinants of the Hankel matrices Hn = (si+j)nj=0. The positiveness of S implies that a corresponding p(A) defined on R has infinitely many growth points. Also, p(A) is a nondecreasing function (this fact follows from its definition and the spectral theorem). To the matrix J we assign the second-order difference equation:
an-iyn-1 + bnyn + anyn+1 = Ayn, A e C, n = 1,2,.... (2.2)
It has two linearly independent systems of solutions:
P(A) := (Pn(A))~o and Q(A) := (Qn(A))~o; (2.3)
P0(A) = 1, Pi(A) = ^—Qo(A) = 0, Qi(A) =-.
ao ao
Thus, Pn(A) and Qn(A) are polynomials in A, and any solution (yn(A))00=o is a linear combination of them: yn = Pn(A)c + Qn(A)d, c,d G C. We write Pn(A) = po,n + Pi,nA +... + Pn,nAn; Pn,n = 0. The polynomials Pn(A) are orthogonal with respect to S (or dp(A)):
I oo
n+m,
Pi,nPj,mSi+j = Pn(A)Pm(A)dp(A) = 5m
ij=0 -oo
This can be proved in the following manner. Using Eq. (2.2), for the vectors en we get en = = Pn(J)eo. Then we have
5m,n = (em,en ) = (Pm (J )eo ,Pn(J )eo ) = (Pn (J )Pm (J )eo, eo) =
Pn(A)Pm(A)d(Exeo,eo) = j Pn(A)Pm(A)dp(A).
)-i ^^ )z-o) — Vx m\
by the spectral theorem
Using the above relations, we get a solution of the inverse problem: given p(A) and the sequence of powers 1, A, A2, ... we construct the polynomials Pn(A) with leading positive coefficients by applying the standard orthogonalization procedure. From this orthogonality and Eq. (2.2) it follows that
œ œ
an = j APn(A)Pn+i(A)dp(A), bn = J AP2n(A)dp(A).
(2.4)
Also, the orthogonality relations lead to the following representation of Pn(A):
Sq Si ... Sn
Pn(A) =
1
An-i(J )An (J )
Si S2
sn-1 sn
1A
sn+i
s2n-1
An
Hence, Pn{A) = JA" + Rn-i(A) and it follows from Eq. (2.2) that
y An (J )
(2.5)
where
Dn (J ) = det
\J Ara+i( J)Ara_i( J) A n(J)
i So ... sn— i Sn+i
\sn . . . s2n-1 S2n+i
bn =
Dn (J ) Dn-i(J )
An(J ) An-i(J ):
A-i = 1, D-i =0, Do (J ) = si.
(2.6)
Thus, Eqs. (2.6) give the reconstruction procedure of the elements of J in terms of the moment sequence S, and this procedure amounts to the solution of the inverse spectral problem. In other words, we are coming to the following formalization of the latter:
a
n
S = {(Jfc)o,o}fc°=o ^ J.
Condition (2.1), together with sQ = 0, gives a criterion for an arbitrary sequence S = {s^}£=q of integers to be the moment sequence of a certain Jacobi matrix J.
Using the second of equations (2.6) and applying induction on n, we obtain the following criterion of "sparsity" of the matrix J in terms of S:
Proposition 1. {bn = 0}~=Q ^ {S2fc+1 = 0}^=q.
Therefore, for the matrix J with zero main diagonal, all odd moments of dp(A) (or the Weyl function of J) are zero. Also, as noted by Moser in [2], for such J the corresponding Weyl function m(z) is odd.
Most of the above results have analogs for nonsymmetric difference operators (e.g., operators generated by infinite three diagonal matrices, see Section 4), including the operators with matrix coefficients.
3. Integration of semi-infinite Volterra lattices
Consider the Cauchy problem for the system:
bi = bi(bi+i — bi-i), i e Z+, bi = bi(t), t e [0,T), 0 <T <
(3.1)
bi(t) G l
oo, b-1
Ö-1 = 0.
Assume that for all t, bi(t) > 0 and set bi = a?, ai > 0. Then the system can be rewritten as
ài — — a.i(a,f, l — a2^).
(3.2)
This system admits the Lax representation L = [L, A] with the following infinite matrices L = L(t) and A = A(t) [1]:
L(t) =
V
.
A(t) =
( 0 ao 0 0 0 \
ao 0 a1 0 0
0 a1 0 a2 0
0 ... ... ... ..
( 0 0 -aoai 0 0 \
0 0 0 -aia2 0 aoa1 0 0 0 -a2a3
0 a1 a2 .. .. ..
(3.3)
Thus, the matrix L(t) is a Jacoby matrix with zero main diagonal, and all previous results are valid here. Note that A(t) is a skew-symmetric part of L?(t)/2, i.e.,
A(t) = skew(L2(i)/2) = - (LL(i) - L2up(t)),
(3.4)
where L?ow(t) and LUp(t) are, respectively, strict lower and upper triangular parts of L2(t). Using the above Lax representation and the evolution of polynomials P(A, t) and Q(A, t) corresponding to L(t), one may get the following equation for its Weyl function, see e.g. [6]:
m(z, t) = m(z, t)(z2 — a?(t)) + z = m(z, t)(z2 — s2(t)) + z. Since m(z,t) is an odd function, we have: m(z,t) = 0.5(m(z,t) — m(—z,t)) = z f
(3.5)
dp(A, t) Ä2T3-
We therefore find:
'S
dp(X, t) A2 — z2
=z
{z2 - c%{t))dp{\,t)
+ 'J dp(A, t) =
-
oo
=z
(A2 - a%(t))dp(\,t) X2-z2 '
Using the uniqueness of the Stieltjes transform, we get the following evolutionary equation for the spectral function of L(t):
dp(A, t) = (A2 - a2(t))dp(A, t) = (A2 - s2(t))dp(A, t). (3.6)
Therefore,
dp(A, t) = ex2tdp(A, 0)e— ^ al(T)dT. Using the normalization condition for dp(A,t), we find
X x
1 = I dp(A,t) = e- $ al(T)dT j ex21dp(A, 0), -( -(
and, finally,
, , ex21dp(A, 0)
J ex21dp(A, 0) -(
Equation (3.6) leads to the corresponding equation for the moments:
S2k = S2(fc+1) - S2S2k, k G Z+.
