Poisson Algebras associated with Constrained Dispersionless Modified KP Hierarchies

15 Pages·0.14 MB·English

by  Jen-Hsu Chang

Poisson Algebras associated with Constrained
Dispersionless Modified KP Hierarchies
Jen-Hsu Chang ∗
Institute of Mathematics, Academia Sinica,
Nankang, Taipei, Taiwan
and
Ming-Hsien Tu†
0
0 Department of Physics, National Chung Cheng University,
0
Minghsiung, Chiayi, Taiwan
2
(February 8, 2008)
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a
J

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1 Abstract

]
I
S
Weinvestigatethebi-Hamiltonian structuresassociatedwithconstraineddis-
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n persionless modified KP hierarchies which are constructed from truncations
i
l of the Lax operator of the dispersionless modified KP hierarchy. After trans-
n
[ forming their second Hamiltonian structures to those of Gelfand-Dickey type,

1 we obtain the Poisson algebras of the coefficient functions of the truncated
v Lax operators. Then we study the conformal property and free-field realiza-
6
tions of these Poisson algebras. Some examples are worked out explicitly to
3
0 illustrate the obtained results.
1
0
0
0
/
n
i
l
n
:
v
i
X
r
a
Typeset using REVTEX
∗E-mail: [email protected]
†E-mail: [email protected]
1
I. INTRODUCTION
The dispersionless integrable hierarchies can be viewed as the quasi-classical limit of
the ordinary integrable systems [1]. A typical example is the dispersionless Kadomtsev-
Petviashvili (dKP) hierarchy which has played an important role in theoretical and math-
ematical physics (see, for example, [2] and references therein). The Lax formulation of the
dKP hierarchy can be constructed by replacing the pseudo-differential Lax operator of KP
with the corresponding Laurent series. On the other hand, an analogue construction can be
made for the modified KP (mKP) hierarchy and thus leads to the dmKP hierarchy.
In the previous work [3], we established the Miura map between the dKP and dmKP
hierarchies, which turns out to be canonical in the sense that the bi-Hamiltonian structure
of the dmKP hierarchy [4] is mapped to the bi-Hamiltonian of the dKP hierarchy [4,5]. We
also studied the solution structure of the dmKP hierarchy using the twistor construction [2].
In this paper we turn to the Poisson algebras of the bi-Hamiltonian structures associated
with the dmKP hierarchy and, particularly, its reductions. For the ordinary mKP hierarchy
the reductions are quite limited [6–8]. However, in the dispersionless limit, we show that
the Lax operator of the dmKP hierarchy can be truncated to any finite order and their
associated bi-Hamiltonian structures can be obtained via the Dirac reduction. To proceed
the formulation of the dmKP hierarchy, we recall some basic facts about the algebra of
Laurent series in the following.
Let Λ be an algebra of Laurent series of the form
N
Λ = A A = a pi ,
i
{ | }
i=−∞
X
with coefficients a depending on an infinite set of variables t x,t ,t , . The algebra Λ
i 1 2 3
≡ ···
can be decomposed into the subalgebras as
Λ = Λ Λ ,
≥k <k
⊕
where
Λ = A Λ A = a pi
≥k i
{ ∈ | }
i≥k
X
Λ = A Λ A = a pi
<k i
{ ∈ | }
i<k
X
and using the notations : Λ = Λ and Λ = Λ for short. Although Λ form a commu-
+ ≥0 − <0
tative and associative algebra under multiplication, we can define a Lie-bracket associated
with Λ such that
∂A∂B ∂A∂B
[[A,B]] = , A,B Λ
∂p ∂x − ∂x ∂p ∈
which canberegardedasthe Poisson bracket defined inthe2-dimensional phase space(x,p).
For a given Laurent series A we define its residue as
resA = a
−1
2
and its trace as
trA = resA.
Z
For any two Laurent series A = a pi and B = b pi we have
i i i i
P P
res[[A,B]] = i(a b )′
i −i
i
X
which implies
tr[[A,B]] = 0,
and
tr(A[[B,C]]) = tr([[A,B]]C).
Finally, given a functional F(A) = f(a) we define its gradient as
R
δF δf
= p−i−1
δA δa
i i
X
where the variational derivative is defined by
δf ∂f
= ( 1)i ∂i ,
δak Xi − ∂a(ki)!
with a(i) (∂ia ),∂ ∂/∂x.
k ≡ k ≡
This paper is organized as follows: In section II, we will derive the bi-Hamiltonian
structures of constrained dmKP hierarchies from the one of the dmKP hierarchy by the
Dirac reduction. In section III, we will investigate the conformal property of the second
Poisson brackets associated with constrained dmKP hierarchies. In section IV, the free-field
realizations of these Poisson algebras will be given through the corresponding Kupershmidt-
Wilson (KW) theorem. We will give some examples to illustrate the obtained results in
section V. Section VI contains some concluding remarks.
