Table Of ContentOn some class of reductions for
Itoh-Narita-Bogoyavlenskii lattice
A K Svinin
9
0
Institute for System Dynamics and Control Theory, Siberian Branch of Russian
0
Academy of Sciences, Russia
2
n E-mail: [email protected]
u
J
Abstract. We show a broad class of constraints compatible with Itoh-Narita-
2
Bogoyavlenskii lattice hierarchy. All these constraints can be written in the form of
] discreteconservationlawI =I withappropriatehomogeneouspolynomialdiscrete
I i+1 i
S function I =I[a].
.
n
i
l
n
PACS numbers: 02.30.Ik
[
Keywords: KP hierarchy, Itoh-Narita-Bogoyavlenskii lattice
2
v
7
1. Introduction
1
5
4
The aim of this paper is to show explicitly some class of constraints compatible with
.
2
extended Volterra lattice
0
9 n n
0 a′ = a a − a , (1)
: i i i−j i+j
v j=1 j=1
X X
Xi which we consider asa single evolution equation on the unknown function a ≡ a(i,x)
i
r
a of discrete variable i ∈ Z and continuous variable x ∈ R. For any n ≥ 1, this equation
is known to be integrable discretization for Kortweg-de Vries equation [5]. Narita in the
work [10], making of use Hirota’s method, showed that extended Volterra lattice admits
soliton solutions. In [7] Itoh considered Lotka-Volterra systems which are equivalent
to equation (1) supplemented by specific periodicity condition a = a . In what
i+2n+1 i
follows we call equation (1) Itoh-Narita-Bogoyavlenskii (INB) lattice. Perhaps the most
interesting case from the point of applications is n = 1, corresponding to Volterra lattice
[18], [8].
Equation (1) is known to admit the hierarchy of pair-wise commuting generalized
symmetries which, as is shown in the paper, can be written in the form
∂ a = (−1)sa S(n,s)(i−(s−1)n+1)−S(n,s)(i−sn) , (2)
s i i sn−1 sn−1
(cid:16) (cid:17)
where ∂ stands for derivative with respect to evolution parameter t with s ≥ 2. The
s s
functions S(n,l)[a] will be explicitly defined in Section 3. The INB equation itself can
s
be written in the form (2) with t = x and S(n,1) = n a . When the hierarchy
1 n−1 j=1 i+j−1
P
On some class of reductions for Itoh-Narita-Bogoyavlenskii lattice 2
is represented as (2), it is clear that stationarity condition attached to some evolution
parameter t , can be written as periodicity condition S(n,s)(i + n + 1) = S(n,s)(i). It
s sn−1 sn−1
turns out that there exist a wide class of homogeneous polynomial discrete functions
I = I[a] defined by‡
I = a ···a
i i+j1 i+jl+1
(j1,...X,jl+1)∈J
for which periodicity condition I = I , with respective period T ∈ Z is consistent
i+T i
with INB lattice and its hierarchy. The set of all invariant constraints considered in the
paper naturally include periodicity conditions a = a .
i+T i
We consider equation (1) and its hierarchy as a simplest case of reduction of the
so-called Darboux-KP (DKP) chain hierarchy which in fact is a bi-infinite sequence of
KP hierarchies and our main goal, in fact, is to approve our previous results [16] on
this simple example. For completeness, we give, in the next section, preliminaries on
our approach to investigate some class of integrable lattices related to the KP hierarchy.
In Section 3, we show compatible constraints for equation (1) in its explicit form in
Theorem 3. The Section 4 is devoted to Volterra lattice and its reductions. We write
down, in this section, attached systems of ordinary differential equations generated by
corresponding constraints and its discrete symmetry transformations. Also we claim the
relations defining spectral curves associated with Lax matrices.
2. Preliminaries on DKP chain hierarchy and its invariant submanifolds
2.1. DKP chain hierarchy
In[14]and[15]wehavedevelopedanapproachinwhichabroadcommunityofintegrable
differential-difference equations (lattices) are related to KP hierarchy. In a paper [16] we
have shown that these integrable lattices admit a wide class of constraints compatible
with all higher flows of its hierarchy. An objective of this section is to provide the reader
by information about the DKP chain hierarchy and its reductions.
