IJMMS 32:10 (2002) 587–614
PII. S0161171202202069
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
ON THE COMPATIBLE WEAKLY NONLOCAL POISSON
BRACKETS OF HYDRODYNAMIC TYPE
ANDREI YA. MALTSEV
Received 5 February 2002
We consider the pairs of general weakly nonlocal Poisson brackets of hydrodynamic type
(Ferapontov brackets) and the corresponding integrable hierarchies. We show that, under
the requirement of the nondegeneracy of the corresponding “first” pseudo-Riemannian
νµ
metric g(0) and also some nondegeneracy requirement for the nonlocal part, it is possible
to introduce a “canonical” set of “integrable hierarchies” based on the Casimirs, momentum functional and some “canonical Hamiltonian functions.” We prove also that all the
“higher” “positive” Hamiltonian operators and the “negative” symplectic forms have the
weakly nonlocal form in this case. The same result is also true for “negative” Hamiltonian
operators and “positive” symplectic structures in the case when both pseudo-Riemannian
νµ
νµ
metrics g(0) and g(1) are nondegenerate.
2000 Mathematics Subject Classification: 35Q58, 37K05, 37K10, 37K25.
1. Introduction. We discuss in this paper the Poisson pencils of weakly nonlocal
Poisson brackets of hydrodynamic type (Ferapontov brackets). This means that we
consider the following pair of Hamiltonian operators:
νµ
νµ
Jˆ(0) = g(0) (U )
g0
d
ζ
νµ
η
η
µ
ν
e(0)k w(0)kη
(U)UX D −1 w(0)kζ (U)UX ,
+ b(0)η (U )UX +
dX
k=1
νµ
νµ
Jˆ(1) = g(1) (U )
g1
d
ζ
νµ
η
η
µ
ν
e(1)k w(1)kη
(U)UX D −1 w(1)kζ (U)UX ,
+ b(1)η (U )UX +
dX
k=1
(1.1)
where e(0)k , e(1)k = ±1 and D −1 = (d/dX)−1 are defined in a “skew-symmetric” way
D −1 =
1
2
X
−∞
dX −
+∞
dX
(1.2)
X
and require that the expression
νµ
νµ
νµ
Jˆλ = Jˆ(0) + λJˆ(1)
(1.3)
defines the Poisson bracket satisfying Jacobi identity for any λ.
We mention here that the brackets of this kind are the generalization of DubrovinNovikov local homogeneous brackets of hydrodynamic type [6, 7, 8]:
ν
νµ
η
U (X), U µ (Y ) = g νµ (U )δ (X − Y ) + bη (U)UX δ(X − Y )
(1.4)
588
ANDREI YA. MALTSEV
with the Hamiltonian operator
νµ
JˆDN = g νµ (U )
d
νµ
η
+ bη (U)UX .
dX
(1.5)
Theorem 1.1 (Dubrovin and Novikov). Consider the bracket (1.4) with nondegenerate tensor g νµ (U ). From the Leibnitz identity, it follows that g νµ (U) and
ξµ
µ
µ
Γνη (U ) = −gνξ (U )bη (U ) (gνξ g ξµ = δν ) transforms as a metric with upper indices and
the Christoffel symbols under the pointwise coordinate transformations Ũ ν = Ũ ν (U).
Bracket (1.4) is skew-symmetric if and only if g νµ is symmetric and the connection
µ
Γνη is compatible with the metric ∇η g νµ ≡ 0.
µ
Bracket (1.4) satisfies the Jacobi identity if and only if the connection Γνη is symmetric
νµ
and has zero curvature Rηξ ≡ 0.
It follows from Dubrovin-Novikov theorem that any bracket (1.4) with nondegenerate g νµ can be written locally in the “constant form”
ν
n (X), nµ (Y ) = ν δνµ δ (X − Y ),
ν = ±1
(1.6)
in the flat coordinates nν = nν (U ).
The functionals
Nν =
+∞
−∞
nν (X) dX
(1.7)
are Casimirs of bracket (1.4) and the functional
P=
+∞
−∞
N
1 ν ν
n (X)nν (X) dX
2 ν=1
(1.8)
is a momentum operator generating the flow UTν = UXν . Form (1.6) can be considered
as the canonical form for the DN-bracket (1.4) with the nondegenerate tensor g νµ .
It can be also seen that any functional of “hydrodynamic type”
H=
+∞
−∞
h(U) dX
(1.9)
generates a “hydrodynamic type system”
µ
UTν = Vµν (U)UX
(1.10)
according to bracket (1.4).
We also mention that bracket (1.4) with degenerate tensor g νµ (U) of constant rank
has more complicated but also nice differential geometric structure (see [18]).
The first generalization of DN-bracket to the weakly nonlocal case was the MokhovFerapontov bracket [29]
νµ
η
µ
U ν (X), U µ (Y ) = g νµ (U )δ (X − Y ) + bη (U)UX δ(X − Y ) + cUXν ν(X − Y )UY ,
(1.11)
ON THE COMPATIBLE WEAKLY NONLOCAL POISSON BRACKETS . . .
589
where ν(X − Y ) = 1/2 sgn(X − Y ), corresponding to the Hamiltonian operator
νµ
JˆDN = g νµ (U )
d
νµ
η
µ
+ bη (U)UX + cUXν D −1 UX .
dX
(1.12)
Theorem 1.2 (Mokhov and Ferapontov). Consider bracket (1.11) with nondegenerate tensor g νµ (U ). Then
(1) bracket (1.11) is skew-symmetric and satisfies Leibnitz identity if and only if the
ξµ
µ
tensor g νµ (U ) is a metric with upper indices and Γνη = −gνξ bη are the connection coefficients compatible with g νµ (U);
µ
(2) bracket (1.11) satisfies the Jacobi identity if and only if the connection Γνη is
symmetric and has the constant curvature equal to c, that is,
ντ
= c δνµ δτη − δτµ δνη .
Rµη
(1.13)
Bracket (1.11) has a weakly nonlocal form. However, any local translationally invariant functional
H=
h(U) dX
(1.14)
generates a local system of hydrodynamic type with respect to (1.11). Indeed, we have
µ
UX
∂h
≡ ∂X h
∂U µ
(1.15)
if h does not depend on X explicitly; so, the application of D −1 gives the local expression for the corresponding flow.
The canonical form of bracket (1.11) was first presented by Pavlov in [31] and can
be written as
nν (X), nµ (Y ) = ν δνµ − cnν nµ δ (X − Y ) − cnνX nµ δ(X − Y )
µ
+ cnνX ν(X − Y )nY ,
(1.16)
where nν = nν (U ) are the annihilators for bracket (1.11) (on the space of rapidly
decreasing functions nν (X) at X → ±∞). Also, the implicit expression for the density
of P was represented in [31].
We will see, however, that the Casimirs and the momentum operator for bracket
(1.11) actually depend on the boundary conditions imposed on the functions U ν (X)
for X → ±∞ (see [25]) (the condition U ν → 0, X → ±∞, in general, is not invariant under
the pointwise transformations Ũ ν = Ũ ν (U )). As pointed out in [25], we cannot speak
about the Casimirs and momentum functional until we fix the boundary conditions
at infinity, and, in the general case, it is better to speak about the invariant set of
N + 1 (for MF-bracket) functionals playing the role of either Casimirs of momentum
operator according to the boundary conditions. We consider this later for the case of
more general Ferapontov brackets.
590
ANDREI YA. MALTSEV
The general Ferapontov bracket [11, 12, 13, 16] has the form
ν
νµ
η
U (X), U µ (Y ) = g νµ (U )δ (X − Y ) + bη (U)UX δ(X − Y )
+
g
η
µ
ζ
ν
ek wkη
(U)UX ν(X − Y )wkζ (U)UY ,
(1.17)
k=1
ek = ±1, which corresponds to the weakly nonlocal Hamiltonian operator
νµ
JˆF = g νµ (U )
g
d
ζ
νµ
η
η
µ
ν
+ bη (U )UX +
ek wkη
(U)UX D −1 wkζ (U)UX .
dX
k=1
(1.18)
Theorem 1.3 (Ferapontov [11, 16]). Consider bracket (1.17) with nondegenerate
tensor g νµ (U ). Then,
(1) bracket (1.17) is skew-symmetric and satisfies Leibnitz identity if and only if tenξµ
µ
sor g νµ (U ) is a metric with upper indices and Γνη = −gνξ bη are the connection
νµ
coefficients compatible with g (U );
µ
(2) bracket (1.11) satisfies the Jacobi identity if and only if the connection Γνη is
ν
symmetric and the metric g νµ and tensors wkη (U) satisfy the equations
µ
ν
g ντ wkτ = g µτ wkτ
,
ντ
=
Rµη
g
µ
µ
∇ν wkη = ∇η wkν ,
ν τ
ν
τ
ek wkµ
wkη − wkµ
wkη
.
(1.19)
k=1
Moreover, this set is commutative [wk , wk ] = 0.
It was pointed out by Ferapontov that the equations written above are Gauss and
Petersson-Codazzi equations for the submanifold ᏹN with flat normal connection in
the pseudo-Euclidean space E N+g . In this consideration, the tensor g νµ is the first
ν
quadratic form of ᏹN , and wkη
are the Weingarten operators corresponding to g
parallel vector fields in the normal bundle Nk , such that Nk , Nm = ek δkm . It was also
proved by Ferapontov that these brackets can be constructed as a Dirac restriction of
the local DN-bracket
I
Z (X), Z J (Y ) = I δIJ δ (X − Y ),
I, J = 1, . . . , N + g, I = ±1
(1.20)
in E N+g to the submanifold ᏹN [12, 16].
As far as we know, the cases of brackets (1.11), (1.17) with the degenerate tensors
g νµ (U ) were not studied in the literature.
All brackets (1.4), (1.11), and (1.17) are closely connected with the diagonalizable
integrable systems (1.10).
The general procedure of integration of the so-called “semi-Hamiltonian” diagonal
systems of hydrodynamic type was constructed by Tsarëv [34, 35]. It can be shown that
any diagonal system (1.10) which is Hamiltonian with respect to bracket (1.4), (1.11),
or (1.17) (with diagonal g νµ (U )) satisfies Tsarëv “semi-Hamiltonian” property, and