Note that this formula is similar (up to a constant) to the formula (2.2) from the classical paper of M. Kac and P. van Moerbeke [1].
In view of the above, we can apply the procedure of recovering the elements a^t) from dp(A,t), thus solving the Volterra system.
Theorem 1. Suppose we have a nondecreasing function p(A,t) defined on the Borel sets
of R, with infinitely many growth points and such that all its moments sk (t) = f Ak dp(A,t)
are finite, s2k+1(t) = 0, k G Z+; s0(t) = f dp(A,t) = 1. Also suppose that dp(A,t) satisfies Eq. (3.6) for all t G [0,T). Then
ai(t)= J APi(A, t)Pi+i (A,t)dp(A,t), (3.7)
where {Pi(A,t)}00=o are the polynomials orthogonal with respect to dp(A,t) and satisfy the system (3.2).
Proof. Since {Pi(A,t)}°
o is an orthonormal system with respect to dp(A,t), we get
APi(A,t) = a— i(t)Pi— i(A, t) + ai(t)Pi+i(A,t), i = 1,2,... (3.8)
(one may consider ai(t) as the Fourier coefficients of polynomials APj,(A,t) with respect to this system. Since s2k+1 = s2k+1(t) = 0, using Eq. (2.5) one can check that the polynomials Pj,(A,t)
X
contain only even powers of A, thus J" APi(A,t)Pi (A, t)dp(A, t) = 0).
Then, differentiating Eq. (3.7), we have
y XPi(X, t)Pi+i (X,t)dp(X,t)\ = j XPi(X, t)Pi+i (X,t)dp(X,t) +
— oo / — oo
œ oo
+ J XPi(X,t)Pi+i(X,t)dp(X,t)+ J XPi(X,t)iPi+i(X,t)dp(X,t) = I + II + III.
— oo —oo
Using Eq. (3.6), we find
oo oo
X3Pi(X,t)Pi+i(X,t)dp(X,t) - s2(t) j XPi(X,t)Pi+i(X,t)dp(X,t) =
— oo —oo
J A3Pi(A,t)Pi+1 (A,t)dp(A,t) - S2(t)ai(t).
— oo
From Eq. (3.8) and orthogonality of {Pj,(X,t)} it follows that
oo oo
II = Oi+1 (t) j P%(X,t)Pi+2(X,t)dp(X,t) + ai(t) J Pi(X,t)Pi(X,t)dp(X,t) =
—oo —oo
oo i oo
= Oi+1 (t) ! (t)Pk(X,t)Pi+2(X,t)dp(X,t) + ai(t) Î Pi(X,t)Pi(X,t)dp(X,t) = k—0
—oo k—0 —oo
= Oi(t) j Pi(X,t)Pi(X,t)dp(X,t), —
oo
where fk(t) = J Î}i(r,t)Pk(r,t)dp(X,t) are the Fourier coefficients of polynomials P}i(X,t) with —
respect to the system {Pi(X, t)}.
oo
Differentiating J Pi(X,t)Pi(X,t)dp(X,t) = 1by t, we get —
oo oo
0 = 2 y Pi(X, t)Pi (\,t)dp(\,t)+ J X2Pi (X,t)Pi(X,t)dp(X,t) - S2(t).
—oo —oo
Thus, for all i
oo / oo
[ P(X,t)P(X,t)dp(X,t) = -i I [ X2P2(X, t)dp(X, t) - s2(t) I ; (3.9)
and for II we finally find
II = ^ I - j X2P2(X,t)dp(X,t)+s2(t)
For III we get from Eq. (3.8)
X x
III = ai(t) J Pi+i(A,i)Pm(X,t)dp(X,t) + a-1 (t) j Pl+i(\,t)P-i(X,t)dp(X,t). -( -(
x
Differentiating by t the relation / Pi+1(X,t)Pi-1 (X,t)dp(X,t) = 0 and using Eq. (3.6), we find
-(
that
X X
J Pi+1 (X,t)Pi-i(X,t)dp(X,t) = -J X2 Pi-i(X,t)Pi+i (X,t)dp(X,t).
-X -X
X
Applying Eq. (3.9) to J" Pi+1(X,t)Pi+1 (X,t)dp(X,t), we finally find that III can be written as
-X
/ X \ X
III = X2P-+i(X,t)dp(X,t)-s2(t) J -Oi_i(i) J X2Pi-i(X, t)Pi+i(X, t)dp(X, t).
-X -X
Summing up I, II and III, we arrive at the following relation for <ii(t):
X
ai(t) = j X3Pi(X,t)Pi+1 (X,t)dp(X,t) -
X3 X
f X2P2+1(X,t)dp(X,t) + i X2P2(X,t)dp(X,t) I -
2
\—oo
X
- a-1 (t) j X2Pi-1(X,t)Pi+1 (X,t)dp(X,t). (3.10)
-X
Again, applying Eq. (3.8) to get rid of X2 and X3 in Eq. (3.10), we finally obtain
• _ 2 ,3,2 ai / 2 l02,2 \ 2 _l/2 2\
ai — (i'i-i(ii + O'i + ai+ia,i —+ ^ai * ai+I) ~ ai~iai — ~~;ai\ai4-i ~ ai-iJ-
2 2 □
Note that in the works of Bogoyavlenskii the first equations of Volterra hierarchy were derived from their common Lax representation with spectral parameter. Also, the links between the Volterra hierarchy and the hierarchy of Korteveg-de Vries (KdV) equations were studied there (see [9] for details). These links were considered in detail in [16], where it was finally established that the equations of the former hierarchy (named in [16] the KM hierarchy after M. Kac and P. van Moerbeke) represent a discretization of the equations of the latter one. Here the shape of Eq. (3.6) and the method of proving the above proposition lead us to another way of constructing this hierarchy. Namely, the following statement holds.