Integrable lattices in our geometric set-up naturally appear as a result of reductions
of a bi-infinite sequence of KP hierarchies whose equations of motion we write in the
form of two evolution generating equations [9]
∂ h(i) = ∂H(s)(i), (3)
s
∂ a(i) = a(i) H(s)(i+1)−H(s)(i) . (4)
s
(cid:16) (cid:17)
The first relation (3) yields evolution equations of KP hierarchy in the form of local
conservation laws [19]. Laurent series: generating functions for conserved densities and
corresponding fluxes of KP hierarchy
h(i) = z + h (i)z−k+1 and H(s)(i) = zs + Hs(i)z−k,
k k
k≥2 k≥1
X X
‡ Here J ⊂Zl+1 is some finite indexing set.
On some class of reductions for Itoh-Narita-Bogoyavlenskii lattice 3
are related with KP wave functions
ψ = 1+ w (i)z−k exp( t zs)
i k s
k≥1 s≥1
X X
as
h(i) = ∂ψ ·ψ−1 and H(s)(i) = ∂ ψ ·ψ−1,
i i s i i
respectively. In turn, the Laurent series a(i) = z + a (i)z−k+1 is calculated as
k≥1 k
a(i) = zψ ·ψ−1. We call equations (3) and (4) DKP chain hierarchy. It is useful to
i+1 i P
rewrite generating equation (4) in the form of differential-difference conservation law
∂ ξ(i) = H(s)(i+1)−H(s)(i)
s
with
2 3
1 1
ξ(i) = lna(i) = lnz + a (i)z−k − a (i)z−k + a (i)z−k −···
k k k
2 3
k≥1 k≥1 k≥1
X X X
≡ lnz + ξ (i)z−k.
k
k≥1
X
Thus, more exactly, equations (3) and (4) can be written as follows:
∂ h (i) = ∂Hs (i), ∂ ξ (i) = Hs(i+1)−Hs(i).
s k k−1 s k k k
To establish relationship of integrable lattices like INB lattice (1), Shabat dressing
lattice [11], Toda lattice [17], Belov-Chaltikian lattice [3] and so on, with KP hierarchy,
the following two theorems are useful.
Theorem 1. [14] The submanifold Sn defined by condition
l−1
zl−na[n](i) ∈ H (i), ∀i ∈ Z (5)
+
is tangent with respect to DKP chain flows defined by (3) and (4).
Theorem 2. [15] The chain of inclusions of invariant submanifolds
Sn ⊂ S2n ⊂ S3n ⊂ ··· ⊂ Skn ⊂ ···
l−1 2l−1 3l−1 kl−1
is valid.
Here, by definition
s a(i+j −1), s ≥ 1
j=1
a[s](i) = Q1, s = 0
|s| a−1(i−j), s ≤ −1
j=1
arediscreteFa`adiBrunoiQteratesofLaurentseries a(i). Itisobviousthatthecoefficients
a[s] defined through the relation§
j
a[s] = zs + a[s]zs−j
j
j≥1
X
§ We use simplified notations a[s] ≡ a[s](i), a[s] ≡ a[s](i) in formulas which contain no shifts with
j j
respect to discrete variable i∈Z.
On some class of reductions for Itoh-Narita-Bogoyavlenskii lattice 4
are the discrete functions of (a ,...,a ). These functions are related with each other by
1 j
obvious relation
k−1
a[s1+s2](i) = a[s1](i)+ a[s1](i)a[s2](i+s )+a[s2](i+s ) = (s ↔ s ). (6)
k k j k−j 1 k 1 1 2
j=1
X
Here the symbol (s ↔ s ) denotes the same right-hand side of this relation but with
1 2
mutually replaced s and s . This relation will be extensively used throughout the
1 2
paper.
We observe that the condition (5) can be written in the form of the following
generating relation:
l
zl−na[n] = H(l) + a[n]H(l−k).
k
k=1
X
2.2. n-th discrete KP hierarchy and its reductions
When restricting DKP chain hierarchy on Sn, all the coefficients Hs become discrete
0 k
polynomial functions of (a ,...,a ) defined by [14], [16]
1 k+s
s−1
Hs = F(n,s) ≡ a[sn] + q(n,sn)a[(s−j)n], (7)
k k k+s j k+s−j
j=1
X
where q(n,r) = q(n,r)[a ,a ,...,a ], by definition, are polynomial discrete functions
k k 1 2 k
defined through generating relation
zr = a[r] + q(n,r)zj(n−1)a[r−jn] (8)
j
j≥1
X
which yields
k−1
a[r] + a[r−jn]q(n,r) +q(n,r) = 0, ∀k ≥ 1. (9)
k k−j j k
j=1
X
Let us write down the first few q(n,r)
k
q(n,r) = −a[r], q(n,r) = −a[r] +a[r]a[r−n],
1 1 2 2 1 1
q(n,r) = −a[r] +a[r]a[r−n] +a[r−2n]a[r] −a[r]a[r−n]a[r−2n].