so, it can be integrated by Tsarëv’s method. Probably, all semi-Hamiltonian systems
are in fact Hamiltonian corresponding to some weakly nonlocal H.T.P.B. with (maybe)
an infinite number of terms in the nonlocal tail. Some investigation of this problem
ON THE COMPATIBLE WEAKLY NONLOCAL POISSON BRACKETS . . .
591
can be found in [3, 16], but, in general, this problem is still open. We also mention
that the examples of nondiagonalizable Hamiltonian integrable (by inverse scattering
methods) systems (1.10) were also investigated in [14, 15].
As was pointed out in [13, 16], if the manifold ᏹN has a holonomic net of lines of
ν
can be written in the diagocurvature, the metric g νµ (U ) and all the operators wkη
ν
ν
nal form in the corresponding coordinates r = r (U). Here, we do not impose this
requirement and consider any brackets of Ferapontov type.
η
ν
(U )UX in the nonlocal part of (1.17) are linearly
We will assume that the flows wkη
independent (with constant coefficients). (The nonlocal part in (1.17) actually repreη
ν
(U)UX ,
sents the nondegenerate quadratic form on the linear space generated by wkη
k = 1, . . . , g written in the canonical form with ek = ±1.) As pointed out by Ferapontov,
the local functional
H = h(U) dX
(1.21)
generates in this case the local flow with respect to bracket (1.17) if and only if the
η
ν
(U)UX such that the exfunctional H is a conservation law for any of the flows wkη
pressions
η
ν
wkη
(U )UX
∂h
∂U ν
(1.22)
represent the total derivatives with respect to X of some functions Qk (U) for any k.
This fact is also true for more general weakly nonlocal Poisson brackets having the
form
G
ij Bk ϕ, ϕx , . . . δ(k) (x − y)
ϕi (x), ϕj (y) =
k=1
+
g
ek Ski
j
ϕ, ϕx , . . . ν(x − y)Sk ϕ, ϕy , . . . ,
(1.23)
k=1
where δ(k) (x − y) = (d/dx)k δ(x − y), ek = ±1, and the set {Ski (ϕ, ϕx , . . .)} is linearly
independent.
As far as we know, the first example written precisely in this form was the Sokolov
bracket [33]
ϕ(x), ϕ(y) = −ϕx ν(x − y)ϕy
(1.24)
for the Krichever-Novikov equation
ϕt = ϕxxx −
2
3 ϕxx
h(ϕ)
+
,
2 ϕx
ϕx
(1.25)
592
ANDREI YA. MALTSEV
where h(ϕ) = c3 ϕ3 + c2 ϕ2 + c1 ϕ + c0 , with the Hamiltonian function
H=
2
1 h(ϕ)
1 ϕxx
+
dx.
2 ϕx2
3 ϕx2
(1.26)
As established in [23, 24], the flows Ski (ϕ, ϕx , . . .) commute with each other for any
general bracket (1.23) and conserve the corresponding Hamiltonian structure (1.23)
on the phase space {ϕi (x)} (this fact was important for the averaging procedure for
such brackets considered there). However, for general brackets (1.23), they are not
necessarily generated by the local Hamiltonian functions having the form
(1.27)
H = h ϕ, ϕx , . . . dx.
Actually, brackets (1.23) are very common for so-called “integrable systems” (like
KdV or NLS) possessing the multi-Hamiltonian structures connected by the recursion
operator according to the Lenard-Magri scheme [20]. As such, it was proved in [10]
that all the higher PB brackets for KdV given by the recursion scheme starting from
Gardner-Zakharov-Faddev bracket
ϕ(x), ϕ(y) = δ (x − y)
(1.28)
ϕ(x), ϕ(y) = −δ (x − y) + 4ϕ(x)δ (x − y) + 2ϕx δ(x − y)
(1.29)
and the Magri bracket
have exactly form (1.23). In [25], the same fact was proved for the case of NLS hierarchy,
and also the weakly nonlocal form of the “negative” symplectic forms for KdV and NLS
was established.
Brackets (1.4), (1.11), and (1.17) appear in these systems as the “dispersionless”
limit of the corresponding bracket (1.23) or, in more general case, as a result of the
averaging of (1.23) on the families of quasiperiodic solutions of corresponding evolution system connected with the Whitham method for slow modulations of parameters
[1, 2, 6, 7, 8, 21, 22, 23, 24, 26, 30, 32].
We consider here the compatible brackets of Ferapontov type and prove the simνµ
ilar facts for the case of the nondegenerate pencils (i.e., det g(0) ≠ 0) with also some
nondegeneracy conditions for nonlocal part of Jˆ(0) + λJˆ(1) .
We also mention that the wide classes of local pencils of hydrodynamic type (DNbrackets) were investigated in detail in [4] (see also [5, 9] and the references therein),
where they play an important role in the structure of Dubrovin-Frobenius manifolds
connected with solutions of WDVV equation for topological field theories. In [16, 27],
some important questions of weakly nonlocal pencils of hydrodynamic type (H.T.)
were also considered. In [17, 28], also the generic diagonal compatible flat pencils in
terms of inverse scattering method (see [19, 36]) were discussed.
2. On the canonical form and symplectic operator for the general F -bracket. We
now formulate the properties of brackets (1.17) established in [25] which we will need
in further consideration.
ON THE COMPATIBLE WEAKLY NONLOCAL POISSON BRACKETS . . .
593
Consider bracket (1.17) with nondegenerate tensor g νµ (U). According to Ferapontov results, we can represent it as a Dirac restriction of DN-bracket in RN+g to the
submanifold ᏹN ⊂ RN+g with flat normal connection. We fix the point z ∈ ᏹN and
introduce the corresponding loop space in the vicinity of z
L ᏹN , z = γ : R1 → ᏹN : γ(−∞) = γ(+∞) = z ∈ ᏹN .
(2.1)
So, we fix the boundary conditions for the functions U ν (X) and require that these
functions rapidly approach their boundary values at X → ±∞.
Theorem 2.1 (Maltsev and Novikov [25]). Consider the bracket (1.17) with nondegenerate g νµ (U ) defined on the loop space L(ᏹN , z). Consider the corresponding
embedding ᏹN ⊂ RN+g with flat normal connection. Consider the flat orthogonal coordinates Z I in RN+g such that
(1) Z I (z) = 0, I = 1, . . . , N + g, and the corresponding DN-bracket in RN+g has the
form
I
Z (X), Z J (Y ) = E I δIJ δ (X − Y ),
I, J = 1, . . . , N + g, E I = ±1;
(2.2)
(2) the first N coordinates Z 1 , . . . , Z N are tangential to ᏹN at the point z;
(3) the last g coordinates Z N+1 , . . . , Z N+g are orthogonal to ᏹN at the point z.
Then,
(1) bracket (1.17) has exactly N local Casimirs given by the functionals
Nν =
+∞
−∞
nν (X)dX,
(2.3)
where nν (U ) are the restrictions of the first N (tangential to ᏹN at z) coordinates
Z 1 , . . . , Z N on ᏹN ,
(2) all the flows
η
ν
(U)UX
Utνk = wkη
(2.4)
are Hamiltonian with respect to (1.17) with local Hamiltonian functionals
Hk =
+∞
−∞
hk (X) dX,
k = 1, . . . , g,
(2.5)
where hk (U ) are the restriction of the last g coordinates Z N+k , k = 1, . . . , g on
ν
ᏹN and wkη
(U ) are the Weingarten operators corresponding to parallel vector
fields Nk (U ) in the normal bundle with the normalization
T
Nk (z) = 0, . . . , 0, −E N+k , 0, . . . , 0 ,
(2.6)
where −E N+k stays at the position N + k, in the coordinates Z 1 , . . . , Z N+g , so
Nk (U ), Nl (U ) = E N+k δkl (there is an arithmetic mistake in the journal variη
ν
ant of [25] where the generated flows are written as E N+k wkη
(U)UX ),
594
ANDREI YA. MALTSEV
(3) bracket (1.17) has in coordinates nν (U) the “canonical form” corresponding to
the space L(ᏹN , z), that is,
g
ν
µ
ν
µ
ν νµ
ek fk (n)fk (n) δ (X − Y )
n (X), n (Y ) = δ −
k=1
g
−
µ
ek fkν (n) X fk (n)δ(X − Y )
(2.7)
k=1
+
g
µ
ek fkν (n) X ν(X − Y ) fk (n) Y ,
k=1
where ν = E ν , ν = 1, . . . , N, ek = E N+k , k = 1, . . . , g, nν (z) = 0, fkν (z) = 0 (i.e.,
fkν (0) = 0). (Actually, we have the equality fkν (U) ≡ Nkν (U), where Nkν (U) are
the first N components of the vectors Nk (U) in the coordinates Z 1 , . . . , Z N+g .)
We note here that the Casimirs and “canonical functionals,” as well as the canonical
forms, depend on the phase space L(ᏹN , z). So, if we do not fix the loop space, it is
better to speak just about the N +g canonical functions (the restrictions of the flat coordinates from RN+g ) playing the role of Casimirs or canonical functionals depending
on the boundary conditions. (Also, we will have many “canonical forms” of bracket
(1.17) with different fkν (U ) in this case.)
For the case of Mokhov-Ferapontov bracket, the canonical form will be (1.16) for
any fixed space L(ᏹN , z) (although the coordinates nν (U) will be different for different loop spaces). The explicit form of canonical functional, which is the momentum
operator in this case, can then be written as (see [25])


