Table Of ContentSymmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 005, 11 pages
Integrable Discrete Equations Derived by Similarity
Reduction of the Extended Discrete KP Hierarchy
Andrei K. SVININ
Institute for System Dynamics and Control Theory,
134 Lermontova Str., P.O. Box 1233, Irkutsk, 664033 Russia
E-mail: [email protected]
6 Received November 16, 2005, in final form January 08, 2006; Published online January 19, 2006
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Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper005/
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2
n Abstract. WeconsidertheextendeddiscreteKPhierarchyandshowthatsimilarityreduc-
a tionofitssubhierarchiesleadtopurelydiscreteequationswithdependenceonsomenumber
J
of parameters together with equations governing deformations with respect to these para-
9 meters. It is written down discrete equations which naturally generalize the first discrete
1
Painlev´eequationdP inasensethatautonomousversionoftheseequationsadmitthelimit
I
to the first Painlev´e equation. It is shown that each of these equations describes B¨acklund
]
I transformationsof Veselov–Shabatperiodic dressing lattices with odd period knownalso as
S
n. Noumi–Yamada systems of type A(21()n−1).
i
l Key words: extended discrete KP hierarchy; similarity reductions; discrete Painlev´e equa-
n
tions
[
2000 Mathematics Subject Classification: 37K10
4
v
4
0
1 Introduction
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7
0
The main goal of the present paper is to exhibit an approach, which aims to construct a broad
5
community of one-dimensional discrete systems over finite number of fields sharing the property
0
/ of having Lax pair. It is achieved by similarity reduction of equations of the so-called extended
n
i discrete KP (edKP) hierarchy [18, 19, 20], which itself is proved to be a basis for construction
l
n of many classes of integrable differential-difference systems (see, for example, [20] and references
:
v therein). By analogy with the situation when Painlev´e equations and its hierarchies arise as
i aresultofsimilarityreductionofintegrablepartialdifferentialevolution equations[2,6],discrete
X
equations (like dP ) and its corresponding hierarchies also can be obtained by application of
r I
a group-theoretic methods.
In this connection it is necessary to mention various approaches aiming to select integrable
purely discrete one-dimensional equations, for instance, by applying the confinement singularity
test[9],fromB¨acklundtransformationsforthecontinuousPainlev´eequations[8,14],byimposing
integrable boundary conditions for two-dimensional discrete equations and reductions to one-
dimensional ones [12, 11, 10, 15], deriving discrete systems from proper chosen representations
of some affine Weyl group [16].
In the article we propose a scheme for constructing purely discrete equations by similarity
reductions of equations of edKP hierarchy and then investigate more specific cases. Namely we
analyze one-component generalizations of dP — discrete equations which turn out to serve as
I
discretesymmetrytransformationforVeselov–Shabat periodicdressinglattices withoddperiod.
(1)
The latter also known in the literature as Noumi–Yamada systems of type A due to series
l
of works by Noumi with collaborators (see, for example, [17]) where they selected the systems
of ordinary differential equations admitting a number of discrete symmetries which realized as
2 A.K. Svinin
automorphisms on field of rational functions of corresponding variables and constitute some
representation of extended affine Weyl group W˜ (A(1)).
l
The paper is organized as follows. In Section 2, we give necessary information on Darboux-
KPchain, itsinvariantmanifoldsandtheedKPhierarchy. InSection3, weshowthatself-similar
ansatzesyieldalargeclassofpurelydiscretesystemssupplementedbyequationsgoverningdefor-
mationswithrespecttoparametersenteringthesesystems. InSection4,weshowone-component
discrete equations naturally generalizing dP . We prove that these discrete equations together
I
with deformation equations are equivalent to Veselov–Shabat periodic dressing chains. Finally,
in this section we show that each of these discrete systems or more exactly its autonomous
version has continuous limit to P .