II. BI-HAMILTONIAN STRUCTURES
The (generalized) dmKP hierarchy is defined by the Lax operator of the form
K = pn +v pn−1 + , (n > 0)
n n−1
···
which satisfies the equations of motion
dK
n = [[B ,K ]], B = (Kk/n) (2.1)
dt k n k n ≥1
k
or zero-curvature conditions
3
∂B ∂B
k l
+[[B ,B]] = 0. (2.2)
k l
∂t − ∂t
l k
For n = 1, the first nontrivial flows (t = y,t = t) of (2.2) are given by
2 3
3 3
v = v (v2) ,
−1x 2 0y − 2 0 x
3
v = 2v (v2) 2v v .
−1y 0t − 2 0 y − −1 0x
which, by eliminating v , yields the dmKP equation
−1
3 3 3
v = v2v + v ∂−1v + ∂−1v .
0t −2 0 ox 2 0x x 0y 4 x 0yy
The Hamiltonian structures associated with K have been obtained by Li [4] using the
n
classical r-matrix formulation. Especially, the second structure can be expressed as
δF δG
(n)
F,G (K ) = res J
{ } n Z 2 δKn! δKn!
(n)
where the Hamiltonian map J is defined by
2
(n)
J (X) = [[K ,X]] K [[K ,(K X) ]] (2.3)
2 n ≥−1 n − n n ≥1
with X = x p−i−1. It is quite natural to define the conserved quantities as
i i
P n
H = trKk/n,
k k n
then the Lax flows (2.1) can be described by the Hamiltonian equations
dK δH
n (n) (n) k
= H ,K (K ) = J .
dtk { k n}2 n 2 δKn!
Based on the above results, we would like to consider reductions of the dmKP hierarchy and
their associated Hamiltonian structures. Let us consider truncations of the Lax operator K
n
as follows
K = pn +v pn−1 + +v p−m, m Z/ 0 . (2.4)
(n,m) n−1 −m
··· ∈ { }
It is quite easy to show that these are consistent truncations with respect to the Lax flows
(2.1). Thus for each pair (n,m), the Lax operator K with infinitely many coefficient func-
n
tions is reduced to a finite-dimensional one K which we refer to the constrained dmKP
(n,m)
−
hierarchy. However, the Hamiltonian map (2.3) can not preserve the form of dK /dt
(n,m) k
since the lowest order term of J(n)(δH/δK ) in p is p−m−1. Hence we shall consider the
2 (n,m)
Lax operator K¯ = K +µp−m−1 and then impose the constraint µ = 0 by the Dirac
(n,m) (n,m)
¯
reduction. It turns out that Hamiltonian flows for K under the condition µ = 0 gives
(n,m)
the second class constraint:
4
δH
res K¯ , = 0 (2.5)
"" (n,m) δK¯(n,m)##!µ=0
where
δH δH δH
= + pm.
δK¯ δK δµ
(n,m) (n,m)
That means the function δH/δµ should be in terms of δH/δv ,i = m, m+1, ,n 1.
i
− − ··· −
Solving the constraint (2.5), we obtain
δH 1 x δH
v = res K ,
−m (n,m)
δµ m Z "" δK(n,m)##
which implies
δH δH δH
(n)
J = K , K K , K ,
2 δK¯(n,m)! "" (n,m) δK¯(n,m)##≥−1 (n,m) − (n,m) (n,m)δK¯(n,m)!≥1
 
δH δH
= K , K K , K
(n,m) (n,m) (n,m) (n,m)
"" δK(n,m)##≥−1 − δK(n,m)!≥1
δH  
+ K ,v ,
(n,m) −m
"" δµ##
δH δH
= K , K K , K
(n,m) (n,m) (n,m) (n,m)
"" δK(n,m)##+ −"" δK(n,m)!+##
δH δH
+ K , K + K , K
(n,m) (n,m) (n,m) (n,m)
"" δK(n,m)!0## "" δK(n,m)##−1
1 x δH
+ K , res K , ,
(n,m) (n,m)
m "" Z "" δK(n,m)####
δH
(n,m)
J . (2.6)
≡ 2 δK(n,m)!