N
1 +∞ 

(2.8)
ν nν nν dX.
P=
1 − 1 − c
c −∞
ν=1
Using the restriction of the symplectic form of DN-bracket on ᏹN , it is also possible to get the symplectic form Ωνµ (X, Y ) for the F -bracket (1.17) with nondegenerate
g νµ (U ) [25]. The symplectic form appears to be also weakly nonlocal and can be written as
N+g
Ωνµ (X, Y ) =
EI
I=1
N
∂Z I (U)
∂Z I (U )
(X)ν(X − Y )
(Y )
ν
∂U
∂U µ
g
∂nτ
∂hk
∂nτ
∂hk
=
(X)ν(X
−
Y
)
(Y
)
+
ek
(X)ν(X − Y )
(Y )
ν
µ
ν
µ
∂U
∂U
∂U
∂U
τ=1
k=1
(2.9)
τ
on ᏹN .
We can also write the corresponding symplectic operator Ω̂νµ on L(ᏹN , z) as
Ω̂νµ =
N
τ=1
∂nτ −1 ∂nτ ∂hk −1 ∂hk
D
+
ek
D
∂U ν
∂U µ k=1 ∂U ν
∂U µ
g
τ
with D −1 defined in the skew-symmetric way.
(2.10)
ON THE COMPATIBLE WEAKLY NONLOCAL POISSON BRACKETS . . .
595
We introduce the space ᐂ0 (z) of vector fields ξ ν (X) rapidly decreasing at X → ±∞
on L(ᏹN , z). It is easy to see that all the hydrodynamic type systems (1.10) satisfy
this requirement as the vector fields on L(ᏹN , z) and so belong to ᐂ0 (z). We can then
naturally define the action of symplectic operator Ω̂νµ on the space ᐂ0 (z).
We will also need the space Ᏺ0 (z) of 1-forms ων (X) rapidly decreasing at X → ±∞
on L(ᏹN , z). We note here that Ω̂νµ ξ µ (X) ∉ Ᏺ0 (z) in the general case.
We will now prove by the direct calculation that the symplectic form Ω̂νµ is the
νµ
inverse of the Hamiltonian operator JˆF on the appropriate functional spaces.
Theorem 2.2. (I) On the functional space ᐂ0 (z), the relation
νξ
JˆF Ω̂ξµ = Î
(2.11)
is true, where Î is the identity operator on the space ᐂ0 (z).
(II) On the functional space Ᏺ0 (z), the relation
ξµ
Ω̂νξ JˆF = Î
(2.12)
is true, where Î is the identity operator on Ᏺ0 (z).
Proof. (I) We have

νξ
JˆF Ω̂ξµ

g
d
νξ
ζ
η
η −1 ξ
ν
νξ

+ bη (U )UX +
= g (U )
ek wkη (U)UX D wkζ (U)UX
dX
k=1


g
N
τ
τ
l
l
∂n
∂n
∂h
∂h
τ
−1
−1
,
D
+ el
D
×
∂U ξ
∂U µ l=1 ∂U ξ
∂U µ
τ=1
(2.13)
ν
(U ) correspond to the vector fields N(k) (U) such that
where the operators wkη
T
N(k) (z) = 0, . . . , 0, −ek , 0, . . . , 0
(2.14)
(−ek stays at the position N + k), and we have identically N(k) (U), N(l) (U) = ek δkl .
νµ
Since the integrals of nτ (U ), and hl (U ) generate the local flows according to JˆF ,
we have
ξ
ζ
wkζ (U )UX
∂nτ
≡ qkτ (U ) X ,
∂U ξ
ξ
ζ
wkζ (U)UX
∂hl
≡ pkl (U) X
∂U ξ
(2.15)
for some functions qkτ (U ) and pkl (U ). Moreover, consider N tangential vectors to ᏹN
in RN+g corresponding to the coordinates U ν
e(ξ) (U ) =
∂n1
∂nN ∂h1
∂hg
,...,
,
,...,
ξ
ξ
ξ
∂U
∂U ∂U
∂U ξ
T
(2.16)
and our parallel orthogonal vector fields N(k) (U) defined by condition (2.14) in the
coordinates (Z 1 , . . . , Z N+g ).
596
ANDREI YA. MALTSEV
Since by the following definition:
d
ξ
ζ
N(k) (U ) = wkζ UX e(ξ) (U),
dX
(2.17)
we have from (2.15)
qkτ
X
τ = N(k)
X,
l
N+l pk X = N(k)
X,
τ = 1, . . . , N, l = 1, . . . g
(2.18)
for the components of N(k) (U ).
We normalize the functions qkτ (U ) and pkl (U) such that
pkl (z) = 0.
qkτ (z) = 0,
(2.19)
We then have
τ
qkτ (U ) = N(k)
(U ),
N+l
pkl (U) = N(k)
(U) + ek δlk .
(2.20)
Now, using the equalities
τ
d
d τ
qk X =
q − qkτ
,
dX k
dX
l
d
d l
pk X =
p − pkl
dX k
dX
(2.21)
on L(ᏹN , z), we can write
νξ
JˆF Ω̂ξµ |ᐂ0 (z) =
g
τ
∂nτ η −1 d
ν
τ
−1 ∂n
g
+
e
w
U
D
q
D
k
X
kη
k
∂U ξ k=1
dX
∂U µ
X
τ=1
g
g
l
d l
∂hl
∂hl
νξ η ∂h
η
ν
+ el g νξ
+ bη UX
+
ek wkη
UX D −1
pk D −1
ξ
ξ
∂U X
∂U
dX
∂U µ
l=1
k=1


g
N
∂nτ d −1 ∂nτ ∂hl d −1 ∂hl 
D
D
+ g νξ 
τ
+ el
ξ
µ
∂U dX
∂U
∂U ξ dX
∂U µ
τ=1
l=1
N
−
τ
νξ
g
N
∂nτ
∂U ξ
νξ η
+ bη UX
η
ν
ek τ wkη
UX D −1 qkτ
k=1 τ=1
−
g
g
k=1 l=1
η
ν
ek el wkη
UX D −1 pkl
d −1 ∂nτ
D
dX
∂U µ
d −1 ∂hl
D
.
dX
∂U µ
(2.22)
We can here replace (d/dX)D −1 by identity, and we have
−1 1
D fX = f (X) − f (−∞) + f (+∞)
2
(2.23)
for any function f (U ) on L(ᏹN , z) according to the definition of D −1 . So, according to
normalization (2.19), we can replace the operators D −1 (d/dX)qkτ and D −1 (d/dX)pkl
just by qkτ and pkl on L(ᏹN , z).
ON THE COMPATIBLE WEAKLY NONLOCAL POISSON BRACKETS . . .
597
According to the same normalization, the expressions within the brackets in the
first two terms are equal to
l νξ ∂h
η
ν
JˆF
= wlη
UX .
ξ
∂U L(ᏹN ,z)
τ
νξ ∂n
= 0,
JˆF
∂U ξ L(ᏹN ,z)
(2.24)
So, we have
νξ
JˆF Ω̂ξµ |ᐂ0 (z)


g
N
τ
τ
l
l
∂hl
∂n
∂n
∂h
∂h

=
+ g νξ 
τ
+ el
∂U µ
∂U ξ ∂U µ l=1 ∂U ξ ∂U µ
τ=1
l=1


g
g
N
τ
l
η −1 
l ∂h 
ν
τ τ ∂n
ek wkη UX D
qk
+ e l pk
−
.
∂U µ l=1
∂U µ
τ=1
k=1
g
η
ν
el wlη
Ux D −1
(2.25)
Using (2.20) and (2.16), we now get
N
l
∂hk
∂hk
∂nτ l ∂h
+
+
e
p
=
N
,
e
=
.
l
(k)
(µ)
k
∂U µ l=1
∂U µ
∂U µ
∂U µ
g
τ qkτ
τ=1
(2.26)
Also, using the evident relation
N
∂nτ ∂nτ ∂hl ∂hl
+ el
≡ gξµ (U),
∂U ξ ∂U µ l=1 ∂U ξ ∂U µ
g
τ
τ=1
(2.27)
we get the statement (I) of the theorem.
(II) We have

