Toda lattice realization of integrable hierarchies

11 Pages·0.13 MB·English

by  L. Bonora

SISSA-ISAS 144/95/EP
IFT–P057/95
Toda lattice realization of integrable hierarchies
L.Bonora
International School for Advanced Studies (SISSA/ISAS)
Via Beirut 2, 34014 Trieste, Italy, and
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9 INFN, Sezione di Trieste.
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C.P.Constantinidis
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J Instituto de F´isica Te´orica – UNESP

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Rua Pamplona 145, 01405 S˜ao Paulo, Brasil
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E.Vinteler
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7 International School for Advanced Studies (SISSA/ISAS)
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Via Beirut 2, 34014 Trieste, Italy
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Abstract
We present a new realization of scalar integrable hierarchies in terms of the Toda lattice
hierarchy. In other words, we show on a large number of examples that an integrable hierarchy,
definedbyapseudodifferentialLaxoperator,canbeembeddedintheTodalattice hierarchy. Such
a realization in terms the Toda lattice hierarchy seems to be as general as the Drinfeld–Sokolov
realization.
1
1 Introduction
Scalarintegrablehierarchiescanbeintroducedintermsof(pseudo)differentialoperatorsbymeans
of a formalism first introduced by Gelfand and Dickey (see [1]). This is the most ‘disembodied’
form in which such hierarchies can appear, and it can be taken as a reference form. One can then
consider realizations of these hierarchies in physical systems. A comprehensive realization is the
one studied by Drinfeld and Sokolov in terms of linear systems defined on Lie algebras, [2]; let
us refer to it as the Drinfeld–Sokolov realization (DSR). In this letter we present a new general
realization of integrable hierarchies in terms of the Toda lattice hierarchy (TLH). We call it Toda
lattice realization (TLR), and it looks as general as the DSR. While the DSR is contiguous to
(reduced) WZNW models and Toda field theories in 2D, the TLR is inspired by matrix models,
see [3],[4].
Theletter is organized as follows. Insection 2weintroducetheTLR.We donotgive ageneral
proof of it, but in section 3 we verify it on a large number of examples among KP, n–KdV and
other classes of hierarchies. Section 5 is devoted to some comments.
2 The Toda lattice realization of integrable hierar-
chies.
In the Gelfand–Dickey (GD) formalism an integrable hierarchy can be entirely specified in terms
of the Lax operator
L = ∂N +Na1∂N−2+Na2∂N−3+...+NaN−1+NaN∂−1+... (2.1)
where ∂ = ∂ . The operator L may be purely differential, in which case a = 0 for k ≥ N,
∂x k
and we get the N-KdV hierarchy. The fields a may be either elementary or composite of more
k
elementary fields, as in the case of the (N,M)–KdV hierarchies studied in [5],[6],[7]. If the
hierarchy is integrable, the flows are given by
∂L
= [(Lk/N) ,L] (2.2)
+
∂t
k
wherethesubscript+ denotes thedifferential partof a pseudodifferentialoperator, t is identified
1
with x and k spans a specific subset of the positive integers.
The Toda lattice hierarchy is defined in terms of a semi–infinite Jacobi matrix Qˆ ∗. We
parametrize it as follows
∞ ∞
Qˆ = E + aˆ (j)E , (E ) = δ δ (2.3)
j,j+1 l j+l,j j,m k,l j,k m,l
jX=0(cid:16) Xl=0 (cid:17)
and consider aˆ as fields defined on a lattice. The flows are given by
l
∂Qˆ
= [(Qˆk) ,Qˆ], k = 1,2,... (2.4)
+
∂t
k
∗InthispaperwelimitourselvestoasimpleversionoftheTLH,inwhichonlyonematrixQˆ andonesetofparameters
intervene, instead of two or more [10],[3]
2
wherethesubscript+ denotes theuppertriangular partof amatrix, includingthemain diagonal.