Theorem 2. There exists a countable set of nonlinear dynamical systems having the same matrix L in their Lax representation as the system (3.2). The corresponding spectral functions satisfy the evolutionary equations:
dp(X,t) = (X2k - S2k(t))dp(X,t), k E N. (3.11) "j^_RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2020, 16(3), 397-419_
It can be proved in the following manner. Assuming that Eq. (3.11) holds true and applying the same arguments as those used in deriving Eq. (3.10), we get the following relation:
0i(t)= J \2k+1Pi(X,t)Pi+i(\,t)dp(\,t) —
X2kP2+i(X,t)dp(X,t)+ [ \2kP2(X,t)dp(X, t)
2
\—oo
- a-i(t) J X2kPi-i(X, t)Pi+i(X,t)dp(X,t) (3.12)
-
(for k = 1 this formula coincides with Eq. (3.10)). From Eq. (3.12) the desired systems can be obtained in the same way as the system (3.2) was derived from Eq. (3.10). This procedure is easy programmable. In particular, we get the following systems (V1 is the initial system (3.2)):
/2 2 ai ' ,2 „2 J2 „2
v2'- ài - -J ( a2+1 ai+k - ai-1 y.
+1 > ai+k 1
i+k ai-1 ai-k k=0 k=0 /
2 k+1 2 k+1
• ^ = — I a,i+1 ^^ a'i+p ~ a'i-i a'i-k a'i-p
k=0 p=0 k=0 p=0
a , 2 k+1 p+1 2 k+1 p+1
: ài = —iai+i ^^ ai+k ^^ ai+p ai+q ~ ai-1 X-/ ai-k ai-p
^ k=-1 p=0 ç=r k=-1 p=0 ç=r
where r = max(0, —k); (3.13)
Then it can be checked that each system Vk satisfies the Lax equation
L(t) = [L(t),Ak (t)], (3.14)
where Ak(t) = skew(L2k(t)/2) (cf. Eq. (3.4)). In other words, Vk are the systems of Volterra (or KM in terms of [16]) hierarchy. Note that the procedure of obtaining Vk from Eq. (3.12) is computationally faster than the one based on Eq. (3.14), because no matrix calculations are needed.
Also, the Volterra hierarchy can be generated via the bi-Hamiltonian structure of the initial system bi = bi(bi+i — bi-1), based on the two Poisson brackets [10]:
{bi+i,bi}2 = bibi+i, {bi+i,bi}3 = bibi+i(bi + bi+i), {bi+2,bi>3 = bibi+ibi+2.
Namely, for the systems of Volterra hierarchy {Vk}£=i, written in coordinates bk, we have
Vk : bi = {Hk-1, bi}3 = {Hk, bi }2,
where Hk are the Hamiltonians, in particular, for Vi: H0 = ^i ln bi, Hi = ^i bi = Tr(L2/2). This bi-Hamiltonian formalism, which dates back from the work of Magri [15], leads to an algorithm (offered in [16]) for generating the systems Vk which can be represented by the diagram:
Hk-i H3 Vk-i H2 Hk H3 Vk H ... (3.15)
(here we tacitly assume that Hk are finite, which holds, e.g., for the finite or periodic lattices). In [16], V2 and V3 were obtained in this manner. In general, to obtain higher systems using this scheme, one should get the whole chain of "lower" Hamiltonians. However, as noted in [16], in this particular case all the Hamiltonians can be calculated independently by the formula
' L2k"
Hk, = Tr. 2
where L(t) is defined by Eq. (3.3) (in the finite case L(t) is a finite Jacobi matrix). The Hamiltonians of Volterra hierarchy are presented in Table 1, where
ai . 2 2 \ ^ (b2
H2 = E(y + «?«m)=E(|+^ i ^ ' i ^ '
= ( -§- + bíbí+lbí+2 ) >
i
m = y^ + ~ + blbl+lbl+2(bl + 2bl+l + bl+2 + 5»+3)) ■
Table 1. Hamiltonians of Volterra hierarchy
System Vi U2 U3 v4
U2 Hi H2 H3 H4 ...
{, }3 Ho Hi H2 H3 ...
Although the proposed approach to generation of Volterra-type systems does not require the knowledge of these Hamiltonians, we can first derive Vk from Eq. (3.12) and then find Hk from the Hamiltonian representation of Vk with respect to {, }2. For example, to obtain V5 given V4, we can calculate H4 in this manner and further on, applying { , }3, find the equations for V5:
, 2 k+l p+l q+l
bi = bA bi+r —
^ k=0 p=0 q=0 r=0
2 k+l p+l q+l
- bi-iJ2 bi-kJ2 bi-pJ2 bi-q^2 bi-r +
k=0 p=0 q=0 r=0
-l 2m+3 n+l ll+k(k-m)(l-\n\)
+ bi+l E bi+m E bi+n^2 bi+k E bi+l —
m=-2 n=m+l k=0 l=ll
-l 2m+3 n+l ll+k(k-m)(l-\n\)
— bi-l E bi-m E bi-n^2 bi-k E bi-l
m=-2 n=m+l k=0 l=ll
where 11 = \k — 11 — m — 2| + 2 j ^™n |. A straightforward application of Eq. (3.12) for k = 5
(with the help from MATLAB Symbolic Toolbox) leads to the same result with the comparable computation time.
4. Nonsymmetric case
4.1. Volterra and Toda hierarchies
First consider the infinite three-diagonal matrices of the form
/0 0 0 0 0 \ 1 ai;i 0,1,2 0 0 0 1 02,2 02,3 0
M —
0
ai,i+i G C, ai,i+i = 0, i G N.