g
τ
τ
l
l
∂n
∂n
∂h
∂h
−1
−1

=
D
+ el
D
∂U ν
∂U ξ l=1 ∂U ν
∂U ξ
τ=1

ξµ
Ω̂νξ JˆF
N
τ
g
d
ξµ η
ξ
ζ
η
µ
ek wkη UX D −1 wkζ UX
+ bη UX +
× g ξµ
dX
k=1
=
N
τ
τ=1
−
N
g
∂nτ −1 d ∂nτ ξµ ∂hl −1 d ∂hl ξµ
D
g
+
D
g
∂U ν
dX ∂U ξ
∂U ν
dX ∂U ξ
l=1
τ
g
∂nτ −1 ∂nτ ξµ
∂hl −1 ∂hl ξµ
D
g
−
D
g
∂U ν
∂U ξ
∂U ν
∂U ξ
X
X
l=1
τ
g
∂nτ −1 ∂nτ ξµ η ∂hl −1 ∂hl ξµ η
D
b
U
+
el
D
bη UX
η
X
∂U ν
∂U ξ
∂U ν
∂U ξ
l=1
τ=1
+
N
τ=1
+
g
N τ ek
τ=1 k=1
+
g
g
l=1 k=1
el ek
(2.28)
∂nτ −1 d τ
ζ
µ
τ d
D
−
q
q
D −1 wkζ UX
k
k
∂U ν
dX
dX
∂hl −1 d l
ζ
µ
l d
p
D
−
p
D −1 wkζ UX .
k
k
∂U ν
dX
dX
Again, we can replace the operators (d/dX)D −1 by identity and D −1 (d/dX)qkτ and
D −1 (d/dX)pkl by qkτ and pkl . Then, according to the definition of coordinates hl , we
598
ANDREI YA. MALTSEV
have (∂hl /∂U ξ )(z) = 0; so, we also put D −1 (d/dX)(∂hl /∂U ξ ) = (∂hl /∂U ξ ). Now, using
the same arguments, We get
ξµ
Ω̂νξ JˆF = Î +
N
τ
τ=1
∂nτ ξµ
∂nτ
−1 d
−
Î
g
D
ν
∂U
dX
∂U ξ
g
∂nτ −1 µξ ∂nτ
∂hl −1 µξ ∂hl
ˆF
ˆ
D
−
e
D
J
J
l
F
∂U ν
∂U ξ L(ᏹN ,z) l=1 ∂U ν
∂U ξ L(ᏹN ,z)
τ=1


g
g
N
τ
l
∂n
∂h
ζ
ζ
µ
µ
l
τ
τ
−1
−1

+
q D wkζ UX + el
p D wkζ UX 
∂U ν k
∂U ν k
k=1 τ=1
l=1
−
N
τ
(2.29)
µξ
(we used the skew-symmetry of the operator JˆF for the action from the right).
Again, using (2.24) and (2.26), we get
ξµ
Ω̂νξ JˆF = Î +
N
τ
τ=1
∂nτ
∂nτ ξµ
−1 d
D
g ,
−
Î
∂U ν
dX
∂U ξ
(2.30)
that is,
ξµ
Ω̂νξ JˆF fµ (X) = fµ (X) −
N
τ
τ=1
∂nτ
∂nτ
(X)
(z)g ξµ (z)fµ (z)
∂U ν
∂U ξ
(2.31)
for any fµ (X). From the definition of Ᏺ0 (z), we now obtain the part (II) of the theorem.
νξ
Remark 2.3. We can also say that the operator JˆF Ω̂ξµ is identity if it is acting from
the left on Ᏺ0 (z) and from the right on ᐂ0 (z). This interpretation will be convenient
later for the consideration of the recursion operator.
We also introduce the momentum functional P generating the flow
UTν = UXν
(2.32)
with respect to the general bracket (1.17) (with nondegenerate g νµ (U)).
Lemma 2.4. Any F -bracket (1.17) with nondegenerate tensor g νµ (U) has the local
momentum operator P generating the flow (2.32) on the space L(ᏹN , z). The functional
P can be written in the form
P=
+∞
−∞
1
p(U ) dX =
2
+∞
−∞