(2.4) represents a hierarchy of differential–difference equations for the fields a . In particular the
l
first flows are
′
aˆ (j) = aˆ (j +1)−aˆ (j)+aˆ (j) aˆ (j)−aˆ (j −l) (2.5)
l l+1 l+1 l 0 0
(cid:16) (cid:17)
where we have adopted the notation ∂ f ≡ f′ ≡ ∂f, for any function f. The parameter t of the
∂t1 k
TLH will be identified later on with the corresponding parameter t in (2.2) whenever the latter
k
exists; therefore, in particular, t will be identified with x.
1
Next, integrability permits us to introduce the function Fˆ(n,t) (the free energy in matrix
models) via
∂2
Fˆ(n,t) = Tr [Qˆk,Qˆl] (2.6)
∂t ∂t +
k l
(cid:16) (cid:17)
where Tr(X) denotes the finite trace n−1X . In particular (2.6) leads to
j=0 j,j
P
∂2
Fˆ(n,t) = aˆ (n) (2.7)
∂t2 1
1
It is clear that by means of (2.4) we can compute the derivatives of any order of Fˆ in terms of the
entries of Qˆ. In general we will denote by Fˆ the derivative of Fˆ with respect to t ,...,t .
k1,...,ks k1 ks
Next we introduce the operator D , defined by its action on any discrete function f(n)
0
(D f)(n)= f(n+1)
0
For later use we remark that, if f = 0, the operation Tr is the inverse of the operation D −1.
0 0
We will also use the notation e∂0 instead of D , with the following difference: D is meant to be
0 0
applied to the nearest right neighbour, while e∂0 acts on whatever is on its right. Now we can
equivalently represent the matrix Qˆ by the following operator
∞
Qˆ(j) = e∂0 + aˆ (j)e−l∂0 (2.8)
l
l=0
X
The contact between (2.8) and (2.3) is made by acting with the former on a discrete function
ξ(j); then Qˆ(j)ξ(j) is the same as the j–th component of Qˆξ, where ξ is a column vector with
components ξ(0),ξ(1),.... We will generally drop the dependence on j in (2.8) and merge the two
symbols.
After this short introduction to the GD formalism and the Toda lattice hierarchy, let us come
to the presentation of the TLR of the integrable hierarchy defined by the Lax operator (2.1), i.e.
to the problem of embedding the latter into the TLH. The prescription consists of several steps.
Step 1. In Qˆ we set aˆ = 0 and replace the first flows (2.5) with
0
′
D0aˆ1 = aˆ1, D0aˆi = aˆi+aˆi−1, i= 2,3,... (2.9)
Step 2. We compute
∂aˆ
1 = ∂Tr [Qˆ ,Qˆk] ≡ ∂Fˆ (2.10)
+ 1,k
∂t
k
(cid:16) (cid:17)
3
The right hand side will be a polynomial of the fields aˆ to which monomials of D and D−1 are
k 0 0
applied. Next we substitute the first flows (2.9) to eliminate the presence of D . Examples:
0
Fˆ = aˆ ,
1,1 1
Fˆ = (D +1)aˆ = 2aˆ +aˆ′, (2.11)
1,2 0 2 2 1
Fˆ = (D2+D +1)aˆ +D aˆ aˆ +aˆ aˆ +aˆ D−1aˆ = 3aˆ +3aˆ′ +aˆ′′+3aˆ2
1,3 0 0 3 0 1 1 1 1 1 0 1 3 2 1 1
and so on.
Next we recall that
∂2
Fˆ = Fˆ
1,k
∂t ∂t
k 1
Usingthisand(2.11), wecanrecursivelywriteallthederivativesofaˆ withrespecttothecouplings
l
t (and in particular the flows) in terms of derivatives of Fˆ, which, in turn, can be expressed as
k
functions of the entries of Qˆ. Example:
∂ 1 1 ∂
aˆ = ∂Tr [Qˆ2,Qˆk] − aˆ′
∂t 2 2 + 2∂t 1
k k
(cid:16) (cid:17)
Ingeneralwe willneedall Fˆ . Herearesomegeneral formulas. Letusintroducethesymbols
k1,...,kn
Aˆ[k] as follows
j
Qˆk = ek∂ +ka e(k−2)∂ +Aˆ[k]e(k−3)∂ +Aˆ[k]e(k−4)∂ +Aˆ[k]e(k−5)∂ +... (2.12)
1 2 3 4
The explicit form of the first few is:
Aˆ[k] = k a′ +kaˆ (2.13)
2 2 1 2
!