We note that the above modified Hamiltonian map for m = 1 are just the classical limit of
± (n,∞)
the second structures of mKP hierarchies obtained in [6–8]. Besides, when m , J
→ ∞ 2
(n)
recovers the Hamiltonian map J , as expected.
2
Finally, we would like to remark that the first Poisson structure of the constrained
(n,m)
dmKP hierarchies can be defined as a deformation of J by shifting K K +λ
2 (n,m) 7→ (n,m)
(n,m)
by a constant parameter λ. Then the second structure induces a linear term J
2 7→
(n,m) (n,m)
J +λJ with
2 1
δH δH δH
(n,m)
J = K , K , (2.7)
1 δK(n,m)! "" (n,m) δK(n,m)##≥−1 − (n,m) δK(n,m)!≥1
 
which, by definition, is compatible with the second structure and is a Laurent series of order
at most n 1. It turns out that (2.7) is nothing but the first Poisson structure defined in
−
5
(n,m)
[4]. Note that the Hamiltonian map J is consistent with the Lax flows (2.1) for m > 0
1
(n,m)
but not for m < 0 due to the fact that the lowest order term of J (δH/δK ) in p is
1 (n,m)
p−1, not p|m|. Hence, just like the second structure, we require the use of Dirac’s theory of
constraints to obtain the consistent result. This will be done in the next section.
III. POISSON ALGEBRAS
Having constructed the Hamiltonian map of the constrained dmKP hierarchies, we are
now ready to calculate the Poisson brackets of the coefficient functions v in (2.4). Before
i
doing that, we would like to show that the complicated form of the modified Hamiltonian
(n,m)
map J defined by K can be transformed to the familiar Gelfand-Dickey (GD) type
2 (n,m)
structures via the following identification
K = L p−m = (pn+m +u pn+m−1 + +u )p−m
(n,m) n+m n+m−1 0
···
where
v = u , i = m, m+1, ,n 1. (3.1)
i i+m
− − ··· −
On the other hand, from the variation
δF δF
δF = res δK = res δL
(n,m) n+m
Z δK(n,m)! Z δLn+m!
we have
δF δF
= pm .
δK δL
(n,m) n+m
Using the above relations, some terms in (2.6) can be rewritten as
δF δF
K , K = p−m L , L
(n,m) (n,m) n+m n+m
"" δK(n,m)##+ "" δLn+m##+
′
δF
mK p−1 K
(n,m) (n,m)
− δK(n,m)!+
′
δF
mK p−1 K ,
(n,m) (n,m)
− δK(n,m)!0
δF δF
K , K = p−m L , L
(n,m) (n,m) n+m n+m
"" δK(n,m)!+## "" δLn+m!+##
′
δF
mp−1K K ,
(n,m) (n,m)
− δK(n,m)!+
1 x δF 1 x δF
K , res K , = p−m L , res L ,
(n,m) (n,m) n+m n+m
m "" Z "" δK(n,m)#### m "" Z "" δLn+m####
δF
p−1K res K ,
(n,m) (n,m)
− "" δK(n,m)##
6
δF
K , K
(n,m) (n,m)
−"" δK(n,m)!0##
′
δF
mp−1K K ,
(n,m) (n,m)
− δK(n,m)!0
which imply
δF δG
(n,m) (n,m)
F,G (K ) = res J ,
{ }2 (n,m) Z 2 δK(n,m)! δK(n,m)!
δF δG
= res ΘGD ,
Z 2+3 δLn+m! δLn+m!
= F,G GD(L ) (3.2)
{ }2+3 n+m
where the Hamiltonian map ΘGD ΘGD + 1ΘGD with
2+3 ≡ 2 m 3
ΘGD(X) = [[L ,X]] L [[L ,(L X) ]], (3.3)
2 n+m + n+m − n+m n+m +
x
ΘGD(X) = L , res[[L ,X]] . (3.4)
3 n+m n+m
(cid:20)(cid:20) Z (cid:21)(cid:21)
Besides the standard second GD structure ΘGD, (3.4) is called the third GD bracket which
2
is compatible with the second one. Hence, under the identification (3.1), the modified
Hamiltonian structure (2.6) has been mapped to the sum of the second and the third GD
structures defined by the polynomial L .