N
τ
τ
τ
n n +
τ=1
g

k
k
ek h h
dX,
(2.33)
k=1
where the functions nτ and hk correspond to the loop space L(ᏹN , z).
Proof. We should here just prove the relation
∂p
ξ
= Ω̂νξ UX
∂U ν
(2.34)
ON THE COMPATIBLE WEAKLY NONLOCAL POISSON BRACKETS . . .
599
on L(ᏹN , z) according to part (I) of Theorem 2.2. So, we have
N
τ=1
τ
g
∂nτ −1 ∂nτ ξ ∂hk −1 ∂hk ξ
D
U
+
e
D
U
k
X
∂U ν
∂U ξ
∂U ν
∂U ξ X
k=1
N
(2.35)
g
∂nτ τ ∂hk k
∂
=
n
+
ek
h =
p.
ν
ν
ν
∂U
∂U
∂U
τ=1
k=1
τ
3. The λ-pencils and the integrable hierarchies. We now consider the operator
! d
!
νµ
νµ
η
+ b(0)η + λb(1)η UX
dX
g0
g1
ζ
ζ
η
µ
η
µ
ν
ν
+
e(0)k w(0)kη
UX D −1 w(0)kζ UX + λ
e(1)k w(1)kη
UX D −1 w(1)kζ UX
νµ
νµ
νµ
νµ
νµ
Jˆλ = Jˆ(0) + λJˆ(1) = g(0) + λg(1)
k=1
(3.1)
k=1
νµ
νµ
for Jˆ(0) and Jˆ(1) having form (1.18). We will call the λ-pencil (3.1) nondegenerate, for
νµ
small λ, if det g(0) (U ) ≠ 0.
We will admit here that the linear spaces ᐃ(0) and ᐃ(1) , generated by the sets
ν
η
η
ν
U ,
w(0)1η UX , . . . , w(0)g
0η X
ν
η
η
ν
U ,
w(1)1η UX , . . . , w(1)g
1η X
(3.2)
can have a nontrivial intersection ᐂ.
We introduce the basis in ᐂ
η
η
ν
ν
UX , . . . , v̂dη
UX ,
v̂1η
(3.3)
where d = dim ᐂ, and consider the linear space ᐃ, generated by all the flows from
ᐃ(0) and ᐃ(1) (dim ᐃ = g0 + g1 − d) with basis
ν
η
η
η
η
η
η
ν
ν
ν
ν
ν
Ꮾ = ŵ(0)1η
UX , . . . , ŵ(0)(g
U , v̂1η
UX , . . . , v̂dη
UX , ŵ(1)1η
UX , . . . , ŵ(1)(g
U
0 −d)η X
1 −d)η X
(3.4)
ν
η
η
(U )UX , . . . , w̃gν0 +g1 −d,η (U )UX ,
= w̃1η
ν
ν
where ŵ(0)kη
and ŵ(1)sη
are some linear combinations of operators w(0) and w(1) ,
respectively, and the flows corresponding to ŵ(0)k , v̂m , and ŵ(1)s are all linearly independent.
The flows
ν
η
η
η
η
ν
ν
ν
U , v̂1η
UX , . . . , v̂dη
UX ,
ŵ(0)1η UX , . . . , ŵ(0)(g
0 −d)η X
(3.5)
ν η
η
η
η
ν
ν
ν
UX , ŵ(1)1η
UX , . . . , ŵ(1)(g
U
v̂1η UX , . . . , v̂dη
1 −d)η X
will then give bases in ᐃ(0) and ᐃ(1) , respectively.
νµ
The nonlocal part of the bracket Jˆλ
g0
k=1
η
µ
ζ
ν
e(0)k w(0)kη
UX D −1 w(0)kζ UX + λ
g1
η
µ
ζ
ν
e(1)s w(1)sη
UX D −1 w(1)sζ UX
(3.6)
s=1
will correspond in our case to some quadratic form Qλks (linear in λ), k, s = 1, . . . , g0 +
g1 − d on the space ᐃ.
600
ANDREI YA. MALTSEV
In our further consideration, the question if Qλks is nondegenerate on ᐃ for λ ≠ 0
or not will be important, and we will mainly consider the pencils (3.1) such that Qλks
is nondegenerate on ᐃ.
We now formulate the properties of nondegenerate pencils (3.1) also satisfying the
requirement of nondegeneracy of the form Qλks for λ ≠ 0 connected with the “canonical” integrable hierarchies.
νµ
Theorem 3.1. Consider the nondegenerate pencil (3.1) (det g(0) ≠ 0) such that the
Qλks
is also nondegenerate on ᐃ for λ ≠ 0 (small enough). Then,
form
(I) it is possible to introduce the local functionals
+∞
nν (U , λ) dX, ν = 1, . . . , N,
N ν (λ) =
−∞
P (λ) =
k
H(0)
(λ)
=
s
(λ) =
H(1)
+∞
−∞
+∞
−∞
+∞
−∞
p(U , λ) dX,
(3.7)
hk(0) (U , λ) dX,
k = 1, . . . , g0 ,
hs(1) (U , λ) dX,
s = 1, . . . , g1 ,
which are the Casimirs, momentum operator and the Hamiltonian functions for
η
η
νµ
ν
ν
(U )UX and w(1)sη
(U )UX for the bracket Jˆλ , respectively;
the flows w(0)kη
(II) all the functions nν (U , λ), p(U , λ), hk(0) (U, λ), and hs(1) (U, λ) are regular at λ →
0 and can be represented as regular series
nν (U , λ) =
+∞
nνq (U )λq ,
p(U, λ) =
q=0
hk(0) (U , λ)
=
+∞
+∞
pq (U)λq ,
q=0
hk(0),q (U )λq ,
hs(1) (U, λ)
q=0
=
+∞
(3.8)
hs(1),q (U)λq .
q=0
Moreover, we can choose these functionals in such a way that
∂pq
(z) = 0,
∂U µ
∂nνq
∂U µ
∂hk(0),q
∂U µ
(z) = 0,
(z) = 0,
q ≥ 1,
∂hs(1),q
(z) = 0,
∂U µ
∂nν0
(z) = eµν ,
∂U µ
q ≥ 0,
(3.9)
where det eµν ≠ 0;
k
s
(0), and H(1)
(0) are the Casimirs, momentum
(III) the integrals N ν (0), P (0), H(0)
η
ν
operator, and the Hamiltonian functions for the flows w(0)kη
(U)UX and
η
νµ
ν
w(1)sη (U )UX with respect to Jˆ(0) , while the flows generated by the functionals
Fq =
+∞
−∞
fq (U) dX
(3.10)
are connected by the relation
νξ
Jˆ(0)
∂f(q+1)
νξ ∂fq
= −Jˆ(1)
∂U ξ
∂U ξ
(3.11)
ON THE COMPATIBLE WEAKLY NONLOCAL POISSON BRACKETS . . .
601
for any functional F (λ) from the set (3.7). All the functionals Fq given by the
expansions (3.8) generate the local flows and commute with each other with
respect to both brackets Jˆ(0) and Jˆ(1) .
νµ
Proof. We now put λ > 0 and write Jˆλ in the form
νµ
νµ
νµ
Jˆλ = g(0) + λg(1)
+
g0
! d
!
νµ
νµ
η
+ b(0)η + λb(1)η UX
dX
η
µ
ζ
ν
e(0)k w(0)kη
UX D −1 w(0)kζ UX
(3.12)
k=1
+
"
"
ζ
η
µ
ν
e(1)k λw(1)kη
UX D −1 λw(1)kζ UX .
g1
k=1
In the case of the nondegenerate form Qλks on ᐃ, we can consider the corresponding
embedding of ᏹN to RN+g0 +g1 depending on λ. Indeed, all the flows w̃s from the set
Ꮾ will in this case satisfy conditions
νξ
µξ
µ
µ
µ
ν
,
∇ν w̃sη = ∇η w̃sν ,
gλ w̃sξ = gλ w̃sξ
νµ
µ
µ
ν
ν
Qλks w̃sζ − w̃kη Qλks w̃sζ
Rηζ =
w̃kη
(3.13)
k,s
µξ
for nondegenerate gλ according to Ferapontov theorem.
ν
ν
ν
Since the operators w(0)kη
(U ) and w(1)sη
(U) are just linear combinations of w̃nη
(U)
(with constant coefficients) and the form Qλks coincides with the nonlocal part of (3.12),
we will have the corresponding Gauss and Petersson-Codazzi equations for these flows
νµ
νµ
and the curvature tensor Rηζ . So, for nondegenerate gλ , we will get the local embedding of ᏹN to RN+g0 +g1 (actually, to some subspace RN+g0 +g1 −d ⊂ RN+g0 +g1 ) depending
on λ.
Since the embedding is defined up to the Poincaré transformation in RN+g0 +g1 , we
can choose, at all λ, the coordinates Z I , I = 1, . . . , N + g0 + g1 in such a way that
(1) all Z I = 0 at the point z ∈ ᏹN , and the metric GIJ in RN+g0 +g1 has the form
GIJ = E I δIJ ,
(3.14)
where E N+k = e(0)k , k = 1, . . . , g0 , E N+g0 +s = e(1)s , s = 1, . . . , g1 ;
(2) the first N coordinates Z ν , ν = 1, . . . , N are tangential to ᏹN at the point z ∈ ᏹN
at all λ;
(3) the last g0 + g1 coordinates are orthogonal to ᏹN at the point z, and the Weingarten operators
"
"
ν
ν
ν
ν
(U ), . . . , w(0)g
(U ), λw(1)1η
(U), . . . , λw(1)g
(U)
w(0)1η
0η
1η
(3.15)
correspond to the parallel vector fields N(0)(k) (U), N(1)(s) (U) in the normal bundle such that
T
N(0)(k) (z) = 0, . . . , 0, −E N+k , 0, . . . , 0
(3.16)
602
ANDREI YA. MALTSEV
(−E N+k stays at the position N + k) k = 1, . . . , g0 ,
T
N(1)(k) (z) = 0, . . . , 0, −E N+g0 +s , 0, . . . , 0
(3.17)
(−E N+g0 +s stays at the position N + g0 + s) s = 1, . . . , g1 .
So, according to [25], the restriction of the first N coordinates Z 1 , . . . , Z N gives the
Casimirs of bracket (3.12) on L(ᏹN , z)
Ñ ν (λ) =
+∞
−∞
ñν (U , λ) dX =
+∞
−∞
Z ν |ᏹN (U, λ) dX,
ν = 1, . . . , N,
(3.18)
while the restrictions of the last g0 + g1 coordinates give the Hamiltonian functions
η
ν
UX
for the flows w(0)kη
k
H̃(0)
(λ) =
and
+∞
−∞
h̃k(0) (U , λ) dX =
+∞
−∞
Z N+k |ᏹN (U, λ) dX,
k = 1, . . . , g0
(3.19)
√
η
ν
λw(1)sη
UX
s
H̃(1)
(λ) =
+∞
−∞
+∞
h̃s(1) (U , λ) dX =
−∞
Z N+g0 +s |ᏹN (U, λ) dX,
s = 1, . . . , g1 .
(3.20)
Remark 3.2. For λ < 0, the signature of RN+g0 +g1 may be different from the case
λ > 0, but all the statements of the theorem will certainly be also true.
We now study the λ-dependence of the functions ñν (U, λ), h̃k(0) (U, λ), and h̃s(1) (U, λ)
at λ → 0.
Consider N tangential vectors to ᏹN , corresponding to the coordinate system {U ν },
that is,
e(ν) =
∂Z 1
∂Z N+g0 +g1
,...,
ν
∂U
∂U ν
T
,
ν = 1, . . . , N.
(3.21)
We have the following relations (in RN+g0 +g1 ) for the differentials of e(ν) (U, λ),
N(0)(k) (U , λ), and N(1)(s) (U , λ) on ᏹN :
µ
de(ν) = Γνη (U , λ)e(µ) dU η −
g0
µ
E N+k gνµ (U, λ)w(0)kη (U, λ)N(0)(k) dU η
k=1
g1
" µ
− λ
E N+g0 +s gνµ (U , λ)w(1)sη (U, λ)N(1)(s) dU η ,
s=1
ν
(U , λ)e(ν) dU η ,
dN(0)(k) = w(0)kη
"
ν
dN(1)(s) = λw(1)sη
(U , λ)e(ν) dU η ,
µ
ξµ
µ
where Γνη = −gνξ bη , gνξ g ξµ ≡ δν .
(3.22)
603
ON THE COMPATIBLE WEAKLY NONLOCAL POISSON BRACKETS . . .
So, for any curve γ(t) on ᏹN (γ(0) = z), we have the evolution system for e(ν) (t),
N(0)(k) (t), and N(1)(s) (t) having the general form
 
∗(t, λ)
e(ν) (t)
 
d 
N(0)(k) (t) =  ∗(t, λ)
 √
dt 
λ ∗ (t, λ)
N(1)(s) (t)



√
e(ν) (t)
λ ∗ (t, λ)


 N(0)(k) (t) ,
0


0
N(1)(s) (t)
∗(t, λ)
0
0
(3.23)
where all ∗(t, λ) are regular at λ → 0 (matrix-) functions of λ.
The formal solution of (3.23) can be written as the chronological exponent
t
T exp
Ĥ(t)dt
0
= Î +
+∞
1
n!
n=1
t1
0
···
tn
0
T Ĥ t1 · · · Ĥ tn dt1 · · · dtn
(3.24)
applied to the initial data e(ν) (0), N(0)(k) (0), and N(1)(s) (0), where Ĥ(t) is the matrix
of system (3.23).
It is now easy to verify that, for any n ≥ 1, we have

∗(t, λ)

T Ĥ t1 · · · Ĥ tn = 
√ ∗(t, λ)
λ ∗ (t, λ)
∗(t, λ)
∗(t, λ)
√
λ ∗ (t, λ)