(4.1)
In the following we will identify each matrix M with the (possibly unbounded) nonsymmetric operator defined as the closure of the operator acting on the dense set of finite vectors from l2, where the action of this operator is described via matrix calculus. In the same manner as above, we can define the Weyl function of M by the formula m(z) = (Rze0, e1), where Rz = (zE — M)_1 is
oo s
the resolvent of M, its formal power expansion at infinity: m^A, L) := ^ ^ , S*fc = Ih j, ,,
fc=o A +
(So = 0, S1 = 1) and its moment sequence S = {Sk}0=o. This moment sequence of the Weyl
i
function generates the following bilinear mapping on the ring of polynomials: for F(A) = FPAp,
p=0
G(\) ^ GiX l=0
(F(A), mo(A, L)G(A)) = (1,F(A)mo(A, L)G(A)) FpSp+l+1Gl.
p=0l=0
Let Ak be determinants of the Hankel matrices (Si+j+1)kJ=0, k G Z+. Similarly to Eqs. (2.6), the elements of M can be found by the formulas
ai,i+1 —
AiAi_
i^i-2
Di-i Di_
ai,i -
i-2
Ai i Ai
i2
(4.2)
where
Di — det
f Si ...Si Si+2 \
VSi+1 ... S2i S2i+2 J
A-1 — 1, D-1 — 0, Do — S2.
The following criterion for the sequence S = {Sk }0=0 to be the moment sequence corresponding to the Weyl function of a certain matrix M holds:
Theorem 3. The sequence S is the moment sequence of the Weyl function of the matrix M defined by Eq. (4.1) iff
S0 = 0, S1 = 1 (normalization condition); for all k G Z+,
Afc — 0.
(4.3)
The latter condition is equivalent to condition (2.1) for the Jacobi matrices. This result follows from Theorem 2 of [17], which establishes a similar criterion for a wider class of infinite band matrices. As in the case of Jacoby matrices, the sparsity criterion holds for the matrices M :
Ki = 0}= ^ {s2k = 0}?=o.
Now consider the Lax representation for the system (3.1) with the nonsymmetric matrix L = L(t):
L(t) =
(0 0 0 0 0 \ 10 bo 0 0 0 1 0 bi 0
0
/0 0 0 0
, A(t) =
.
0
\
0 0 0 -bobi 0 0 0 0 0 -bib2 0 0 0 ... ...
= -Lup(t).
(4.4)
.
Also assume that bj (0) = 0, i G Z+. Obviously, L(t) is a (sparse) matrix of the same structure as M, so all above results are applicable to this case.
For the matrix L we define the polynomials (Pk)kL0, Pk = Pk(X,t) satisfying the equation
Pk-i + bk-iPk+1 = Xyk, X G C, k = 1,2,...
with the initial conditions P0 = 0, P1 = 1, and the polynomials (Tk)X=0, Tk = Tk(X,t) satisfying the "adjoint" difference equation:
Tk-ibk-2 + Tk+i = XTk, k G N, b-i = 1; To = 0, Ti = 1.
One can check (by induction) the orthogonality relations with respect to m^(X, L, t) corresponding to L(t):
(1,Pi (X,t)mx (X, L, t)Tj (X,t)) = , i,j G N. (4.5)
Therefore, we get
bk = (1,XPk+i(X,t)mx (X,L,t)Tk+2 (X,t)).
Using the equation (Lk(t))' = LkA — ALk, which follows from the Lax representation, we find the evolutionary equations for the moments Sk:
Sk = Sk+2 — S3Sk, k G Z+, (4.6)
from which it follows that
(1,XPk+i (X,t)m x (X,L,t)Tk+2(X,t)) = (1,X3 Pk+i(X,t)mx (X,L,t)Tk+2 (X,t)) — Sb.
Thus, as in the proof of Theorem 1, we are coming to the equation which is equivalent to that of the Volterra lattice (3.1) (cf. Eq. (3.10)):
bk = (1,X3 Pk+i (X,t)mx(X,L,t)Tk+2 (X,t))
— (1, X2Pk+i(X, t)mx(X, L, t)Tk+i(X, t))bk —
— (1, X2Pk(X, t)mx(X, L, t)Tk+2(X, t)).
Similarly, we get the equations for the moments of another systems of Volterra hierarchy:
Vm
Sk = Sk+2m — S2m+1Sk, m = 1, 2,...
(4.7)
and the equations for Vm, similar to Eqs. (3.12):
bk — (1,X2m+1 Pk+1(X,t)m00(X,L,t)Tk+2(X,t)) -
- (1, X2mPk+1(X, t)mo(X, L, t)Tk+1(X, t))bk -
- (1, X2mPk(X, t)mo(X, L, t)Tk+2(X, t)).
(4.8)
Here we mention that unlike in the symmetric case, where the uniform boundedness of initial data guarantees the existence of a global solution of the corresponding Volterra system, here we can claim that for the bounded initial data the solution exists and is unique only on the interval [0,5) for some 5 > 0, see [14] for details.
Also note that the above formulas are valid in the non-Abelian case, e.g., when bk are matrices or operators (details of the inverse problem method applicable to this case can be found in [12]). For example, using Eqs. (4.8), we can derive equations for V3 both in the Abelian and non-Abelian case (cf. the third of Eqs. (3.13)):
bk = bk bk+1bk+2bk+1 + bk bk+1bk+2bk+2 + bk bk+1bk+2bk+3 + bk bk bk bk+1 +
+ bk bk+1bk bk+1 + bk bk bk+1bk+1 + bk bk bk+1bk+2 + bk bk+1bk+1bk+1 +
+ bk bk+1bk+1bk+2 — bk-1bk-2bk-1bk — bk-2bk-2bk-1bk — bk-3bk-2bk-1bk —
— bk-1bkbkbk — bk-1bkbk-1bk — bk-1bk-1bkbk — bk-2bk-1bkbk —
— bk-1bk-1bk-1bk — bk-2bk-1bk-1bk + bkbk-1bkbk+1 — bk-1bkbk+1bk.