3 3 1 2 1 2 1 1 1
It can be checked that a more general relation than (9), namely [16]
k−1
a[r](i)+ a[r−jn](i)q(n,r−p)(i+p)+q(n,r−p)(i+p) = a[p](i) (10)
k k−j j k k
j=1
X
with any p ∈ Z is valid. Solving this in favor of q(n,r−p)(i+p) yields
k
k−1
a[p](i)+ q(n,r−(k−j)n)(i)a[p] (i)+q(n,r)(i) = q(n,r−p)(i+p).
k j k−j k k
j=1
X
One sees that, when restricting on Sn, DKP chain hierarchy is reduced to evolution
0
equations in the form of differential-difference conservation law
∂ ξ (i) = F(n,s)(i+1)−F(n,s)(i), (11)
s k k k
On some class of reductions for Itoh-Narita-Bogoyavlenskii lattice 5
where F(n,s) is given by (7). We refer to these equations with some fixed n ≥ 1 as
k
n-th discrete KP hierarchy. It is worth to notice that these equations, in fact, appear
as a result of restriction of DKP chain hierarchy on Sn and subsequent projection of
0
dynamics on the space M whose points are defined by infinite number of functions of
discrete variable (a ,a ,...). One can say that M has infinite functional dimension.
1 2
Let us denote by M the space whose points are defined by finite number (a ,...a )
k 1 k
of functions of discrete variable i and π : M 7→ M being a natural projection.
k k
It is obvious that reduction of DKP chain hierarchy on the intersection Sn ∩ Sp is
0 l−1
equivalent to restriction of the flows given by (11) on some submanifold M ⊂ M.
n,p,l
This submanifold is defined by infinite number of algebraic equations [16]
J(n,p,l)[a ,...,a ] = 0, k ≥ 1 (12)
k 1 k+l
with
l−1
J(n,p,l)(i) = a[p] (i)−a[ln](i)− q(n,ln−p)(i+p)a[(l−j)n](i).
k k+l k+l j k+l−j
j=1
X
Observe, that in the case p = ln the relations J(n,ln,l) = 0 are identities and therefore
k
produce no nontrivial submanifold of M. This is so because Sn ⊂ Sln thanks to the
0 l−1
Theorem 2.
Taking into account (10) we can also write
k−1
J(n,p,l)(i) = q(n,ln−p)(i+p)+ a[−(k−j)n](i)q(n,ln−p)(i+p). (13)
k k+l j k+l−j
j=1
X
Let us denote Q(n,p,l)(i) = q(n,ln−p)(i+p) and therefore one can rewrite the relation (13)
k k+l
as
k−1
J(n,p,l) = Q(n,p,l)+ a[−(k−j)n]Q(n,p,l). (14)
k k j k−j
j=1
X
It is evident that the submanifold M can be equivalently defined by algebraic
n,p,l
equations Q(n,p,l)[a ,...,a ] = 0. Solving (14) in favor of Q(n,p,l) yields
k 1 k+l k
k−1
Q(n,p,l) = J(n,p,l)+ q(n,−(k−j)n)J(n,p,l). (15)
k k j k−j
j=1
X
By definition, one has Q(n,p+n,l+1)(i) = Q(n,p,l)(i+n). Making of use this relation, (10)
k k+1
and (15), one can prove that
k
J(n,p+n,l+1)(i) = J(n,p,l)(i+n)+ a[n](i)J(n,p,l)(i+n). (16)
k k+1 j k−j+1
j=1
X
For Q(n,p,l) we are able to write down following evolution equations
k
s
D Q(n,p,l)(i) = Q(n,p,l)(i+sn)+ q(n,sn)(i+p)Q(n,p,l)(i+(s−j)n)
ts k k+s j k+s−j
j=1
X
s
−Q(n,p,l)(i)− q(n,sn)(i−(s+k −j)n)Q(n,p,l)(i)
k+s j k+s−j
j=1
X
On some class of reductions for Itoh-Narita-Bogoyavlenskii lattice 6
which easily derived from
s
D q(n,r)(i) = q(n,r)(i+sn)+ q(n,sn)(i)q(n,r) (i+(s−j)n)
ts k k+s j k+s−j
j=1
X
s
−q(n,r)(i)− q(n,sn)(i+r −(k +s−j)n)q(n,r) (i). (17)
k+s j k+s−j
j=1
X
The latter, in turn, comes from Lax equation attached to linear problem for wave
function Ψ = {ψ : i ∈ Z} coded in (8) [13].