√
λ ∗ (t, λ)
√

λ ∗ (t, λ)
,
λ ∗ (t, λ)
(3.25)
where all ∗(t, λ) are regular matrix-functions at λ → 0.
For the densities of Casimirs (3.18) and Hamiltonian functions (3.19) and (3.20), we
can now write the equations
ν
µ
dñν (t, λ) µ
= Ut e(µ) (t, λ) = ν Ut e(µ) (t, λ), eν (0)
dt
dh̃k(0) (t, λ)
dt
dh̃s(1) (t, λ)
dt
µ
= − Ut e(µ) (t, λ), N(0)(k) (0)
(3.26)
µ
= − Ut e(µ) (t, λ), N(1)(s) (0)
along the same curve γ(t) on ᏹN . Since γ(t) is just the arbitrary curve, we have
ñν (U , λ) = ∗(U , λ),
h̃k(0) (U , λ) = ∗(U , λ),
"
h̃s(1) (U, λ) = λ ∗ (U, λ),
(3.27)
where ∗(U , λ) are regular at λ → 0.
It is easy to see that expression (2.33) for the momentum operator is regular at
λ → 0 in this case, and we can put
hk(0) (U , λ) = h̃k(0) (U , λ),
1
hs(1) (U, λ) = √ h̃s(1) (U, λ)
λ
k
s
(λ) and H(1)
(λ).
to be regular at λ → 0 densities of Hamiltonian functions H(0)
(3.28)
604
ANDREI YA. MALTSEV
According to the geometric construction, we have
#
∂hk(0) (U , λ) #
# ≡ 0,
#
µ
∂U
z
#
∂hs(1) (U, λ) #
# ≡0
#
µ
∂U
(3.29)
z
and also
#
∂p(U , λ) #
# ≡ 0.
∂U µ #z
(3.30)
Since the functions ñν (U , λ) locally give the coordinate system on ᏹN at every λ,
we have
∂ ñν
(z,
λ)
≠ 0,
(3.31)
det
∂U µ
and we can put
nν (U , λ) =
∂ ñν
∂U ξ
(z, 0)
(z, λ)ñη (U, λ)
ξ
∂U
∂ ñη
(3.32)
such that
#
ν
∂nν (U , λ) #
# ≡ ∂n (z, 0) = e(µ) ν ,
#
µ
∂U µ
∂U
z
ν = 1, . . . , N,
(3.33)
where e(µ) are the tangent vectors at the point z for λ = 0, nν (U, 0) = ñν (U, 0). So, we
get parts (I) and (II) of the theorem.
For part (III), we first prove here the following lemma.
Lemma 3.3. Under the conditions formulated in Theorem 3.1 the functionals
+∞
+∞
nνp (U ) dX,
Pq =
pq (U) dX,
Npν =
−∞
k
=
H(0)l
+∞
−∞
−∞
hk(0)l (U ) dX,
s
H(1)t
=
+∞
−∞
(3.34)
hs(1)t (U) dX,
p, q, l, t = 0, 1, . . . , generate the local flows with respect to both Jˆ(0) and Jˆ(1) and comµ
m
n
, and H(1)0
with respect to the bracket Jˆ(0) .
mute with the functionals N0 , P0 , H(0)0
µ
Proof. Since the functionals N0 are the annihilators of Jˆ(0) , they commute with
all other functionals with respect to Jˆ(0) on L(ᏹN , z). Also, from the translational
k
s
, and H(1)t
, we get that they also commute
invariance of all functionals Npν , Pq , H(0)l
ˆ
with the momentum operator P0 of J(0) with respect to Jˆ(0) .
k
s
We then know that the functionals N ν (λ), P (λ), H(0)
(λ), and H(1)
(λ) generate the
ks
local flows with respect to Jˆλ . In the case of nondegenerate form Qλ (on ᐃ), this means
η
ν
(U)UX introduced in (3.4).
that they are the conservation laws for all the flows w̃nη
η
η
ν
ν
(U)UX and
So, they are the conservation laws for the flows w(0)kη (U)UX and w(1)sη
η
ν
(U)UX
generate the local flows with respect to Jˆ(0) and Jˆ(1) . Now, since the flows w(0)mη
η
m
n
ν
and w(1)nη
(U )UX are generated by the functionals H(0)0
and H(1)0
with respect to Jˆ(0) ,
k
s
m
n
, and H(1)t
should commute with H(0)0
and H(1)0
with
we get that all Npν , Pq , H(0)l
respect to Jˆ(0) on L(ᏹN , z).
ON THE COMPATIBLE WEAKLY NONLOCAL POISSON BRACKETS . . .
605
k
s
, and H(1)t
with respect to both brackets, we
For the commutativity of all Npν , Pq , H(0)l
can now use just the common approach for the bi-Hamiltonian systems [20] writing
δFq Jˆ(0) δGk = −δFq−1 Jˆ(1) δGk = δFq−1 Jˆ(0) δGk+1 = · · · = δF0 Jˆ(0) δGk+q = 0
(3.35)
for any two functionals Fq and Gk from the set (3.34) (the same for the bracket Jˆ(1) ).
Remark 3.4. We point out here that the requirement of nondegeneracy of the form
Qλks on ᐃ is important, and, in general, Theorem 3.1 is not true without it. As example,
we consider here the Poisson pencil Jˆλ = Jˆ(0) + λJˆ(1) where
1
1
1
0
U2
UX
0
UX
U1
d
−1 1
D
+
−U 2 −U 1
UX2
UX2
0 −1 dX
0 1
1
1
U2
U1
UX
1 0
UX
−1
,
+
2 D
2
UX
−U
−U 1
UX2
0 1
1
1
UX
0 1 d
0
UX
0
0
−1 1
ˆ
J(1) =
D
+
UX2
UX2
1 0 dX
0 1
2U 1 0
1
1
1 0
0
0
UX
UX
+
D −1
.
UX2
UX2
0 1
2U 1 0
Jˆ(0) =
(3.36)
We have here the three-dimensional space ᐃ generated by the flows
η
ν
UX
w̃1η
1
1
UX
U2
UX
η
ν
,
w̃2η UX =
,
−U 1
UX2
UX2
1
UX
0
0
η
ν
.
w̃3η UX =
2U 1 0
UX2
U1
=
−U 2
(3.37)
Operator Jˆλ can be written as
1
1
U2
U1
UX
λ
d
−1 UX
+
D
−U 2 + 2λU 1 −U 1
UX2
UX2
−1 dX
U2
U1
UX1
UX1
−1
,
+
2 D
2
1
1
UX
−U + 2λU
−U
UX2
1
Jˆλ =
λ
(3.38)
and the form

1
0
λ

0

λ

0
1
=
λ
λ
−1
0


Qλ =  1
0
(3.39)
is degenerate on ᐃ.
The metric
νµ
gλ (U )
(3.40)
606
ANDREI YA. MALTSEV
νµ
here is just the flat metric on R2 , det gλ ≠ 0, and it is easy to check the equations
νξ
µξ
µ
νξ
ν
(λ),
gλ W1ξ (λ) = gλ W1ξ
µξ
µ
ν
gλ W2ξ = gλ W2ξ
,
(3.41)
where
U2
,
−U 1
µ
W1ξ (λ) =
U1
2
−U + 2λU 1
µ
W2ξ (λ) =
1
0
0
.
1
(3.42)
Also,
ξ
ξ
ξ
∂ν W1µ (λ) = ∂µ W1ν (λ),
ξ
∂ν W2µ = ∂µ W2ν
(3.43)
in the flat coordinates U 1 , U 2 , and
12
1
2
1
2
2
1
2
1
R12
= W11
(λ)W22
+ W21
W12
(λ) − W11
(λ)W22
− W21
W12
(λ)
1
2
(λ) + W12
(λ) ≡ 0;
= W11
(3.44)
so, Jˆλ represents a Poisson bracket of Ferapontov type for all λ.
νµ
νµ
η
ν
UX is not
Both g(0) and g(1) are nondegenerate in this case. However, the flow w̃1η
η
ν
Hamiltonian with respect to Jˆ(1) , and w̃3η
UX is not Hamiltonian with respect to Jˆ(0)
as follows from
νξ
µ
µξ
ν
,
g(0) w̃3ξ ≠ g(0) w̃3ξ
νξ
µ
µξ
ν
g(1) w̃1ξ ≠ g(1) w̃1ξ
.
η
(3.45)
η
ν
ν
It is also easy to check that the flows w̃1η
UX and w̃3η
UX do not commute with each
other.
4. The recursion operator and the higher Hamiltonian structures. We now use the
symplectic operator (2.10) for nondegenerate bracket Jˆ(0) and consider the recursion
ντ
Ω̂(0)τµ under the assumptions of Theorem 3.1.
operator R̂µν = Jˆ(1)
We put v τ (U , λ) = nτ (U , λ), τ = 1, . . . , N, v N+k (U, λ) = hk(0) (U, λ), k = 1, . . . , g0 ,
v0s (U )
N+k
τ
≡ v s (U , 0), E(0)
= τ , τ = 1, . . . , N, and E(0)
= e(0)k , k = 1, . . . , g0 (where nτ (U, λ)
hk(0) (U , λ)
and
Ω̂(0)τµ as
are the functions from Theorem 3.1) and write the symplectic form
N+g0
Ω̂(0)τµ =
k=1
k
E(0)
∂v0k −1 ∂v0k
D
.
∂U τ
∂U µ
(4.1)
νµ
We can also write the operator Jˆ(0) as
νµ
νµ
Jˆ(0) = g(0)
N+g0
k
k
d
νµ
η
k
ντ ∂v0
−1 ˆµσ ∂v0
Jˆ(0)
J
+ b(0)η UX +
E(0)
D
.
(0)
dX
∂U τ
∂U σ
k=1
(4.2)
ON THE COMPATIBLE WEAKLY NONLOCAL POISSON BRACKETS . . .
607
For R̂µν , we have the expression
N+g0
ντ
R̂µν = Jˆ(1)
Ω̂(0)τµ =
k
ντ
E(0)
g(1)
k=1
N+g0
+
k=1
N+g0
+
η
k
ντ
E(0)
b(1)η
UX
k=1
D −1
X
k
k
ντ ∂v0
E(0)
g(1)
∂U τ
g1 N+g0
+
∂v0k
∂U τ
∂v0k
∂U µ
∂v0k
∂U µ
(4.3)
∂v0k −1 ∂v0k
D
∂U τ
∂U µ
ζ
η
k
ν
τ
e(1)s E(0)
w(1)sη
UX D −1 w(1)sζ
UX
s=1 k=1
∂v0k −1 ∂v0k
D
,
∂U τ
∂U µ
where
ζ
τ
UX
w(1)sζ
∂v0k
d k
d
Q − Qsk
≡ Qsk X =
∂U τ
dX s
dX
(4.4)
for some Qsk (U ) according to Theorem 3.1.
We normalize the functions Qsk (U ) such that Qsk (z) = 0, and we have
· · · D −1
d k
Q · · · = · · · Qsk · · ·
dX s
(4.5)
on L(ᏹN , z).
Also, d/dX D −1 ≡ Î on L(ᏹN , z), and
N+g0
k
ντ
E(0)
g(1)
k=1
+
∂v0k
∂U τ
g1 N+g0
s=1 k=1
N+g0
=
k=1
D −1
X
N+g0
k
∂v0k
∂v0k
η ∂v0
k
ντ
+
E(0)
b(1)η
UX
D −1
µ
τ
∂U
∂U
∂U µ
k=1
η
k
ν
e(1)s E(0)
w(1)sη
UX Qsk D −1
∂v0k
∂U µ
(4.6)
k
∂v0k
k
ντ ∂v0
Jˆ(1)
E(0)
.
D −1
τ
∂U
∂U µ
Also, using the relation
N+g0
k
E(0)
k=1
∂hs(1) (U, 0)
∂hs(1)0 (U)
∂v0k k
η
τ
Qs = Ω̂(0)µτ w(1)sη
UX =
=
µ
µ
∂U
∂U
∂U µ
(4.7)
on L(ᏹN , z) (since ∂hs(1)q /∂U µ (z) = 0, q ≥ 0), we can write
g1
k
k
∂hs(1)0
η
k
ντ ∂v0
ν
−1 ∂v0
Jˆ(1)
E(0)
−
e(1)s w(1)sη
UX D −1
D
τ
µ
∂U
∂U
∂U µ
s=1
k=1
N+g0
g1
k
k
∂hs(1)0
∂hs(1)0
k
ντ ∂v1
ντ
−1 ∂v0
ˆ
Jˆ(0)
J
= Vµν −
E(0)
−
e
,
D
D −1
(1)s
(0)
τ
µ
τ
∂U
∂U
∂U
∂U µ
s=1
k=1
N+g0
ντ
R̂µν = g(1)
g(0)τµ +
(4.8)
608
ANDREI YA. MALTSEV
where
v k (U , λ) =
+∞
hs(1) (U, λ) =
vqk (U )λq ,
q=0
+∞
hs(1)q (U)λq
(4.9)
q=0
ντ
(U )g(0)τµ (U ).
and Vµν (U ) = g(1)
We mention here that, according to this definition, R̂µν will act from the left on the
vector-fields (on ᐂ0 (z)) and from the right on the gradients of functionals on L(ᏹN , z).
νµ
Theorem 4.1. Consider the nondegenerate pencil (3.1) with det g(0) ≠ 0 such that
Qλks
is also nondegenerate on ᐃ for small enough λ ≠ 0. Then,
the form
(I) any power [R̂ n ], n ≥ 1 of the recursion operator can be written in the form
ν
R̂ n µ