Also, consider the Toda lattice To1:
A k = Ak Bk+1 — Bk Ak, Bk = Ak — Ak-l, (4.9)
where Ak, Bk G C and Ak = 0 (the non-Abelian To1 can be written similarly). In the semiinfinite case it admits the Lax representation with the following infinite matrices L and A:
L — L(t) —
A — A(t) —
0 0 0 0 0
1 Bo Ao 0 0
0 1 B1 A1 0
0
0 0 0 0
0 0 -Ao 0
0 0 0 -A 1
0 0
\
(4.10)
— -Lup(t).
Similarly as above, define (Pk(A,t))0=0 as the polynomials satisfying
Pk-1 + Bk-1Pk + Ak-1Pk+1 = APk, k = 1, 2,...; P0 = 0, P1 = 1; and the "adjoint" polynomials (Tk(A,t))0O=0 as the ones satisfying the equation
Tk-1Ak-2 + TkBk-1 + Tk+1 = ATk, k = 1,2,...; A-1 = 1; T0 = 0, T1 = 1.
These two polynomial systems are mutually orthogonal with respect to m^(X, L,t), e.g., the relations similar to Eq. (4.5) hold for them. For To\, the equation for the moments similar to Eq. (4.6) for the Volterra system takes the form
S k = Sk+i - Bo (t)Sk = Sk+i - S2 Sk, Sk = (Lk (t))i,o. (4.H)
Using Eq. (4.11), one may check that Eqs. (4.9) can be written as
Ak = (1, X2Pk+i(\t)m^(\,S,t)Tk+2(X,t))-
- (1, XPk+1 (X, t)m^(X,S, t)Tk+i(X, t))Ak -
- Bk(1, XPk+i(X, t)mx(X, 1, t)Tk+2(X, t))
- (1, XPk(X, t)m^(X,S, t)Tk+2(X, t));
Bk = (1,X2 Pk+i(X,t)mx (X,S,t)Tk+i (X,t))-
- (1,XPk+i(X,t)m^(X,S,t)Tk(X,t))Ak-i -
- (1, XPk+i(X, t)mUX,S, t)Tk+i(X, t))Bk -
- (1, XPk(X, t)m^(X, 1, t)Tk+i(X, t)).
The last term in the equation for Ak is actually zero, but we write it here for convenience to unify the shape of all equations of Toda hierarchy {Tom}„=i. Namely, the evolutionary equations for the moments of the elements Tom of this hierarchy are written as
Sk = Sk+m - Si+m Sk, m = 1 Z^.^ (4.12)
using which, we obtain
Tom: (4.13)
Ak = (1,Xm+iPk+i,t(X,t)m^(X,S,t)Tk+2(X,t)) -
- (1, XmPk+i(X, t)mx(X,S, t)Tk+i(X, t))Ak -
- Bk(1, XmPk+i(X, t)mx(X,S, t)Tk+2(X, t))
- (1, XmPk(X, t)mx(X,S, t)Tk+2(X, t));
Bk = (1,Xm+iPk+i(X,t)m^(X,S,t)Tk+i(X,t))-
- (1,XmPk+i(X,t)m^(X,S,t)Tk(X,t))Ak-i -
- (1, XmPk+i(X, t)mx(X,S, t)Tk+i(X, t))Bk -
- (1, XmPk(X, t)m<x(X,S, t)Tk+i(X, t)).
One can check that the systems defined by Eqs. (4.13) satisfy the Lax equations S(t) = = [S(t), Am(t)], where S(t) is defined by Eq. (4.10) and Am(t) = -^(t).
Theorem 4. The equations of Volterra hierarchy (4.8) can be transformed to their Toda counterparts defined by Eqs. (4.13). The corresponding mapping for the moments is defined as
S2k-i ^ Sk, k e N, So = 0, (4.14)
which implies
Bk = b2k + b2k-i, Ak = b2kb2k+i; k e Z+; b-i = 0. (4.15) . RUSSIAN JOURNAL OF NONLINEAR DYNAMICS, 2020, 16(3), 397 419_
Proof. Let S = {Sk}£=0 be the moment sequence corresponding to Vm. Then, according to (4.14), we get
S _ A by q Q Q _ q S S
S k = S2k—1 = S2k+2m—1 — S2m+1S2fc— 1 = Sk+m — Sm+1 Sk j
which is exactly Eqs. (4.12), so we can derive from them Eqs. (4.13). To establish Eqs. (4.15), we first find from Eqs. (4.2) that
/ S1 0 S3 0 .
0 S3 0 S5 .
bk = 1 where Ak = det
Al
S3 0 S5 0 0 S5 0 S7
Sk+1 \ Sk+2 Sk+3 Sk+4
\Sk+1 Sk+2 Sk+3 ... S2k S2k+1/
(Sk = Lk 0). Using the sparse structure of Ak, we get the following factorization of the latter (can be proved by direct check):
Ak — Lm Rm/,
k e N,
(4.16)
where
Lm
S1
S3
S3 S5
S2m+1 S2m+3
S2m+1 S2m+3 . . . S4m+1 k'
Rm' —
S3 S5
S5 S7
S2m'+1 S2m'+3
S2m'+1 S2m'+3 ... S4m'-1
m —
2
m —
k1
+ 1.
Set Ai = det(i°i) where Hi = {,S"i+i=0. According to (4.14), Ai = det({S2(i+i)+1 }lij=0) and therefore, Ai = Li for l £ N. By Theorem 3, Ak = 0 for all k, and it follows from formula (4.16) that Ai = 0 for all l as well; therefore, by the same theorem, S is the moment sequence corresponding to a certain matrix M. If we denote in this matrix ak+1 ,k+2 and ak+1, k+1, k £ Z+, by Ak and Bk, respectively, then it follows from Eqs. (4.2) that
Ak —
Ak+iAk_!
Àf. ^
Dk D
k = ~--
k1
AA k A
D
k—
k1
S1
Sk+1
Sk Si
k+2
S2k S2k+2
(D0 = S2, D—1 = 0, AA—1 = 1). Also note that according to (4.14),
Rk —
SS2 SS3
Sk+1 Sk+2
SSk+1
S2k
We have
An —
Ai
A2 A0
S1 S3 S3 S5
— &2 + bob1 - b2 — bob1.