I
2 Darboux-KP chain and edKP hierarchy
2.1 Darboux-KP chain and its invariant submanifolds
In this section we give a sketch of the edKP hierarchy on the basis of approach using the notion
of Darboux-KP (DKP) chain introduced in [13]. Equations of DKP chain are defined in terms
of two bi-infinite sets of formal Laurent series. The first set
∞
h(i) = z+ h (i)z−s+1 :i ∈ Z
s
( )
s=2
X
consists of generating functions for Hamiltonian densities of KP hierarchy, and second one
∞
a(i) = z+ a (i)z−s+1 :i ∈ Z
s
( )
s=1
X
is formed by Laurent series a(i) each of which relates two nearest neighbors h(i) and h(i+1)
by Darboux map h(i) → h(i +1) = h(i) +a (i)/a(i). The DKP chain can be interpreted as
x
a result of successive iterations of Darboux map applying to any fixed solution of KP hierarchy,
say h(0), in forward and backward directions. It is given by two equations
∂ h(i) = ∂H(k)(i),
k
∂ a(i) = a(i)(H(k)(i+1)−H(k)(i)), ∂ ≡∂/∂t , ∂ ≡ ∂/∂x (1)
k k k
first of which defines evolution equations of KP hierarchy in the form of local conservation
laws, and the second one serves as compatibility condition of KP flows with Darboux map. The
Laurentseries H(k) = H(k)(i)in(1)isthecurrentof correspondingconservation law constructed
as special linear combination over Fa`a di Bruno polynomials h(k) = (∂ +h)k(1) by requiring to
be projection of zk on H = h1,h,h(2),...i [4].
+
In [19, 20] we have exhibited two-parameter class of invariant submanifolds of DKP chain Sn
l
each of which is specified by condition
zl−n+1a[n](i) ∈ H (i) ∀i∈ Z. (2)
+
Herea[k] = a[k](i)isthediscreteFa`adiBrunoiterationcalculated withthehelpoftherecurrence
relation a[k+1](i) = a(i)a[k](i+1) with a[0] ≡ 1.
The invariant submanifoldsS1 were presented in [13]. It was shown there that the restriction
l
of DKP chain on S1 is equivalent to discrete KPhierarchy [21]. Restriction ofDKP chain on S1,
0 0
S2, S3 and so on leads to the notion of edKP hierarchy which is an infinite collection of discrete
0 0
KPlikehierarchiesattached tomulti-timest(n) ≡ (t(n),t(n),...)withn ≥ 1. Allthesehierarchies
1 2
Integrable Discrete Equations 3
“live” on the same phase-space M which can be associated with hyperplane whose points are
parametrized by infinite number of functions of discrete variable: {a = a (i) :k ≥ 1}. One can
k k
treat these functions as analytic ones whose the domain of definition is restricted to Z. We have
shown in [19] that the flows on Sn are given in the form of local conservation laws as
0
zk(n−1)∂(n)lna(i) = K[kn](i+1)−K[kn](i), (3)
k (n) (n)
that is on Sn one has H(k) = zk(1−n)K[kn] with
0 (n)
p
K[pn] ≡ q(n,pn)zk(n−1)a[(p−k)n].
(n) k
k=0
X
Here the coefficients q(n,r) are some polynomials in variables {a(s) ≡ Λs(a )} (for definition, see
k l l
below (5)). In what follows, we refer to (3) as nth subhierarchy of edKP hierarchy. In the
following subsection we show Lax pair for edKP hierarchy and write down evolution equations
(n,r)
on the functions q generated by (3) in its explicit form.
k
2.2 Lax representation for edKP hierarchy
First, let us recall the relationship between Fa`a di Bruno iterations and formal Baker–Akhiezer
function of KP hierarchy [4,13]ψ = ψ = 1+ w (i)z−s eξ(t,z) withξ(t,z) = t zk. One
i s k
s≥1 ! k≥1
has P P
∂k(ψ) ∂ (ψ) Ψ
h(k) = , H(k) = k , a[k](i) = i+k, Ψ ≡ ziψ .
i i
ψ ψ Ψ
i
These relations allow representing the DKP chain constrained by relation (2) as compatibility
condition of auxiliary linear systems. As was shown in [19, 20], the restriction of DKP chain
on Sn leads to linear discrete systems (see also [18])
0
Qr Ψ = zrΨ, zk(n−1)∂(n)Ψ = (Qkn) Ψ (4)
(n) k (n) +
with the pair of discrete operators
k
Qr ≡ q(n,r)zs(n−1)Λr−sn, (Qkn) ≡ q(n,kn)zs(n−1)Λ(k−s)n.