Aˆ[k] = k a′′+ k aˆ′ +kaˆ + k a2
3 3 1 2 2 3 2 1
! ! !
Aˆ[k] = k a′′′+ k aˆ′′+ k aˆ′ +kaˆ +(3 k − k )a a′ +2 k a aˆ
4 4 1 3 2 2 3 4 3 2 1 1 2 1 2
! ! ! ! ! !
Aˆ[k] = k a(4)+ k aˆ′′′+ k aˆ′′+ k aˆ′ +kaˆ + k aˆ aˆ +2 k a aˆ
5 5 1 4 2 3 3 2 4 5 2 2 2 2 1 3
! ! ! ! ! !
+ k a3+(3 k − k )a′a′ +(4 k −2 k + k )a a′′+
3 1 4 3 1 1 4 3 2 1 1
! ! ! ! ! !
k k ′ k k ′
(3 − )a aˆ +(3 −2 )aˆ a
3 2 1 2 3 2 2 1
! ! ! !
and so on. In terms of these coefficients we can compute all the derivatives of Fˆ. For example
Fˆ = Aˆ[k]
1,k k
Fˆ = (D +1)Aˆ[k]
2,k 0 k+1
Fˆ = (D2+D +1)Aˆ[k] +3a Aˆ[k] (2.14)
3,k 0 0 k+2 1 k
Fˆ = (D3+D2+D +1)Aˆ[k] +4a (D +1)Aˆ[k] +a(4)Aˆ[k]
4,k 0 0 0 k+3 1 0 k+1 2 k
4
This procedure allows us to compute all the derivatives of the fields aˆ in terms of the same fields
l
and their derivatives with respect to x≡ t – therefore, in particular, the flows.
1
So far all our moves have been completely general (except for setting aˆ = 0, but see the
0
comment at the end of this section). The next step is instead a ‘gauge choice’, that is we make
a particular choice for the matrix Qˆ. The word ‘gauge’ is not merely colorful. In fact gauge
transformations play here a role analogous to gauge transformations in [2]. The relevant gauge
transformations in the present case are defined by Qˆ → G−QˆG−−1, where G− is a strictly lower
triangular semi–infinite matrix.
Step 3. We fix the gauge by imposing the condition
QˆN = eN∂0 + a e(N−1−l)∂0 (2.15)
l
l=1
X
where the a are the same as in eq.(2.1). The matrix Qˆ that satisfies such condition will be
l
referred to as Q¯. It is clear that Q¯N exactly mimics the Lax operator L. The condition (2.15)
recursively determines aˆ in terms of the fields a that appear in L.
k l
aˆ = a¯ ≡ P (a )
k k k l
where P are differential polynomials of a . In particular we always have aˆ = a¯ ≡ a .
k l 1 1 1
Step 4. Then we evaluate both sides of the flows found in Step 2 at aˆ = a¯ . The order here
k k
is crucial. The gauge fixing of the flows must be the last operation.
Now we claim:
Claim. The flows obtained in this way coincide with the flows (2.2) for corresponding cou-
plings.
We will substantiate this claim with a large number of examples in the next section.
It is perhaps useful to summarize our method: start from the TLH flows, use the first flows
(2.9) and impose the relevant gauge fixing; the resulting flows are the desired differential integrable
flows.
We would like to end this section with a remark concerning the restriction aˆ = 0 we imposed
0
at the very beginning. This can be avoided at the price of working with very encumbering
formulas. One can keep aˆ 6= 0 provided one uses the first flows (2.5) instead of (2.9) in Step 1.
0
In this way it is possible, in general, to eliminate D in the flows only when it acts over aˆ , l 6= 0
0 l
(see the last section for an additional comment on this point). We obtain in this way the same
equations as above with the addition of terms involving aˆ . We can suppress all these additional
0
terms at the end (Step 5) by imposing aˆ = 0 as part of the gauge choice. The final result is of
0
course the same as before. This justifies our having imposed aˆ = 0 from the very beginning.