n+m
Since the Poisson algebras associated with the second GD structure have been obtained
[5], we only need to treat the third one. Therefore, by (3.2), we can now use (3.3) and (3.4)
(n,m) (n,m)
instead of (2.6) to read off the Poisson brackets v (x),v (y) = J (v) δ(x y)
{ i j }2 − 2 ij −
(n,m) (cid:16) (cid:17)
where the operators J (v) are taken at the point x. After some straightforward
2
ij
algebras we have (cid:16) (cid:17)
n(n+m)
(n,m)
J = ∂,
2 n−1,n−1 m
(cid:16) (cid:17)
n(i+m+1)
(n,m)
J = v ∂,
2 i,n−1 m i+1
(cid:16) (cid:17)
n(j +m+1)
(n,m)
J = ∂v ,
2 n−1,j m j+1
(cid:16) (cid:17)
(n,m)
J = (n i 1)v ∂ +(n j 1)∂v
2 i,j − − i+j+2−n − − i+j+2−n
(cid:16) (cid:17)
n−1
+ [(k i 1)v ∂v +(k j 1)v ∂v ]
i+j+2−k k k i+j+2−k
− − − −
k=j+2
X
(i+m+1)(j +1)
+ v ∂v (3.5)
i+1 j+1
m
where i,j = m, m+1, ,n 2 and v = 0. We refer the above Poisson algebra to
i<−m
− − ··· −
w(n,m)-algebra.
For the Poisson algebra associated with the first structure, it can be directly obtained
from the Hamiltonian map (2.7) for the case of m > 0 as follows:
7
(i+1)v ∂ +(j +1)∂v , 1 i,j n 1
i+j+2 i+j+2
− ≤ ≤ −
J(n,m>0) =  (i+1)v ∂ (j +1)∂v , m i,j 2
(cid:16) 1 (cid:17)ij 0− i+j+2 − i+j+2 −other≤wise.≤ −
However the case for m <0 requires the Dirac reduction and turns out to be
|m|−1
(n,m<0) (n,1) (n,1) (n,1) −1 (n,1)
J = J J J J , m i,j n 1.
1 ij 1 ij − 1 ik 1 kl 1 lj | | ≤ ≤ −
(cid:16) (cid:17) (cid:16) (cid:17) k,Xl=−1(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Note that the bi-Hamiltonian structures of constrained dmKP hierarchies can be cast into
the following recursive formula:
δH δH
(n,m) k+n (n,m) k
J = J .
1 ij δv 2 ij δv
j j
(cid:16) (cid:17) (cid:16) (cid:17)
Next, let us focus on the algebraic structures of the w(n,m)-algebra (3.5). The first few of
them are
n(n+m)
(n,m)
v (x),v (y) = ∂ δ(x y),
{ n−1 n−1 }2 − m · −
n(n+m 1)
(n,m)
v (x),v (y) = − ∂v (x) δ(x y), (3.6)
{ n−1 n−2 }2 − m n−1 · −
(n 1)(n+m 1)
(n,m)
v (x),v (y) = v (x)∂ +∂v (x)+ − − v (x)∂v (x) δ(x y).
{ n−2 n−2 }2 −" n−2 n−2 m n−1 n−1 #· −
In spite of the fact that v satisfies the U(1)-Kac-Moody algebra, the algebraic structure
n−1
shown above is still unclear. However, if we define
n 1
w (x) = v (x) − v2 (x) (3.7)
2 n−2 − 2n n−1
then the second and the third equations in (3.6) can be rewritten as
v (x),w (y) (n,m) = [v (x)∂ +v′ (x)] δ(x y),
{ n−1 2 }2 − n−1 n−1 · −
w (x),w (y) (n,m) = [2w (x)∂ +w′(x)] δ(x y)
{ 2 2 }2 − 2 2 · −
where w , being a generator, is a DiffS1 tensor with weight 2 and v a tensor of weight 1.
2 n−1
In fact, using (3.5) and (3.7) we have
v (x),w (y) (n,m) = [iv (x)∂ +v′ (x)] δ(x y)
{ n−i 2 }2 − n−i n−i · −
that means, except v , each coefficient v in the Lax operator K , with respect to
n−2 n−i (n,m)
the generator w , is already a DiffS1 tensor with weight i. Hence the Poisson algebra w(n,m)
2
defined in (3.5) is isomorphic to w -U(1)-Kac-Moody-algebra generated by the primary
(n+m)
fields
n 1
w v , w v − v2 , w v , (3 i n+m)
1 ≡ n−1 2 ≡ n−2 − 2n n−1 i ≡ n−i ≤ ≤
Note that the DiffS1 flows can be viewed as the Hamiltonian flows generated by the Hamil-
tonian ǫ(x)h due to the fact that h = nres(K )1/n = w . That is the reason why the
1 1 (n,m) 2
w-algebraic structure is encoded in the constrained dmKP hierarchy. Finally when we take
R
the limit m , (3.7) still holds and the Poisson algebra (3.5) recovers to w(n,∞) w(n)
→ ∞ ≡ dmKP
defined by the Lax operator K .