For k > 1
b2k b2k+l
A2k-lA2k+2 _ Lk-lRkLk+lRk+l _ Lk+lLk-l Ak+lAk-l
Aok A
2k A2k+l
Lk Rk Lk Rk+l
Lk
A k
= A
k ■
The formula for Bk follows from the identity
2l
J- = Y, bm, I > 0.
Al m=0
(4.17)
D o
It can be proved by induction on I. For I = 0, — = ¿>2 = S3 = bo. Suppose the identity defined
A 0
by formula (4.17) is proved for l = 0,...,k - 1. It means that
2k
Ebk
Dk-l . , . , Dk-l . AkRk-l . Ak-lRk+l
+ &2fc-l+&2fc = ^----h
m=0
Ak-i Ak-i Ak-iRk Ak{Dk-lRk + AkRk-l) + ^k-lRk+l A k-iA k Rk
For the expression in brackets the following representation holds:
A k Rk
AkRk-l + Dk-lRk = Ak-lD{k1ll, where D^ =
S2
Sk+l Sk+3
Sk+2 ■ ■ ■ S2k+l S2k+3
It follows from one general identity for determinants proved in the Appendix. To establish the identity (4.17) for l = k, we have to check that
A k D k_l + A k-lRk+l = Dk Rk,
or
Sl S2 ■ ■ ■ Sk+l S2 S3 ■ ■■ Sk+2
52 S3 ■ ■ ■ Sk Sk+2
53 S4 ■ ■ ■ Sk+l Sk+3
Sk+l Sk+2 ■ ■ ■ S2k+l Sk+l Sk+2 ■ ■ ■ S2k-l S2k+l
+
+
SSl SS2 SSk
S2 S3 ■ ■ ■ Sk+l
52 S3 ■ ■ ■ Sk+2
53 S4 ■ ■ ■ Sk+3
00 O 00 O
Sk Sk+l ■ ■ ■ S2k-l Sk+2 Sk+3 ■ ■ ■ S2k+2
Sl ■ ■ ■ Sk Sk+2 S2 ■ ■ ■ Sk+l Sk+3
s O O O O O
Sk ■ ■ ■ S2k S2k+2 Sk+l Sk+2 ■ ■ ■ S2k
52 S3 ■ ■ ■ Sk+l
53 S4 ■ ■ ■ Sk+2
The latter formula can be written as
k — 1 times
Sj
S2
0 0
j2 S3 S3 S4
Sk+1 Sk+2 Sk+2 Sk+3
Sk 0. ■ ■ 0 Sk+1 Sk+2 ■ ■ ■ S2k S2k+1
S1 S2 ■ ■ ■ Sk 0 ■■■ 0 Sk+1 Sk+2
S2 S3 ■ ■ ■ Sk+1 0 ■■■ 0 Sk+2 Sk+3
Sk+1 Sk+2 ■ ■ ■ S2k
0
0 S2k+1 S2k+2
k — 1 times
Using elementary matrix transformations, we can write this determinant as
k — 1 times
j1 j2
S?2
S3
S4
Sk+1 Sk+2 Sk+2 Sk+3
Sk 0. ■ ■ 0 Sk+1 Sk+2 ■ • ■ ■ S2k S2k+1
0 S2 ■ ■ ■ Sk 0 0 0
0 S3 ■ ■ ■ Sk+1 0 0 0
0 Sk+1 ■ • ■ ■ S2k—1 0 0 0
Sk+1 Sk+2 ■ • ■ ■ S2k Sk+2 Sk+3 ■ • ■ ■ S2k+1 S2k+2
Expanding the determinant to the last row and applying the Laplace formula to the minors (of order 2k) appearing after the expansion, we get zero. Thus, the identity (4.17) is proved for all l. □
4.2. Finite case
Now consider the finite systems Vm and Tom. In this case, the matrices L(t) and A(t) in the corresponding Lax representations are finite, so we first need to establish a "finite" version of Theorem 3. Namely, for N ^ 2 consider the matrix
/0000 0 \
Mn =
0 0
1 a1;1 a1)2 0 0
0 1 02,2 a2,3 0
0 '•• '•• '•• '••
0 ■■■ 0 1 aN N
ai,i+1 S C, ai,i+1 =
(4^18)
For MN we define its Weyl function mN(z) and the moment sequence S of the latter, similarly as for the matrix M. Then the following statement holds.
Theorem 5. The sequence S is the moment sequence of the Weyl function mN(z) of the matrix MN iff
• So = 0, Si = 1;
• Afc = 0, for k = 0,...,N - 1.
• there exist C0,...,CN-1 G C, such that
N-1
Si CjSi-1-j, i = N + 1,...,ro. (4.19)
j=o
The proof of this theorem for a wider class of finite band matrices is contained in [13]. It is proved in a similar manner to Theorem 3, and the proof relies on the fact that an infinite Hankel matrix H = (Si+j+1)fj=0 has a finite rank N if and only if the condition defined by Eq. (4.19) is fulfilled (see [18, Chapter 15]). In the finite case mN(z) is a rational function.
Now assume that we have a finite dynamical system which admits a Lax representation with the matrix L = Mn . Then it often happens that the quantities Cj are the first integrals of the system. To be more precise, the following proposition holds.
Proposition 2. Assume that a dynamical system F can be written as MN = [MN,A], where MN is defined by formula (4.18) and the moments of mN(z) satisfy the equations
Sk = f (t) ^ diSk+1 - b(t)S^j , k G N, (4.20)
for some f (t), b(t), dj G C, p G N. Then
(Cj =0, j = 0,...,N - 1. (4.21)
Proof. According to Eq. (4.20),
SN +1 = f (t) ^E diSN+1+l — b(t)SN .