0
3 Examples.
In this section we present a large number of examples in support of the claim of the previous
section. Of course for obvious reasons of space we can explicitly exhibit a few cases only, and for
each case only a few flows among those we have checked.
5
The KP hierarchy
The KP case corresponds to n = 1 in (2.1). Therefore there is no gauge fixing: aˆ = a . The flows
l l
obtained with our method are simply those in Step 3. Examples:
∂aˆ ′ ∂aˆ ′
1 = 2aˆ +aˆ′ , 1 = 3aˆ +3aˆ′ +aˆ′′+3aˆ2
∂t 2 1 ∂t 3 2 1 1
2 3
(cid:16) (cid:17) (cid:16) (cid:17)
∂aˆ ′ ∂aˆ ′
2 = 2aˆ +aˆ′ +aˆ2 , 2 = 3aˆ +3aˆ′ +aˆ′′+6aˆ aˆ
∂t 3 2 1 ∂t 4 3 2 1 2
2 3
(cid:16) (cid:17) (cid:16) (cid:17)
and so on. Setting aˆ = a , these are exactly the KP flows.
l l
N
The -KdV hierarchy case
In [4] we have explicitly shown that our claim is true for the 3–KdV hierarchy. In this section we
generalize that result. To start with we pick a generic N. The relevant differential operator is
L = DN +Na1DN−2+Na2DN−3+...+NaN−1 (3.1)
We also write
Lk/N = Dk +ka Dk−2+b[k]Dk−3+...+b[k] Dk−j +... (3.2)
1 2 j−1
[k]
The coefficients bj are differential polynomials in al, l = 1,...,aN−2.
Working out the commutator in relation (2.2), we can write down the general formula for
arbitrary flow t :
m
m−1 m−2
∂aj−1 = m a(m−k) − 1 N (b(m) )(j+k)+ (3.3)
∂tm k ! j+k−1 N j +k ! m−k−1
k=0 k=0
X X
m−3 m−2
+ ( mk−2 bk(m−)1aj(m−k−−2−1k)− N −l−jk−k ak−1(bm(m−)l−1)(l−k))
! !
k=0 l=0
X X
j−1m−2
− jN−−k+kl ak−1(bm(m−)l−1)(j−k+l)
!
k=2 l=0
X X
Now let us pass to the TLR of this hierarchy. We recall eqs.(2.13) and (2.14). We fix the
gauge by imposing Aˆ[N] = Na . We solve the equations for aˆ in terms of a and obtain a¯ . Next
j j j j j
we insert back the result in the formulas of the coefficients Aˆ[k] so that they become functions of
j
a . We call the result A¯[k]. Examples:
j j
N −k
A¯[k]/k = a − a′
2 2 2 1
N −k (N −2k+3)(N −k) N −k
A¯[k]/k = a − a′ + a′′− a2 (3.4)
3 3 2 2 12 1 2 1
N −k (N −2k+3)(N −k) (N −k+2)(k−2)(N −k)
A¯[k]/k = a − a′ + a′′− a′′′
4 4 2 3 12 2 24 1
(N −k+2)(N −k)
′
−(N −k)a a + a a
1 2 2 1 1
6
Then, using our recipe, we obtain
∂a ∂2F
∂−1 1 = | = A¯[k]
∂t ∂t ∂t aˆ=a¯ k
k 1 k
∂a 1 ∂2F N −2∂a N −1
∂−1 2 = | + 1 = A¯[k] + (A¯[k])′
∂t 2∂t ∂t aˆ=a¯ 2 ∂t k+1 2 k
k 2 k k
∂a (cid:16)1 ∂2F (cid:17) N −3∂a (N −3)2∂a′ ∂a
∂−1 3 = | + 2 − 1 +(N −3)∂−1(a 1)
aˆ=a¯ 1
∂t 3∂t ∂t 2 ∂t 12 ∂t ∂t
k 3 k k k k
(cid:16) (cid:17)
N −1 (N −1)(N −2)
= A¯[k] + (A¯[k] )′+ (A¯[k])′′+a A¯[k]+(N −3)∂−1(a (A¯[k])′)
k+2 2 k+1 6 k 1 k 1 k
∂a 1 ∂2F N −4∂a (N −4)(N −5)∂a′ ∂(a a )
∂−1 4 = + 3 | − 2 +(N −4)∂−1 1 2
aˆ=a¯
∂t 4∂t ∂t 2 ∂t 12 ∂t ∂t
k 4 k k k k
(cid:16) ′′ (cid:17)
(N −4)(N −2) 1∂a ∂a N −1
− ( 1 +a 1) = A¯[k] + (A¯[k] )′
2 6 ∂t 1∂t k+3 2 k+2
k k
(3N −11)(N −1)(N −2) (N −2)(N −4)
+ (A¯[k])′′′− a (A¯[k])′+
24 k 2 1 k
(5N −16)(N −1) N −1
+ (A¯[k] )′′+(N −4)∂−1(a (A¯[k] )′+ (a (A¯[k])′′+a (Aˆ[k])′)
12 k+1 1 k+1 2 1 k 2 k
and so on, where aˆ = a¯ denotes gauge fixing.