(n) s (n) + s
s≥0 s=0
X X
Here Λ is a usual shift operator acting on arbitrary function f = f(i) of the discrete variable
(n,r) (n,r)
as (Λf)(i) = f(i + 1). The coefficients q = q (i) are uniquely defined as polynomial
s s
functions on M in coordinates a through the relation1
l
zr = zs(n−1)q(n,r)a[r−sn], r ∈ Z. (5)
s
s≥0
X
(s) (s)
We assign to a its scaling dimension: [a ] = l. One says that any polynomial Q in a is a
l l k l
homogeneous one with degree k if Q → ǫkQ when a → ǫla .
k k l l
1As was mentioned above, in fact, these are polynomials in a(s). For notational convenience we assume that
l
q(n,r) ≡1 for all possible n and r.
0
4 A.K. Svinin
The consistency condition for the pair of equations (4) reads as Lax equation
zk(n−1)∂(n)Qr = [(Qkn) ,Qr ]
k (n) (n) + (n)
and can be rewritten in its explicit form as
k
∂(n)q(n,r)(i) = Q(n,r)(i) ≡ q(n,kn)(i)·q(n,r) (i+(k−j)n)
k s s,k j s−j+k
j=0
X
k
(n,kn) (n,r)
− q (i+r−(s−j +k)n)·q (i). (6)
j s−j+k
j=0
X
It is important also to take into account algebraic relations
k s
q(n,r1+r2)(i) = q(n,r1)(i)·q(n,r2)(i+r −jn)= q(n,r2)(i)·q(n,r1)(i+r −jn) (7)
k j k−j 1 j k−j 2
j=0 j=0
X X
coded in permutability operator relation Qr1+r2 =Qr1 Qr2 = Qr2 Qr1 .
(n) (n) (n) (n) (n)
It is quite obvious that equations (6) and (7) admit reductions with the help of simple
(n,r)
conditions q ≡ 0 (∀s ≥ l + 1) for some fixed l ≥ 1. As was shown in [19, 20], these
s
reductions can be properly described in geometric setting as double intersections of invariant
manifolds of DKP chain: S = Sn ∩ Sln−r. The formula (6) when restricting to S is
n,r,l 0 l−1 n,r,l
proved to be a container for many integrable differential-difference systems (lattices) which can
be found in the literature (for reference see e.g. [20]).
2.3 Conservation laws for edKP hierarchy
The conserved densities for edKP hierarchy can be constructed in standard way as residues
h(n) = Res Qsn = q(n,sn). (8)
s (n) s
(cid:0) (cid:1)
Corresponding currents are easily derived from (6). One has
∂(n)q(n,sn)(i) = Q(n,sn)(i) = J(n)(i+n)−J(n)(i) = I(n)(i+1)−I(n)(i) (9)
k s s,k s,k s,k s,k s,k
with
k−1k−l
(n) (n,kn) (n,sn)
J (i) = q (i+(j +l−k−1)n)·q (i+(j −1)n)
s,k l s+k−l
l=0 j=1
XX
and
n k−1(k−l)n
(n) (n) (n,kn) (n,sn)
I (i) = J (i+j −1) = q (i+j+(l−k)n−1)·q (i+j −1).
s,k s,k l s+k−l
j=1 l=0 j=1
X X X
We observe that by virtue of relations (7) with r = ln, r = sn and k = l+s, the relation
1 2
Q(n,ln) = Q(n,sn) with arbitrary l,s ∈ N is identity and therefore we can write down exactness
l,s s,l
property relation
∂(n)h(n) = ∂(n)h(n) ∀s,l ≥ 1. (10)
s l l s
Integrable Discrete Equations 5
(n)
Looking at (9), it is natural to suppose that there exist some homogeneous polynomials ξ
s
in q such that
k
n
h(n)(i) = ξ(n)(i+j −1) ∀s ≥ 1. (11)
s s
j=1
X
(n)
If so, then one must recognize that {ξ : s ≥ 1} is an infinite collection of conserved densities
s
of nth subhierarchy of edKP hierarchy, i.e.