i
As was shown in [16], there exist weaker than (12) conditions invariant with
respect to n-th discrete KP hierarchy equations (11). Desired constraints are written as
periodicity conditions
I(n,p,l)(i+n) = I(n,p,l)(i) (18)
k k
with
k−1
I(n,p,l)(i) = J(n,p,l)(i)+ a[−p](i+p)J(n,p,l)(i)
k k j k−j
j=1
X
k−1
= Q(n,p,l)(i)+ a[−p−(k−j)n)](i+p)Q(n,p,l)(i).
k j k−j
j=1
X
Moreover, if the conditions (18) are valid then I(n,p,l) do not depend on evolution
k
parameters t , i.e., D I(n,p,l) ≡ 0, ∀s ≥ 1. For these functions, we can write
s ts k
I(n,p+n,l+1)(i) = I(n,p,l)(i+n)−a[−p−n](i+p+n)I(n,p,l)(i+n)
k k+1 k 1
in parallel with (16). It can be shown that relationship of I(n,p,l)’s with wave KP
k
functions is given by
1 l
I(n,p,l)(i)z−j = zl− zlψ + zl−jq(n,ln−p)(i+p)ψ (19)
j ψ i+ln j i+(l−j)n
j≥1 i+p j=1
X X
If d any divisor of n, then the set of conditions I(n,p,l)(i + d) = I(n,p,l)(i) also
k k
produce invariant submanifold of n-th discrete KP hierarchy. We denote corresponding
submanifold by Nd . It is evident that M ⊂ Nd ⊂ Nn .
n,p,l n,p,l n,p,l n,p,l
In the following section we consider invariant constraints for Narita-Bogoyavlenskii
lattice which correspond to restriction of n-th discrete KP hierarchy on M . Our
n,n+1,1
aim to write invariant conditions appeared as a result of intersection of M and
n,n+1,1
Nd in its explicit form.
n,p,l
3. Reductions for INB lattice
3.1. Restriction of n-th discrete KP hierarchy on M . INB lattice
n,n+1,1
Let us consider invariant submanifold M of phase-space M defined by algebraic
n,n+1,1
equations
J(n,n+1,1) = −J(n+1,n,1) = a[n+1] −a[n] = 0, k ≥ 1 (20)
k k k+1 k+1
On some class of reductions for Itoh-Narita-Bogoyavlenskii lattice 7
for some fixed positive integer n. Taking into account (6) we can rewrite (20) as
k−1
a[n] (i)a (i+n)+a (i+n) = 0. (21)
k−j j k
j=1
X
One can easily check that these equations are solved by
a (i) = a[−n](i)a (22)
k k−1 i
where a ≡ a (i). Indeed, substituting the latter in (21) we have
i 1
k−2
a a[n] (i)+ a[n] (i)a[−n](i+n)+a[−n](i+n)
i+n k−1 k−j−1 j k−1
j=1
X
= a a[0] (i) = 0.
i+n k−1
Here we have used (6). Following technical proposition is valid.
Proposition 1. In virtue of relations (20)
s
a[s](i) = a[−n+j−1](i)a , for s ≥ 1 (23)
k k−1 i+j−1
j=1
X
and
|s|
a[s](i) = − a[−n−j](i)a , for s ≤ −1.
k k−1 i−j
j=1
X
Proof. Taking into account (6) and (22), one has
k−1
a[s+1](i) = a[s](i)+ a[s] (i)a (i+s)+a (i+s)
k k k−j j k
j=1
X
k−2
= a[s](i)+a a[s] (i)+ a[s] (i)a[−n](i+s)+a[−n](i+s)
k i+s k−1 k−j−1 j k−1
j=1
X
= a[s](i)+a[−n+s](i)a .
k k−1 i+s
Making ofusethisformulaonecansuccessively toprove (23)fork = 2,3,...byinduction
with respect to s.
With (23), we are able to calculate all a as discrete functions of a to obtain
k
n n n+j1
a (i) = −a a , a (i) = a a a ,
2 i i−j 3 i i−j1 i−j2
jX=1 jX1=1 jX2=1
n n+j1 n+j2
a (i) = −a a a a
4 i i−j1 i−j2 i−j3
jX1=1 jX2=1 jX3=1
and so on.