We give a few concrete examples of the second and third flows:
∂a
∂−1 1 = 2a −(N −2)a′
∂t 2 1
2
∂a (N −1)(N −2)
∂−1 2 = 2a +a′ − a′′−(N −2)a2
∂t 3 2 3 1 1
2
∂a (N −1)(N −2)(N −3)
3 ′ ′′ (4) ′′ ′
= 2a +a − a −(N −2)(N −3)a a −2(N −3)a a
∂t 4 3 12 1 1 1 2 1
2
∂a 3 (N −3)2 3
∂−1 1 = 3a − (N −3)a′ + a′′− (N −3)a2
∂t 3 2 2 4 1 2 1
3
∂a N(N −3) (N −1)(N −2)(N −3)
∂−1 2 = 3a +3a′ − a′′+ a′′′−3(N −3)a a
∂t 4 3 2 2 8 1 1 2
3
∂a3 ′ ′′ ′′′ 3 N (4) 3(3N −7) N (5) ′ ′
= 3a +3a +a − a + a +3a a −3(N −4)a a
∂t3 5 4 3 N 4 ! 2 10N 4 ! 1 1 3 1 3
′ 3 ′′ 3 ′′ 6 N ′′′
−3(N −3)a a − (N −2)(N −3)a a + (N −3)a a + a a
2 2 2 1 2 2 2 1 N 4 ! 1 1
These are flows pertinent to the N–KdV hierarchy with N > 3. In general the formulas of the
N–KdV hierarchy and the corresponding formulas obtained with our method coincide since
b[m] = A¯[m]
k k
We have checked these identities case by case up to the 5-KdV and for m ≤ 5. For the disper-
sionless case we have verified the correspondence up to the 8–KdV flows.
7
The DS hierarchies
Drinfeld and Sokolov, [2], introduced a large set of generalized KdV systems in terms of the pair
(G,c ), where G is a classical Kac-Moody algebra and c is a vertex of the Dynkin diagram
m m
of G. From each choice of the pair (G,c ) they were able to construct a pseudo–differential
m
operatorLwhichgiverisetoahierarchyofintegrableequations. Wehavestudiedalltheexamples
correspondingtotheoperatorLof orders3,4,5 andfoundacomplete agreementwithourmethod.
For simplicity here we present a few examples of order 4 and 5, corresponding to the cases with a
pseudodifferential Lax operator. The cases with a differential Lax operator are restriction of the
4– and 5–KdV hierarchies, and will be omitted.
In each case we give the explicit form of the (pseudo–)differential operator L, the gauge–fixed
matrix Q¯ and the first significant flows:
Order 4.
(1)
Case B :
2
c ,c :L = D4+2u D2+u′D+2(u +u′′)−D−1(u +u′′)′ (3.5)
0 1 1 1 0 1 0 1
′ ′′
v v 1 v 3
Qˆ = e∂ + 1e−∂ − 1e−2∂ +( v + 1 − v2)e−3∂ +
4 4 4 0 8 32 1
3 1
+( v v′ − v′)e−4∂ +...