∂(n)ξ(n)(i) = J(n)(i+1)−J(n)(i).
k s s,k s,k
(n) (n)
Thentheformula(11)says thateach ξ is equivalent toh /nmoduloaddingtrivial densities.
s s
Substituting (11) into (7) with r = 1 and r =kn leaves us with the following equation2:
1 2
s s
ξ(n)(i+n)−ξ(n)(i) = q (i+jn)·q(n,sn)(i)− q (i)·q(n,sn)(i+1−jn).
s s j s−j j s−j
j=1 j=1
X X
One can check that solution of this equation is given by
s l
ξ(n)(i) = q (i+(j −1)n)·q(n,sn−1)(i+1+(j −l−1)n). (12)
s l s−l
l=1 j=1
XX
For example, we can write down the following:
n−1
(n) (n)
ξ (i) = q (i), ξ (i) = q (i)+q (i+n)+q (i)· q (i+j),
1 1 2 2 2 1 1
j=1−n
X
n−1
(n)
ξ (i) = q (i)+q (i+n)+q (i+2n)+q (i)· q (i+j)
3 3 3 3 2 1
j=1−2n
X
2n−1 2n−1
+q (i+n)· q (i+j)+q (i)· q (i+j)
2 1 1 2
j=1−n j=1−n
X X
n−1 n−1 n−1n−1
+ q (i+l)·q (i+j +l)+ q (i+l−n)·q (i+j) .
1 1 1 1
l=1−nj=1−n l=1 j=l
X X XX
(n)
Now let us show that conserved densities ξ satisfy some characteristic equations. From (8)
s
one derives
(n) (n)
δh δξ
s s (n,sn−1)
(i) = n (i) = snq (i+1−ln).
δq δq s−l
l l
Substituting the latter in (12) one gets
s l (n)
δξ
sξ(n)(i) = q (i+(j −1)n) s (i+(j −1)n).
s l δq
l
l=1 j=1
XX
2Hereandinwhatfollowsqk(i)≡qk(n,1)(i). Itisimportanttonotethatthemapping{ak}→{qk}isinvertible
and one can use {qk} as coordinates of M.
6 A.K. Svinin
3 Self-similar ansatzes for edKP hierarchy
and integrable discrete equations
In what follows, we consider nth edKP subhierarchy restricted to the first p time variables
(n) (n)
{t ,...,t } with p ≥ 2. This system is obviously invariant under the group of scaling trans-
1 p
formations
G = g : q → ǫkq (k ≥1), t(n) → ǫ−lt(n) (l = 1,...,p) .
p k k l l
n o
(n,r) (n,r) (n)
Since q , Q , ξ are homogeneous polynomials in q of corresponding degrees, the trans-
k k,s k k
formation g yields
q(n,r) → ǫkq(n,r), Q(n,r) → ǫk+sQ(n,r), ξ(n) → ǫkξ(n).
k k k,s k,s k k
Let us consider similarity reductions of nth edKP subhierarchy requiring
p p
(n) (n) (n) (n,1)
k+ jt ∂ q = kq + jt Q = 0 ∀k ≥ 1. (13)
j j k k j k,j
j=1 j=1
X X
Then any homogeneous polynomial in q satisfies corresponding self-similarity condition. In
l
particular, one has
p p
(n) (n) (n,r) (n,r) (n) (n,r)
k+ jt ∂ q = kq + jt Q = 0. (14)
j j k k j k,j
j=1 j=1
X X
More explicitly, one can rewrite (14) as
p p
(n,r) (n,n) (n,n) (n,r) (n) (n,jn)
q (i) k+α −α + q (i+sn) jt q (i)
k i i+r−kn k+s j j−s
(cid:16) (cid:17) Xs=1 Xj=s
p p
(n,r) (n) (n,jn)
− q (i) jt q (i+r−(k+s)n) = 0
k+s j j−s
s=1 j=s
X X
with
p n
(n,n) (n) (n) (n)
α ≡ jt h (i) = α ,
i j j i+s−1
j=1 s=1
X X
p
where α(n) ≡ jt(n)ξ(n). We observe that α(n,n) do not depend on evolution parameters
j j
j=1
(n) (n) P
{t ,...,t } provided that (13) is valid. Indeed, taking into account exactness property (10)
1 p
one has
p p
∂(n)α(n,n) = sh(n)+ jt(n)∂(n)h(n) = s+ jt(n)∂(n) h(n) = 0.