Whatwelearnfromtheabovecalculationsisthattherestrictionofn-thdiscreteKP
hierarchy on M and subsequent projection π : M 7→ M generate the hierarchy
n,n+1,1 1 1
of evolution equations in the form of differential-difference conservation laws
∂ a = F(n,s)(i+1)−F(n,s)(i)
s i 1 1
On some class of reductions for Itoh-Narita-Bogoyavlenskii lattice 8
together with conservation laws (11), where conserved densities ξ = ξ [a] and fluxes
k k
F(n,s)[a] are some homogeneous polynomials of k-th and (k+s)-th power, respectively.
k
For the first flow we have
a′ = F(n,1)(i+1)−F(n,1)(i)
i 1 1
= a[n](i+1)−a[n](i).
2 2
To calculate the right-hand side of this equation as discrete function of a = a , it is
i
convenient to use
a[n+1](i) = a[n](i)+a[n](i)a +a (i+n)
2 2 1 i+n 2
= a[n](i+1)+a[n](i+1)a +a (i),
2 1 i 2
where a (i) = a[−n](i)a = −a[n](i−n)a . Taking this into account, we get
2 1 i 1 i
a′ = a a[n](i−n)−a[n](i+1)
i i 1 1
(cid:16) (cid:17)
n n
= a a − a ,
i i−j i+j
j=1 j=1
X X
which is nothing but INB equation (1).
3.2. The functions S(n,l)[a] and T(n,l)[a] and its properties
s s
Let us prepare, for further use, the discrete functions S(n,l)[a] and T(n,l)[a] through
s s
l−1
S(n,l)(i) = a (24)
s i+kn+jk
!
0≤jl−1X≤···≤j0≤s kY=0
with l ≥ 0 and s ≥ 0 and
l−1
T(n,l)(i) = a
s i+kn+jk
!
0≤j0<·X··<jl−1≤s kY=0
with l ≥ 0 and s ≥ l−1. We observe that the functions S(n,l) satisfy following relations:
s
d
S(n,l)(i)−S(n,l)(i) = a S(n,l−1)(i+n), (25)
s s−d i+s−j+1 s−j+1
j=1
X
d
S(n,l)(i)−S(n,l)(i+d) = a S(n,l−1)(i+j −1) (26)
s s−d i+(l−1)n+j−1 s−j+1
j=1
X
for d = 1,...,s and
s+1
S(n,l)(i) = a S(n,l−1)(i+n)
s i+s−j+1 s−j+1
j=1
X
s+1
= a S(n,l−1)(i+j −1).
i+(l−1)n+j−1 s−j+1
j=1
X
For T(n,l) we have the identities
s
d
T(n,l)(i)−T(n,l)(i+d) = a T(n,l−1)(i+n+j), (27)
s s−d i+j−1 s−j
j=1
X
On some class of reductions for Itoh-Narita-Bogoyavlenskii lattice 9
d
T(n,l)(i)−T(n,l)(i) = a T(n,l−1)(i) (28)
s s−d i+(l−1)n+s−j+1 s−j
j=1
X
with d = 1,...,s−l+1 and
s−l+2
T(n,l)(i) = a T(n,l−1)(i+n+j) (29)
s i+j−1 s−j
j=1
X
s−l+2
= a T(n,l−1)(i). (30)
i+(l−1)n+s−j+1 s−j
j=1
X
3.3. A class of reductions for INB lattice
With S(n,l) and T(n,l) in hand, we are in position to prove
s s
Proposition 2. On M
n,n+1,1
k−1
J(n,ln+s+1,l)(i) = T(n,k+l)(i−(k −1)n)+ a[−(k−j)n](i)T(n,k+l−j)(i−(k −j −1)n) (31)
k s j s
j=1
X
for s ≥ l and J(n,ln+s+1,l) ≡ 0 for s = 0,...,l−1 and
k
k−1
J(n,ln−s−1,l)(i) = R(n,k+l)(i−(k −1)n)+ a[−(k−j)n](i)R(n,k+l−j)(i−(k −j −1)n)(32)
k s j s
j=1
X
for s ≥ 0, where R(n,k)(i) ≡ (−1)kS(n,k)(i−s−1).
s s
Let us give some remarks. It is accepted in (31), that T(n,k) ≡ 0 with s ≤ k −2.
s
For example J(n,ln+l+1,l) = a[−n]T(n,l+1). As a corollary of this proposition, one has
k k−1 l
M ⊂ M , with l ≥ 1 and s = 0,...,l − 1. We observe comparing (31)
n,n+1,1 n,ln+s+1,l
and (32) with (13) that this proposition can be reformulated in the following equivalent
form.