n
8
IV. KW THEOREM AND FREE-FIELD REALIZATIONS
It has been shown [5,9,10] that the second GD structure defined by (3.3) has nice prop-
erties with respect to the factorization of the associated Lax operator. For example, if we
factorize L = L L , then
1 2
F,G GD(L) = F,G GD(L )+ F,G GD(L ). (4.1)
{ }2 { }2 1 { }2 2
On the other hand, if L = Lα,α N then we have
1 ∈
1
F,G GD(L) = F,G GD(L ). (4.2)
{ }2 α{ }2 1
Eqs.(4.1) and (4.2) are just the corresponding KW theorem for the classical limit of the
second GD bracket. More generally, we shall consider the factorization of the polynomial
L of the form
n+m
l l
L = (p+φ )αi, α = n+m, (4.3)
n+m i i
i=1 i=1
Y X
where the Miura variables φ are zeros of L with multiplicities α , then (4.1) and (4.2)
i n+m i
imply that
l 1 δF ′ δG
F,G GD(L ) = . (4.4)
{ }2 n+m −i=1 αi Z δφi! δφi!
X
To complete the discussion of the KW theorem we have to treat the third GD structure
under the factorization (4.3). In fact, we can show that the third GD structure enjoys the
following property [for the proof, see Appendix A]:
F,G GD(LαL ) = F,G GD(L L ). (4.5)
{ }3 1 2 { }3 1 2
That means the multiplicities α do not involve in the KW theorem with respect to the third
i
GD structure. Hence,
l
F,G GD(L ) = F,G GD( (p+φ )),
{ }3 n+m { }3 i
i=1
Y
l δF ′ δG
= . (4.6)
i,j=1Z δφi! δφj!
X
Combining (4.4) and (4.6) we have
l 1 1 δF ′ δG
F,G GD(L ) = δ .
{ }2+3 n+m −i,j=1(cid:18)αi ij − m(cid:19)Z δφi! δφj!
X
Among other things, the fundamental brackets for the Miura variables φ are
i
9
φ (x),φ (y) (n,m)(K ) = φ (x),φ (y) GD(L )
{ i j }2 (n,m) { i j }2+3 n+m
1 1
= δ ∂ δ(x y) (4.7)
ij
α − m · −
(cid:18) i (cid:19)
where i,j = 1,2,...,l.
Since the above Poisson matrix is symmetric and hence can be diagonalized by linearly
combining the Miura variables φ to obtain the free fields. For example, suppose all zeros are
i
simple, i.e. α = 1,i = 1,2, ,n+m, then v = S (φ ) being the symmetric functions of
i i n−i j
···
φ and the Poisson matrix (4.7) becomes P = 1 1hhT where T denotes the transpose
{ j} 2+3 −m
operation, 1 is a (n+m) (n+m) identity matrix and hT = (1, ,1). It is quite easy to
× ···
construct n+m 1 orthonormal eigenvectors h as follows
i
−
hT = (1, 1,0 ,0)/√2,
1 − ···
hT = (1,1, 2 ,0)/√6,
2 − ···
···
hT = (1,1, ,1, n m+1)/ (n+m)(n+m 1)
n+m−1 ··· − − −
q
which satisfy P h = h and hence have eigenvalue +1. Finally, from orthogonality, the
2+3 i i
remaining orthonormal eigenvector has the form
hT = (1,1, ,1)/√n+m
n+m ···
with eigenvalue n/m. Now if we rewrite the Miura variables φT = (φ ,φ , ,φ ) as
1 2 n+m
− ···
φ = He where H is a (n+m) (n+m) matrix defined by H = [h ,h , ,h ], then
1 2 n+m
× ···
(n,m)
e (x),e (y) = λ ∂ δ(x y) (4.8)
{ i j }2 i · −
with λ = 1 (i = 1,2, ,n+m 1), λ = n/m. Therefore (4.8) provides a free-field
i n+m
··· − −
realization of the w(n,m)-algebras (3.5) and the Lax operator K can be expressed as
(n,m)
n+m
K = (p+(He) )p−m,
(n,m) i
i=1
Y
where the free fields e satisfy the Hamiltonian flows
i
′
∂e δH
i k
= λ .
i
∂tk − δei !
In the case of m , the Poisson matrix of (4.7) becomes diagonal, which provides the
→ ∞
(n)
free-field realization of the w -algebra.
dmKP
V. EXAMPLES
Example 1 : For the Lax operator K = p2 +v p+v +v p−1 the first nontrivial
(2,1) 1 0 −1
equations are given by
10