s s j s j j j s
j=1 j=1
X X
For Baker–Akhiezer function, one has the corresponding self-similarity condition in the form
p
(n) (n)
z∂ − jt ∂ ψ = 0
z j j
j=1
X
Integrable Discrete Equations 7
or
p p
(n,n) (n) (n,jn)
z∂ ψ = α ψ + ψ jt q .
z i i i i+sn j j−s
s=1 j=s
X X
One can rewrite the above formulas in terms of self-similarity ansatzes
(n)
t
T = l , l = 1,...,p−1, ξ = (pt(n))1/pz,
l (pt(n))l/p p
p
1 1 1
q(n,r) = x(n,r), Q(n,r) = X(n,r), ξ(n) = ζ(n).
s (pt(n))s/p s s,k (pt(n))(s+k)/p s,k s (pt(n))s/p s
p p p
At this point we can conclude that similarity reduction of nth edKP subhierarchy yields the
system of purely discrete equations
p−1 p−1
(n,r) (n,n) (n,n) (n,r) (n,jn) (n,pn)
x (i) k+α −α + x (i+sn) jT x (i)+x (i)
k i i+r−kn k+s j j−s p−s
(cid:16) (cid:17) Xs=1 Xj=s
p−1 p−1
(n,r) (n,jn) (n,pn)
− x (i) jT x (i+r−(k+s)n)+x (i+r−(k+s)n)
k+s j j−s p−s
s=1 j=s
X X
(n,r) (n,r)
+x (i+pn)−x (i) = 0, (15)
k+p k+p
p−1
α(n) = jT ζ(n)+ζ(n) (16)
j j p
j=1
X
supplemented by deformation equations
∂ x(n,r) = X(n,r), k = 1,...,p−1.
Tk s s,k
Theseequationsappearasaconsistencyconditionofthefollowinglinearisomonodromyproblem
ξψ + ξ1−kx(n,r)ψ = ξψ ,
i+r k i+r−kn i
k≥1
X
k
∂ ψ = ξkψ + ξk−sx(n,kn)ψ , k = 1,...,p−1,
Tk i i+kn s i+(k−s)n
s=1
X
p p−1
(n,n) (n,jn) (n,pn)
ξ∂ ψ = α ψ + ψ jT x (i)+x (i) .
ξ i i i i+sn j j−s p−s
s=1 j=s
X X
The pair of equations (15) and (16) when restricting dynamics on S becomes a system
n,r,l
(n,r)
over finite number of fields. If one requires x (i) ≡0 (∀k ≥ l+1) then l-th equation in (15)
k
is specified as
n n
(n,r) (n) (n)
x (i)· l+ α − α = 0.
l i+s−1 i+r−ln+s−1
( )
s=1 s=1
X X
(n,r) (n)
Sinceitissupposedthatx 6≡ 0thentheconstants α areforcedtobesubjectsofconstraint
l i
n n
(n) (n)
l+ α − α = 0. (17)
i+s−1 i+r−ln+s−1
s=1 s=1
X X
Remark that this equation makes no sense in the case r = ln. In the following section we
consider restriction on S corresponding to Bogoyavlenskii lattice.
n,1,1
8 A.K. Svinin
4 Examples of one-field discrete equations
4.1 One-field discrete system generalizing dPI and its hierarchies
Let us consider reductions corresponding to S with arbitrary n ≥ 2. For p = 2 one obtains
n,1,1
one-field system3
n−1 (n) n−1 n−1
α
T + x = i , 1+ α(n) − α(n) = 0 (18)
i+s x i+s i−s
i
s=1−n s=1 s=1
X X X
together with deformation equation
n−1 n−1
x′ = x x − x (19)
i i i+s i−s
!
s=1 s=1
X X
which is nothing else but Bogoyavlenskii lattice. The pair of equations (18) is a specification
of (16) and (17). In the case n = 2, (18) becomes dP while (19) turns into Volterra lattice.