Proposition 3. On M
n,n+1,1
q(n,s+1)(i) = (−1)kS(n,k)(i−(k −1)n)
k s
for s ≥ 0 and
q(n,−s−1)(i) = T(n,k)(i−(k −1)n−s−1)
k s
for s ≥ k −1 and q(n,−s−1) ≡ 0 for s = 0,...,k −2.
k
From this proposition and the relation [13]
k−1
q(n,s1+s2)(i) = q(n,s1)(i)+ q(n,s1)(i)q(n,s2)(i+s −jn)+q(n,s2)(i+s ) = (s ↔ s )
k k j k−j 1 k 1 1 2
j=1
X
we get two identities
l−1
S(n,l)(i)+ (−1)jS(n,l−j)(i)T(n,j)(i+(l−j)n)+(−1)lT(n,l)(i) = 0
s s s s
j=1
X
and
l−1
T(n,l)(i)+ (−1)jT(n,l−j)(i)S(n,j)(i+(l−j)n)+(−1)lS(n,l)(i) = 0
s s s s
j=1
X
On some class of reductions for Itoh-Narita-Bogoyavlenskii lattice 10
establishing the relationship between the discrete functions S(n,l)[a] and T(n,l)[a].
s s
Proof of Proposition 2. To save the space we restrict ourselves by the sketch of
the proof. We prove, by induction with respect to k, the validity of (31) for l = 0, i.e.
that on M
n,n+1,1
k−1
a[s+1](i) = T(n,k)(i−(k −1)n)+ a[−(k−j)n](i)T(n,k−j)(i−(k −j −1)n). (33)
k s j s
j=1
X
In the case k = 1 one has a[s+1](i) = s a = T(n,1)(i), by definition. Suppose now
1 j=0 i+j s
that the relation (33) is already proved for k = 1,...,k . Then, for these values of k
P 0
and arbitrary m ∈ Z we can show that
a[m+s+1](i)−a[m](i) =
k k
k−1
= T(n,k)(i+m−(k −1)n)+ a[m−(k−j)n](i)T(n,k−j)(i+m−(k −j −1)n).
s j s
j=1
X
In particular
k−1
a[−n+s+1](i)−a[−n](i) = T(n,k)(i−kn)+ a[−(k−j+1)n](i)T(n,k−j)(i−(k −j)n). (34)
k k s j s
j=1
X
Taking into account (23), we have
s
a[s+1](i)−a[−n](i)T(n,1)(i) = a[−n+j](i)−a[−n](i) a .
k+1 k s k k i+j
jX=1(cid:16) (cid:17)
Then taking into account (30) and (34) we get
k−1
a[s+1](i)−a[−n](i)T(n,1)(i) = a[−(j+1)n](i)T(n,j+1)(i−jn)+T(n,k+1)(i−kn).
k+1 k s k−j s s
j=1
X
Therefore we prove that if (33) is valid for k = 1,...,k , then it is true for k = k +1.
0 0
Thus, by induction, the relation (31) is proven for l = 0. For remaining values l ≥ 1 the
functions J(n,ln+s+1,l) are calculated with the help of recurrence relation (16). Similar
k
reasonings are used to prove (32).
With Proposition 3 we can easily write equations of INB lattice hierarchy. To this
aim, we use the fact that on M one has
n,n+1,1
Q(n,n+1,1)(i) = q(n,−1)(i+n+1) = 0, ∀k ≥ 1.
k k+1
Then from (17) we have
∂ q(n,−1)(i) = q(n,−1)(i) q(n,sn)(i)−q(n,sn)(i−n−1) , (35)
s 1 1 s s
(cid:16) (cid:17)
where q(n,−1)(i) = a , by definition. According to Proposition 3, on M one has
1 i−1 n,n+1,1
q(n,sn)(i) = (−1)sSs (i−(s−1)n). Substituting the latter in (35) we obtain evolution
s sn−1
equations of INB hierarchy (2).
Nowwewouldliketowritedowninvariantconstraintscorrespondingtosubmanifold
M ∩ Nd , where d is any divisor of n or n + 1. From Proposition 2, we
n,n+1,1 n,p,l
know that I(n,ln+s+1,l)(i) = T(n,l+1)(i), for s ≥ l and I(n,ln−s−1,l)(i) = R(n,l+1)(i) =
1 s 1 s