N+g0
n
k
k
n ν
k 
ντ ∂vs
n
−1 ∂vn−s 
ˆ
= V̂ µ + (−1)
E(0)
J(0)
D
∂U τ
∂U µ
s=1
k=1


g1
n
∂hk(1),s−1
∂hk(1),n−s
ντ
n
−1
;
Jˆ(0)
e(1)k 
+ (−1)
D
∂U τ
∂U µ
s=1
k=1
(4.10)
ξµ
νµ
(II) the higher Hamiltonian structures Jˆ(n) = [R̂ n ]νξ Jˆ(0) can be written on Ᏺ0 (z) in
the following weakly nonlocal form:
ν ξµ d
ν ξµ
νµ
η
Jˆ(n) = V̂ n ξ g(0)
+ V̂ n ξ b(0)η UX
dX


N+g0
n
k
k
∂vn−s
ξµ 
k 
ντ ∂vs
n
ˆ
J(0)
+ (−1)
E(0)
g
∂U τ
∂U ξ (0)
s=1
k=1


g1
n
∂hk(1),s−1 ∂hk(1),n−s ξµ
ντ
n

Jˆ(0)
+ (−1)
e(1)k
g(0) 
τ
ξ
∂U
∂U
s=1
k=1


N+g0
n−1
k
∂vsk
µξ ∂vn−s 
k 
ντ
n+1
−1
Jˆ(0)
Jˆ(0)
E(0)
+ (−1)
D
∂U τ
∂U ξ
s=1
k=1


g1
k
k
n
∂h
∂h
µξ
(1),s−1
(1),n−s
ντ

+ (−1)n+1
Jˆ(0)
e(1)k 
D −1 Jˆ(0)
∂U τ
∂U ξ
s=1
k=1
(4.11)
for n ≥ 2;
(III) all the “negative” symplectic forms
ξ
Ω̂(−n)νµ = Ω̂(0)νµ R̂ n µ ,
n≥1
(4.12)
can be represented on ᐂ0 (z) in the form

n
k
∂vsk −1 ∂vn−s

= (−1)
D
∂U ν
∂U µ
s=0
k=1


g1
n
∂hk(1),s−1 −1 ∂hk(1),n−s
n
.
e(1)k 
D
+ (−1)
∂U ν
∂U µ
s=1
k=1
N+g0
Ω̂(−n)νµ
n

k 
E(0)
(4.13)
ON THE COMPATIBLE WEAKLY NONLOCAL POISSON BRACKETS . . .
609
Proof. (I) By induction, we have
R̂ξν
ξ
R̂ n µ


N+g0
n
k
k
n+1 ν
k 
ν ˆξσ ∂vs
n
−1 ∂vn−s 
R̂ξ J(0)
= V̂
E(0)
D
µ + (−1)
∂U σ
∂U µ
s=1
k=1


g1
n
∂hk(1),s−1
∂hk(1),n−s
ν ˆξσ
n
−1


R̂ξ J(0)
e(1)k
+ (−1)
D
σ
µ
∂U
∂U
s=1
k=1


N+g0 N+g0
q
n
k
q
∂v
∂v
qk
n−s
1
ντ
n
−1

Jˆ(0)
+ (−1)
E(0) 
D P0s
∂U τ
∂U µ
q=1 k=1
s=1


N+g0 g1
q
n
q
∂hk(1),n−s
ντ ∂v1
n
−1 qk


ˆ
J(0)
E(0)
+ (−1)
D Q0s
∂U τ
∂U µ
q=1 k=1
s=1
N+g0
q
q
q
n ξ
∂v1
ντ
−1 ∂v0
V̂ µ
E(0) Jˆ(0)
−
D
τ
ξ
∂U
∂U
q=1