I
Hierarchy of dP appears to be related with a matrix model of two-dimensional gravity (see, for
I
example, [7] and references therein). In [5] Joshi and Cresswell found out representation of dP
I
hierarchy with the help of recursion operator.
We arein position to exhibithierarchy of more general equation (18) by consideringthe cases
p = 3,4 and so on. From (12) one has
ζ(n)(i) = x x(n,sn−1)(i+1−n).
s i s−1
Substituting this into (16) leads to
p−1 (n)
α
T + jT x(n,jn−1)(i+1−n)+x(n,pn−1)(i+1−n)= i , (20)
1 j j−1 p−1 x
i
j=2
X
(n)
where α ’s are constants which are supposed to solve the same algebraic equations as in (18).
i
As an example, for p = 3, one has
n−1 n−1 n−1 n−1n−1 (n)
α
T +2T x + x x + x x = i .
1 2 i+s i+s i+j+s i+s−n i+j
x
i
s=1−n s=1−nj=1−n s=1 j=s
X X X XX
Equation (20) should be complemented by deformation ones
n−1 n−1
(n) (n)
∂ x = x ζ (i+s)− ζ (i−s) , k = 1,...,p−1
Tk i i k k
!
s=1 s=1
X X
which are higher members in Bogoyavlenskii lattice hierarchy.
4.2 Equivalence (18) and (19) to Veselov–Shabat periodic dressing lattices
Herewerestrictourselvesbyconsideration only(18)and(19)withouttheirhierarchies. Ouraim
is to show equivalence of this pair of equations to well-known Veselov–Shabat periodic dressing
lattices. To this end, we introduce the variables {r ,...,r } by identifying
0 2(n−1)
x = −r −r , x = −r −r , k = 1,...,n−1,
i+k−1 2k 2k+1 i+n+k−2 2k−1 2k
3Here xi≡x(1n,1)(i) and T ≡T1.
Integrable Discrete Equations 9
2(n−1)
x +T = r +r ,
i+s−1 0 1
s=1
X
where subscripts are supposed to be elements of Z/(2n − 1)Z. One can show that with the
constants
(n) (n)
c = α , c = −α , k = 1,...,n−1,
2k i+k−1 2k−1 i+k+n−2
n−1 n−1
(n) (n)
c = 1− α + α
0 i+s−1 i+s+n−2
s=1 s=1
X X
the equations (18) and (19) can be rewritten as Veselov–Shabat periodic dressing chain [22, 1]
r′ +r′ = r2 −r2+c , r = r , c = c (21)
k k+1 k+1 k k k+N k k+N k
with N = 2n−1. For N = 3 this system is known to be equivalent to P . In the variables
IV
f =r +r the system (21) has the form [22]
k k k+1
n−1 n−1
f′ = f f − f +c , k = 0,...,2(n−1). (22)
k k k+2r−1 k+2r k
!
r=1 r=1
X X
The system (22) for n ≥ 3 can be considered as higher-order generalization of P written
IV
in symmetric form. It was investigated in [17] where Noumi and Yamada showed its invariance
with respect to discrete symmetry transformations (or B¨acklund–Schlesinger transformations)
which are realized as a number of automorphisms {π,s ,...,s } on the field of rational
0 2(n−1)
functions in variables {c ,...,c } and {f ,...,f } as follows:
0 2(n−1) 0 2(n−1)
s (c ) = −c , s (c ) = c +c , l = k±1, s (c )= c , l 6= k,k±1,
k k k k l l k k l l
c
k
s (f ) = f , s (f ) = f ± , l = k±1, s (f ) = f , l 6=k,k±1,
k k k k l l k l l
f
k
π(f ) = f , π(c )= c .