At the same time, it follows from Eq.(4.19) that
N-1 N-1
SN+1 = E Cj SN-j + E Cj SN-j. j=0 j=0
Applying Eq. (4.20) to S1,..., SN in the latter formula and using Eq. (4.19), we find
/ P \ N-1
SN+1 = f (t) I E dlSN+1+l — b(t)SN+1 I + E CjSN-j.
\l=1 J j=0
Thus,
N -1
E Cj SN-j =
j=o
Acting similarly with SN+2,... , S2N—1, we arrive at the system
C n —1S1 + . . . + CoSN = 0,
C N — ISN + ... + C0S2N —1 = 0.
According to Theorem 5, its determinant An — 1 = 0, and the claim follows immediately. □
As we see, Eqs. (4.7) and (4.12) are special cases of Eq. (4.20). Therefore, for the finite systems of Volterra and Toda hierarchy, the corresponding quantities Cj are their first integrals (the same is true for the finite non-Abelian Toda and Volterra lattices, see [19] for details).
Using the Cartan-Weyl classification of semisimple Lie algebras, Bogoyavlenskij (see, e.g., [9]) found the finite integrable systems of Volterra and Toda type, related to each class of algebras. In his terminology, the above Volterra and Toda lattices in the finite case are the An-Volterra and Toda systems. Using Theorem 4, one may check that the mapping defined by Eqs. (4.14)-(4.15) transforms the An Volterra systems (for n ^ 2) into their Am Toda counterparts, where m = |_n/2_|. As an example, consider the A3 Volterra system:
bo = bob1, b 1 = 61 (62 - bo), b2 = -&2&1. It admits the Lax representation (4.4) with the following matrix L(t):
/0 0 0 0 0\ 10 bo 0 0
L = 0 10 b1 0 0 0 1 0 b2 0 0 0 1 0
(4^22)
From Theorem 5 we find that for the moments of its Weyl function the following relations hold:
4
Si = ^ CjSi-1-j = (bo + bi + b2)Si-2 + (bob2)Si-4, i > 5. j=0
By Proposition 2, (b0 + b1 + b2) and b0b2 are the first integrals of the system. Now, applying Eqs. (4.14)-(4.15), we transform Eqs. (4.22) into the Toda A1 system with the matrix L(t):
L
/00 0 \
1 Bo Ac y0 1 B1J
/00 0 \ 1 bo bcb1 y0 1 b1 + b2 y
and the quantities (b0 + b1 + b2) = tr(L) and b0b2 are the first integrals of the transformed system.
Also, consider the following Volterra system B3 [11] from the class Bn: bo = bob1, b 1 = b1(b2 - bo), b2 = -b2b1 - b2.
As noted in [11], the equations for B3 can be obtained from the equations of Volterra system A6 with the following Lax matrix L:
L(t) =
fo o o o o o o o
1 o bo o o o o o
o 1 o bi o o o o
o o 1 o bl o o o
o o o 1 o -bi o o
o o o o 1 o -bi o
o o o o o 1 o -bo
o o o o o o 1 o
Again, applying Theorem 5, we find that
S = C3S-4 = (bl + b\ + bobi + 2bb)S-4, i > 8
(the other Cj here are equal to zero). Thus, C3 is the first integral of the system A6 (this is a known fact, since one can check that C3 = tr(L4)/4) and, therefore, of B3.
As to the transformations between Volterra and Toda systems belonging to the classes other than An and Bn, their detailed study remains an open problem. In particular, it involves the study of inverse spectral problems for the matrices L appearing in the Lax representations for such systems, as well as the proof of results similar to Theorem 4. We plan to address these issues in our future work.
The author thanks Prof. Damianou for useful discussions and reference to [11].
Appendix
Proposition. The following identity holds for k ^ 2 and aitj G C:
ai,i ai,i al,i al,l
ak+i,i ak+i,i
ai,k+i ai,k+i
ak+i,k+i
ai,i ■ ■ ■ ai,k-1
ak,i
ak,k-1
+
+
ai,i ■■■ ai,k-i ai,k+i ai,i ■■■ ai,k-i ai,k+i
ak,i ■ ■ ■ ak,k-i ak,k+i
ai,i ai,i a3,l a3,l
al,k a3,k
ai,i ai,l al,l al,l
al,k al,k
ak,l ak+l,l ■ ■ ■ ak,k
ak+l,l ak+l,l ■ ■ ■ ak+l,k
al,l ■ ■ ■ al,k-l al,k+l a3,l ■ ■ ■ a3,k-l a3,k+l
ak+l,l ■ ■ ■ ak+l,k-l ak+l,k+l
(A 1)
Proof. Using the expansion of the first determinant to the last row, we rearrange (A.1) as
(-1)
k+2
a1,2 a1,3 • • • a1,k+1
a2,2 a2,3
a2,k+1
\ ak,2 ak,3 • • • ak,k+1
a2,1 • • • a2,k—1 0
ak,1
ak,k-1 0 0 ak+1,1
a1,1 a1,3 a2,1 a2,3
a1,k+1 a2,k+1
(-1)
k1
01,1 02,1
ak,1
ak,1 ak,3 • • • ak,k+1
• • a1,k-2 a1,k a1,k+1 • • a2,k-2 a2,k a2,k+1
• • ak,k-2 ak,k ak,k+1
a1,1 a1,2 • • • a1,k a2,1 a2,2 • • • a2,k
02,1
a2,k-1 0
ak,1 ••• ak,k-1
0
0..
02,1 •
ak,1 • 0.