8 1 1 2 0
c :L = D4+2u D2+u′D+u2−u D−1u′
2 1 1 0 0 0
Forc wehave, uptotheordere−4∂,thesameexpressionforQˆ withv = u2. Thefirstnon–trivial
2 0 0
flows are:
∂v 1 3
1 ′′′ ′ ′
= − v − v v +3v
∂t 2 1 4 1 1 0
3
∂v 3
0 ′′′ ′
= v + v v (3.6)
∂t 0 4 1 0
3
where for c ,c :v = 2u ,v = 2(u +u′′) and for c : v = 2u ,v = u2.
0 1 1 1 0 0 1 2 1 1 0 0
(1)
Case D :
3
c ,c : L = D4+2u D2+u′D+2u′′+2u −D−1(u′ +u′′)′+(D−1u )2 (3.7)
0 1 2 2 2 1 1 2 0
u 1 3 3 1
Qˆ = e∂ + 2e−∂ − u′e−2∂ +( u′′− u2− u )e−3∂ +
2 2 2 4 2 8 2 2 1
3
+( u u′ −u′ −u′′′)e−4∂ +...
2 2 2 1 2
c ,c : L = D4+2u D2+3u′D+(2u +3u′′)+(u′ +u′′′)D−1+u D−1u D−1
2 3 2 2 1 2 1 2 0 0
u 1 3 1
Qˆ = e∂ + 2e−∂ +( u′′− u2+ u )e−3∂ +
2 4 2 8 2 2 1
3 1 1
+( u u′ − u′ − u′′′)e−4∂ +...
4 2 2 2 1 4 2
The first non–trivial equations for c ,c are:
0 1
∂u 3 3 9
1 ′ (5) ′′′ ′ ′′′ ′ ′′
= 3u u − u −2u + u u +3u u + u u (3.8)
∂t 0 0 2 2 1 2 1 2 2 2 2 2 2
3
∂u 5 3
2 ′′′ ′ ′
= u − u u +3u
∂t 2 2 2 2 2 1
3
8
and for c ,c are
2 3
∂u 3 3 9
1 ′ (5) ′′′ ′ ′ ′′ ′′′
= 3u u − u −2u − u u + u u +3u u (3.9)
∂t 0 0 2 2 1 2 1 2 2 2 2 2 2
3
∂u 5 3
2 ′′′ ′ ′
= u − u u +3u
∂t 2 2 2 2 2 1
3
Order 5.
(2)
Case A :
5
c ,c : L = D5+2u D3+2u′D2+(2u +4u′′)D+D−1(2u +u′′+u′′′) (3.10)
0 1 2 2 1 2 0 1 2
2 2 2 4
Qˆ = e∂ + u e−∂ − u′e−2∂ + (u +2u′′− u2)e−3∂ +...
5 2 5 2 5 1 2 5 2
c : L = D5+2(v +u )D3+(6v′ +u′)D2+(6v′′+u2+4v u )D+
2 0 1 0 1 0 0 0 1
+(2v′′′ −u u′ +4u v′ +2v u′)+u D−1(u′′+2u v )
0 0 0 1 0 0 1 0 0 0 0
2 1
Qˆ = e∂ + (u +v )e−∂ + (2v′ −3u′)e−2∂ +
5 1 0 5 0 1
1 4 8 8
+ (2u′′ −2v′′ + v u +u2− u2− v2)e−3∂ +...
5 1 0 5 0 1 0 5 1 5 0
c : L = D5+2u D3+3u′D2+(2u +3u′′)D+u′ +u′′′+u D−1u
3 2 2 1 2 1 2 0 0
2 1 1 8
Qˆ = e∂ + u e−∂ − u′e−2∂ + (2u +u′′− u2)e−3∂ +...