k k+1 k k+1
One can easily rewrite the formulas of action of automorphisms on generators r as follows:
k
c c
k k
s (r )= r − , s (r ) = r + ,
k k k k k+1 k
r +r r +r
k k+1 k k+1
s (r ) = r , l 6= k,k+1, π(r )= r .
k l l k k+1
It is known by [16] that this set of automorphisms define a representation of the extended
affine Weyl group W˜ (A(1)) = hπ,s ,...,s i with l = 2(n − 1) whose generators satisfy the
l 0 l
relations
s2 = 1, s s = s s , m 6= k±1, (s s )3 = 1, m = k±1
k k m m k k m
for k,m = 0,...,l and
πl+1 = 1, πs = s π.
k k+1
The shift governed by discrete equations (18) affects on the system (21) as B¨acklund trans-
formation
c
1
r = r + , r = r , k = 1,...,2n−3,
0 2 k k+2
r +r
1 2
c
1
r = r − ,
2(n−1) 1
r +r
1 2
c = c +c , c =c , k = 1,...,2n−4,
0 1 2 k k+2
c = c +c , c = −c .
2n−3 0 1 2(n−1) 2(n−1)
By direct calculations one can check that it coincides with τ = s π2 ∈ W˜ A(1) .
1 2(n−1)
(cid:0) (cid:1)
10 A.K. Svinin
4.3 Continuous limit of stationary version of (18)
Let us show in this subsection that stationary version of the equation (18)
n−1
α
T + x = , (23)
i+s
x
i
s=1−n
X
for any n, is integrable discretization of P : w′′ = 6w2+t.
I
One divides the real axis into segments of equal length ε. One considers discrete set of values
{t = iε : i ∈ Z}. Values of the function w, respectively, are taken for all such values of the
variable t. Therefore one can denote w(t) = w . Let
i
x = 1+ε2w , α= 1−2n−ε4t, T = −2n+1. (24)
i i 1
Substituting (24) in the equation (23) and taking into account the relations of the form
ε2
x = 1+ε2w = 1+ε2 w+εw′ + w′′ +···
i+1 i+1
2
(cid:26) (cid:27)
and turning then ε to zero we obtain, in continuous limit, the equation
n−1
(n−s)2·w′′ = −t−(2n−1)w2
s=1
X
which, by suitable rescaling, can be deduced to canonical form of P . This situation goes in
I
parallel with that when Bogoyavlenskii lattice (19), for all values of n ≥ 2, has continuous limit
to Korteweg–de Vries equation [3].
5 Conclusion
(n)
We have considered in the paper edKP hierarchy restricted to p evolution parameters {t ,...,
1
(n)
t }. It was shown that self-similarity constraint imposed on this system is equivalent to purely
p
discrete equations (15) and (16) supplemented by (p−1) deformation equations which in fact
are evolution equations governing (p−1) flows of nth edKP subhierarchy. What is crucial in our
approach is that we selected quantities α(n,n) which enter these discrete equations and turn out
to be independent on evolution parameters due to exactness property for conserved densities.
Discrete systems over finite number of fields arise when one restricts edKP hierarchy on in-
variant submanifold S . In the present paper we considered only one-field discrete equations
n,r,l
correspondingtoS withn ≥ 2. Itisourobservation thatinthiscasediscreteequation under
n,1,1
consideration supplemented by deformation one is equivalent to Veselov–Shabat periodic dres-
sing lattice with odd period. It is known due to Noumi and Yamada that this finite-dimensional
system of ordinary differential equations admits finitely generated group of B¨acklund transfor-
mations whichrealizes representation of extendedaffineWeylgroupW˜ (A(1) ). Itisnaturalto
2(n−1)
expectthatrestrictionofedKPhierarchytootheritsinvariantsubmanifoldsalsocanyieldfinite-
dimensional systems invariant under some finitely generated groups of discrete transformations.
We are going to present relevant results on this subject in subsequent publications.
Acknowledgements
The author thanks the organizers of Sixth International Conference “Symmetry in Nonlinear
Mathematical Physics” (June 20–26, 2005, Kyiv) for hospitality and their work which allows to
create friendly atmosphere during this Conference. The present paper is the written version of
the talk delivered by the author at this conference.