q
g1 N+g0
n
k
∂h(1),0
∂v
qk
n−s
ντ
n
−1

Jˆ(0)
+ (−1)
e(1)q 
D S0s
∂U τ
∂U µ
q=1 k=1
s=1


q
g 1 g1
n
∂h(1),0
∂hk(1),n−s
ντ
n
−1 qk


ˆ
J(0)
e(1)q
+ (−1)
D T0s
∂U τ
∂U µ
q=1 k=1
s=1
q
q
g1
∂h(1),0
∂h(1),0 n ξ
ντ
−1
ˆ
V̂ µ ,
e(1)q J(0)
D
−
∂U τ
∂U ξ
q=1
(4.14)
where we have
q
k
qk ∂v0
d qk
qk d
ξσ ∂vs
ˆ
J
≡ P0s X =
P − P0s
,
∂U ξ (0) ∂U σ
dX 0s
dX
q
k
qk ∂v0
d qk
qk d
ξσ ∂h(1),s
ˆ
J
Q − Q0s
,
≡ Q0s X =
e(1)k
∂U ξ (0) ∂U σ
dX 0s
dX
q
∂h(1),0 ξσ ∂vsk
qk d qk
qk d
k
ˆ
J(0)
S − S0s
,
E(0)
≡ S0s X =
∂U ξ
∂U σ
dX 0s
dX
q
∂h(1),0 ξσ ∂hk(1),s
qk d qk
qk d
ˆ(0)
e(1)k
T − T0s
J
≡ T0s X =
∂U ξ
∂U σ
dX 0s
dX
k
E(0)
qk
qk
qk
(4.15)
qk
for some functions P0s (U ), Q0s (U ), S0s (U ), T0s (U) according to Theorem 3.1, and
we use the normalization
qk
qk
qk
qk
P0s (z) = Q0s (z) = S0s (z) = T0s (z) = 0.
(4.16)
From Theorem 3.1, we have
∂vsk
(z) = 0,
∂U τ
∂hk(1),s−1
(z)
∂U τ
= 0,
s≥1
s ≥ 1;
actually
∂hk(0),0
∂U τ
(z) = 0 also ,
(4.17)
610
ANDREI YA. MALTSEV
so, for s ≥ 1, we can write
ξτ
Ω̂(0)νξ Jˆ(0)
∂vsk
∂U τ
=
∂vsk
,
∂U ν
ξτ
Ω̂(0)νξ Jˆ(0)
∂hk(1),s−1
∂U τ
=
∂hk(1),s−1
∂U ν
(4.18)
on L(ᏹN , z) and
k
∂vs+1
,
∂U τ
∂hk(1),s−1
∂hk(1),s−1
∂hk(1),s
ν ˆξτ
ντ
ντ
ˆ
ˆ
= J(1)
= − J(0)
R̂ξ J(0)
∂U τ
∂U τ
∂U τ
ξτ
R̂ξν Jˆ(0)
∂vsk
∂U τ
ντ
= Jˆ(1)
∂vsk
∂U τ
ντ
= − Jˆ(0)
(4.19)
(4.20)
according to Theorem 2.2.
We now multiply the equalities
ξτ
Jˆ(1)
∂vsk
∂U τ
k
ξτ ∂v
= − Jˆ(0) s+1
,
∂U τ
ξτ
Jˆ(1)
∂hk(1),s
∂U τ
k
ξτ ∂h(1),s+1
= − Jˆ(0)
∂U τ
(4.21)
by Ω̂(0)νξ from the left. Since
k
∂hk(1),s+1
∂vs+1
(z)
=
(z) = 0,
∂U τ
∂U τ
s ≥ 0,
(4.22)
we get again from Theorem 2.2
k
k
∂vs+1
∂vsk τ
ξτ ∂vs
ˆ
J
R̂
=
−
Ω̂
=
−
, s ≥ 0,
(0)νξ (1)
∂U ν
∂U τ
∂U τ ν
k
k
∂h(1),s τ
∂hk(1),s+1
ξτ ∂h(1),s
ˆ(1)
J
R̂
=
−
Ω̂
=
−
, s≥0
(0)νξ
∂U ν
∂U τ
∂U τ ν
(4.23)
(action from the right).
Using the relation
fX D −1 = −f (X)
(4.24)
for the action from the right of D −1 on any f (U) such that f (z) = 0, we get that the
ξ
last six terms in the expression for R̂ξν [R̂ n ]µ can be written as
g1
q
q
q
q
∂h(1),0 n ξ
∂h(1),0
n ξ
∂v1
ντ
−1 ∂v0
−1
R̂ µ −
R̂ µ
D
D
e(1)q Jˆ(0)
−
∂U τ
∂U ξ
∂U τ
∂U ξ
q=1
q=1
q
q
N+g0
g1
q
q
q
∂h(1),0
∂h(1),n
ντ ∂v1
ντ
−1 ∂vn
n+1
−1
ˆ
J
= (−1)n+1
E(0) Jˆ(0)
+
(−1)
e
.
D
D
(1)q
(0)
∂U τ
∂U µ
∂U τ
∂U µ
q=1
q=1
N+g0
q
E(0)
ντ
Jˆ(0)
(4.25)
Now, using relations (4.19) and (4.20), we get part (I) of the theorem.
ON THE COMPATIBLE WEAKLY NONLOCAL POISSON BRACKETS . . .
611
(II) To avoid much calculations, we just write that according to Theorem 3.1 we have
the relations
q
k
qk d qk
d
qk
ξσ ∂v0
k ∂vn−s
ˆ
J(0)
≡ Pn−s,0 X =
− Pn−s,0
P
,
E(0)
∂U ξ
∂U σ
dX n−s,0
dX
(4.26)
q
k
∂h(1),n−s
qk d
d qk
qk
ξσ ∂v0
k
ˆ(0)
J
=
−
S
≡
S
E(0)
S
n−s,0 X
n−s,0
∂U ξ
∂U σ
dX n−s,0
dX
qk
qk
qk
qk
for some Pn−s,0 (U ), Sn−s,0 (U ), Pn−s,0 (z) = 0, and Sn−s,0 (z) = 0, and so, the expression
Jˆ(n) = R̂ n Jˆ(0) can be written according on Ᏺ0 (z) to (4.10) and (4.2) and Theorem 2.2
as
νµ
Jˆ(n) = local part of R̂ n × local part of Jˆ(0)


N+g0
n
k
∂vsk ∂vn−s
ξµ 
k 
ντ
n
Jˆ(0)
E(0)
g
+ (−1)
∂U τ
∂U ξ (0)
s=1
k=1


k
g1
k
n
∂h
∂h
ξµ
(1),s−1
(1),n−s
ντ
Jˆ(0)
e(1)k 
g(0) 
+ (−1)n
∂U τ
∂U ξ
s=1
k=1
N+g0
q q ν ξτ ∂v q
(4.27)
0
n
−1 ˆµσ ∂v0
ˆ
J(0)
E(0) R̂ ξ J(0)
+
D
τ
σ
∂U
∂U
q=1


N+g0
n
k
∂vsk
µσ ∂vn−s 
k 
ντ
n
−1
Jˆ(0)
D
Jˆ(0)
E(0)
− (−1)
∂U τ
∂U σ
s=1
k=1


g1
n
∂hk(1),s−1
∂hk(1),n−s
µσ
ντ
n
−1

Jˆ(0)
Jˆ(0)
e(1)k 
− (−1)
D
∂U τ
∂U σ
s=1
k=1
(since Jˆ(0) is skew-symmetric, its action from the right differs by sign from the action
from the left). Now, using (4.19), we get part (II) of the theorem. We also mention that
it is important that we consider the space Ᏺ0 (z) to use the equality
D −1
∂v0k
d ∂v0k
=
dX ∂U ξ
∂U ξ
(4.28)
for k = 1, . . . , N.
(III) We have
ξ
Ω̂(−n)νµ = Ω̂(0)νξ R̂ n µ .
qk
(4.29)
qk
Again, using the functions P0s (U ) and Q0s (U), we can write
∂v0k −1 ∂v0k n ξ
R̂ µ
=
D
∂U ν
∂U ξ
k=1


N+g0
n
k
k
∂vn−s
ξτ ∂vs
k 
n
−1

E(0)
+ (−1)
Ω̂(0)νξ Jˆ(0)
D
∂U τ
∂U µ
s=1
k=1


g1
n
∂hk(1),s−1
∂hk(1),n−s
ξτ
n
−1
.
e(1)k 
+ (−1)
Ω̂(0)νξ Jˆ(0)
D
∂U τ
∂U µ
s=1
k=1
N+g0
Ω̂(−n)νµ
k
E(0)
(4.30)
612
ANDREI YA. MALTSEV
Since
∂hk(1),s−1
∂vsk
(z) = 0,
∂U τ
∂U τ
= 0,
s ≥ 1,
(4.31)
we get part (III) using Theorem 2.2 and (4.19) on L(ᏹN , z).
νµ
νµ
We also mention that if both det g(0) ≠ 0 and det g(1) ≠ 0 (and the form Q in nondegenerate on ᐃ), then also the series of “negative” Hamiltonian operators Jˆ(−n) =
R̂ −n Jˆ(0) and “positive” symplectic forms Ω̂(n) = Ω̂(0) R̂ −n will be weakly nonlocal. This
situation takes place, for example, in the Hamiltonian structures of Whitham systems
for KdV, NLS, and SG hierarchies. The local bi-Hamiltonian structure for the averaged
KdV hierarchy was constructed in [6] (see also [7, 8]) using the (Dubrovin-Novikov) procedure of averaging of local field-theoretical brackets for Gardner-Zakharov-Faddeev
and Magri brackets. Both metrics of the corresponding DN-brackets are nondegenerate in this case (and there is no requirements on Q). Also, in [7, 8], the local bracket
for the averaged “V-Gordon” equation
ϕtt − ϕxx + V (ϕ) = 0,
(4.32)
which is the generalization of SG system using the same procedure was constructed.
In [32], all the brackets for averaged KdV, NLS, and SG hierarchies, having local (DN) or
constant curvature (MF) form, were enumerated using a nice differential-geometrical
approach. It can be shown that all the pencils represented in [6, 7, 8, 32] actually
satisfy the requirements of Theorems 3.1 and 4.1. The recursion operator approach
for the local bi-Hamiltonian structure for the averaged KdV hierarchy (in the diagonal
form) was investigated in [1, 2], and all the “positive” operators Jˆ(n) were explicitly
found in [1] in this case. In [23, 24], the general procedure of averaging of brackets (1.23), which gives the weakly nonlocal Hamiltonian operators for the averaged
systems with Hamiltonian structure (1.23), was constructed. For many “integrable”
systems, this method also gives all the “positive” weakly nonlocal Poisson brackets of
Ferapontov type for the corresponding Whitham hierarchy. However, we see here that
the “negative” Hamiltonian operators and Symplectic structures for the averaged KdV,
NLS, and SG also should be weakly nonlocal according to Theorems 3.1 and 4.1. We
believe that there should be a good procedure of averaging of “negative” Hamiltonian
operators for the integrable systems giving the brackets of Ferapontov type and the
general averaging procedure for the weakly nonlocal symplectic structures
Ω̂νµ (x, y) =
N
G
δH(k)
δH(s)
(k)
Cνµ
dks
ν(x − y)
ϕ, ϕx , . . . δ(k) (x − y) +
δϕ(x)
δϕ(y)
k=1
k,s=1
(4.33)
giving the weakly nonlocal symplectic structures and the corresponding F-brackets
for the Whitham systems.
We also believe that the general compatible F-brackets should be important for the
integration of nondiagonalizable (bi-Hamiltonian) systems which cannot be integrated
by Tsarëv’s method.
ON THE COMPATIBLE WEAKLY NONLOCAL POISSON BRACKETS . . .
613
We also mention that the requirement of nondegeneracy of form Q on ᐃ is also
important in Theorem 4.1. As such, it is possible to show that the “negative” and
“positive” Poisson structures corresponding to the pair of operators (3.36) will not be
weakly nonlocal.
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Andrei Ya. Maltsev: L. D. Landau Institute for Theoretical Physics, Kosygina Street
2, Moscow 117940, Russia
Current address: University of Maryland, College Park, MD 20742, USA
E-mail address: maltsev@itp.ac.ru, maltsev@math.umd.edu