02,1 03,1
0 ak+1,2
+ ••• +
a2,k
-1 0 \
ak,k-1 0 0 ak+1,k-1
a2,k-1 a2,k+1 a3,k-1 a3,k+1
ak,1 ak+1,2 • • • ak,k ak+1,1 • • • ak+1,k-1 0
01,1 02,1
a1,k-1 a1,k+1 a2,k-1 a2,k+1
a2,1 • • • a2,k-1 a2,k+1 a3,1 • • • a3,k-1 a3,k+1
ak,1 ••• ak,k-1 ak,k+1 ak+1,1 ••• ak+1,k-1 0
Denote its left-hand and right-hand sides by I and II, respectively. For I and II we get the following representations:
a1,1 02,1
ak,1 0
01,2 02,2
ak,2 0
a1,k-1 a1,k a2,k— 1 a2,k
0 a1,k+1 0 a2,k+1
ak,k-1 ak,k 0 0 0 a21
0
a2,k— 1
ak,k+1 0
0 0 • • 0 0 ak,1 • • • ak,k-1 0 ak+1,1 ak+1,2 • • • ak+1,k-1 0 0 • • • 0 0
0
II
a1,1 a1,2 ■ ■ ■ a1,k-1 a1,k 0 a2,1 a2,2 ■ ■ ■ a2,k—1 a2,k 0
ak,1 ak,2 0 0
0 0 0 0
ak,k-1 ak,k
0
0
a2,k a2,1
ak,k ak,1
0 a1,k+1
0 a2,k+1
0 ak,k+1
a2,k-1 a2,k+1
ak,k-1 ak,k+1
0 ak+1,1 ■ ■ ■ ak+1,k-1
Applying elementary transformations, we consequently find
II
a1,1 a1,2 a2,1 a2,2
ak,1 ak,2
a1,k-1 a1,k a2,k-1 a2,k
ak,k-1 ak,k
0 0
0
0 0
a2,1 a2,2 ■ ■ ■ a2,k— 1 0 a2,1
a2,k— 1
0
a1,k+1 a2,k+1
ak,k+1 0
-ak,1 ak,2 ■ ■ ■ -ak,k-1 0 ak,1 ■ ■ ■ ak,k-1
0
0
0
0 ak+1,1 ■ ■ ■ ak+1,k-1
□
0
0
0
0
0
References
[1] Kac, M. and van Moerbeke, P., On an Explicitly Soluble System of Nonlinear Differential Equations Related to a Certain Toda Lattice, Adv. Math., 1975, vol. 16, no. 2, pp. 160-169.
[2] Moser, J., Three Integrable Hamiltonian Systems Connected with Isospectral Deformations, Adv. Math., 1975, vol. 16, no. 2, pp. 197-220.
[3] Akhiezer, N. I., The Classical Moment Problem and Some Related Questions in Analysis, Edinburgh: Oliver & Boyd, 1965.
[4] Akhiezer, N.I. and Glazman, I.M., Theory of Linear Operators in Hilbert Space, New York: Dower, 1993.
[5] Babelon, O., Continuum Limit of the Volterra Model, Separation of Variables and Non-Standard Realizations of the Virasoro Poisson Bracket, Comm. Math. Phys., 2006, vol. 266, no. 3, pp. 819-862.
[6] Berezanskii, Ju. M., Expansions in Eigenfunctions of Selfadjoint Operators, Transl. Math. Monogr., vol.17, Providence, R.I.: AMS, 1968.
[7] Berezanski, Yu. M., The Integration of Semi-Infinite Toda Chain by Means of Inverse Spectral Problem, Rep. Math. Phys., 1986, vol.24, no. 1, pp. 21-47.
[8] Barrios Rolania, D., Branquinho, A., and Foulquie Moreno, A., Dynamics and Interpretation of Some Integrable Systems via Multiple Orthogonal Polynomials, J. Math. Anal. Appl., 2010, vol. 361, no. 2, pp. 358-370.
[9] Bogoyavlensky, O.I., Breaking Solitons. Nonlinear Integrable Equations, Moscow: Nauka, 1991 (Russian).
[10] Damianou, P.A., The Volterra Model and Its Relation to the Toda Lattice, Phys. Lett. A, 1991, vol. 155, nos. 2-3, pp. 126-132.
[11] Damianou, P. A. and Loja Fernandes, R., From the Toda Lattice to the Volterra Lattice and Back, Rep. Math. Phys, 2002, vol. 50, no. 3, pp. 361-378.
[12] Osipov, A., On Some Issues Related to the Moment Problem for the Band Matrices with Operator Elements, J. Math. Anal. Appl, 2002, vol. 275, no. 2, pp. 657-675.
[13] Osipov, A. S., On a Determinant Criterion for the Weak Perfectness of Systems of Functions Holo-morphic at Infinity and Their Connection with Some Classes of Higher-Order Difference Operators, Fundam. Prikl. Mat., 1995, vol.1, no. 3, pp. 711-727 (Russian).
[14] Osipov, A. S., Discrete Analog of the Korteweg-de Vries (KdV) Equation: Integration by the Method of the Inverse Problem, Math. Notes, 1994, vol. 56, no. 6, pp. 1312-1314; see also: Mat. Zametki, 1994, vol. 56, no. 6, pp. 141-144.
[15] Magri, F., A Simple Model of the Integrable Hamiltonian Equation, J. Math. Phys., 1978, vol. 19, no. 5, pp. 1156-1162.
[16] Morosi, C. and Pizzocchero, L., On the Continuous Limit of Integrable Lattices: 1. The Kac-Moerbeke System and KdV Theory, Comm. Math.. Phys., 1996, vol. 180, no. 2, pp. 505-528.
[17] Yurko, V. A., On Higher-Order Difference Operators, J. Differ. Equations Appl., 1995, vol.1, no. 4, pp. 347-352.
[18] Gantmacher, F. R., The Theory of Matrices: Vol. 2, New York: Chelsea, 2000.
[19] Gekhtman, M., Hamiltonian Structure of Non-Abelian Toda Lattice, Lett. Math. Phys., 1998, vol. 46, no. 3, pp. 189-205.
[20] Volkov, A.Yu., Hamiltonian Interpretation of the Volterra Model, J. Soviet Math., 1989, vol.46, no. 1, pp. 1576-1581; see also: Zap. Nauchn. Sem. Leningrad.. Otdel. Mat. Inst. Steklov. (LOMI), 1986, vol. 150, pp. 17-25, 218.