5 2 5 2 5 1 2 5 2
The equations are for c ,c :
0 1
∂u 12
2 ′′′ ′ ′
= 4u +3u − u u (3.11)
∂t 2 1 5 2 2
3
∂u 7 54 6 51
1 ′′′ ′′′ ′ ′′ ′ ′ (5) ′
= − u +6u u + u u + (u u −u u )− u +3u
∂t 2 1 2 2 5 2 2 5 2 1 1 2 10 2 0
3
for c :
2
∂v 3
0 ′′′ ′′′ ′ ′ ′
5 = −v + u −12v v +6v u +12v u (3.12)
∂t 0 2 1 0 0 0 1 0 1
3
∂u
1 ′′′ ′′′ ′ ′ ′ ′
5 = 6v −4u +15u u +6v u +12v u −12u u
∂t 0 1 0 0 0 1 0 1 1 1
3
and for c are:
3
∂u 12
2 ′′′ ′ ′
= u +3u − u u (3.13)
∂t 2 1 5 2 2
3
∂u 27 12 6 3
1 ′′′ ′ ′′ ′′′ ′ ′ (5) ′
= −2u + u u + u u + (u u −u u )− u +3u u
∂t 1 5 2 2 5 2 2 5 2 1 1 2 5 2 0 0
3
The (N,M)–KdV hierarchies
The (N,M)–KdV hierarchies are defined by the pseudodifferential operator
N−1 M
1 1 1
L = ∂N +N al∂N−l−1+N aN+l−1 ... , N ≥ 1, M ≥ 0 (3.14)
∂ −Sl∂−Sl−1 ∂ −S1
l=1 l=1
X X
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The case (N,0) coincides with the N–KdV case. These hierarchies were studied in [5],[6],[7],[9].
In [7] it was shown that they can be embedded in the DS construction. Now we show that this
class of integrable hierarchies can be entirely embedded in the TLH. Let us see, for example, the
(2,1) case. The Lax operator is
1
L = ∂2+2a +2a
1 2
∂ −S
The gauge fixing gives
1 1 1 1
a¯ = a , a¯ = a − a′, a¯ = − a′ + a′′− a2+a S
1 1 2 2 2 1 3 2 2 4 1 2 1 2
and so on. It leads, via our recipe, to the following flows
∂a ∂a ∂S
∂−1 1 = 2a , ∂−1 2 = a′ +2a S, ∂−1 = S2+2a −S,′
∂t 2 ∂t 2 2 ∂t 1
2 2 2
∂a 3 1 3 ∂a
∂−1 1 = a′ + a′′+ a2+3a S, ∂−1 2 = a′′+3a a +3a′S +3a S2
∂t 2 2 4 1 2 1 2 ∂t 2 1 2 2 2
3 3
and so on. These are exactly the flows of the (2,1)–KdV hierarchy.
4 Comments and conclusion
.
Theexamples wehave considered intheprevioussection donotexhaustall possibleintegrable
hierarchies (for an updating on this subject see [11]). However they are very numerous and they
leave very little doubt that whatever scalar Lax operator (2.1), defining an integrable hierarchy,
we may think of, it can be embedded in the Toda lattice hierarchy in the way we showed above.
Anyhow, thus far we have not found any counterexample. Therefore our construction looks at
least as general as the DS realization. The fact that we are dealing with semi–infinite matrices
may suggest additional possibilities.
We also remark that the TLH, in its general formulation, may encompass several Qˆ matrices
(not only one, as in this paper). Therefore there is room for ‘tensor products of integrable
hierarchies in interaction’.
We endthepaperby recalling thatin thecase of the(1,M)–KdVhierarchies thereisa variant
to the realization of section 2. This was already pointed out in section 6.2 of [4] and, implicitly,
in [5]. If one does not set aˆ = 0 and replaces the first flows (2.5) in the Toda lattice flows, one
0
gets exactly the (1,M) hierarchies if the gauge fixing simply consists of setting aˆ = 0 for l > M.
l
It was shown in [5] that (N,M)–KdV hierarchies can then be extracted from the (1,M) via a
cascade Hamiltonian reduction. However it is not clear whether this method can be generalized
to other hierarchies, and, anyhow, it does not seem to be appropriate to call it a realization of
differential hierarchies, at least in the same sense this terminology has been used in this paper.
Acknowledgements. Oneofus(C.P.C.)wouldliketothankCNPqandFAPESPforfinancial
support.
References
[1] L.A.Dickey, Soliton Equations and Hamiltonian Systems World Scientific, Singapore (1991).
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