On the compatible weakly nonlocal Poisson brackets of hydrodynamic type

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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 metricgνµ₍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, momen-tum 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 metricsgνµ₍₀₎andg₍νµ₁₎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:

ˆ

J₍νµ₀₎=gνµ₍₀₎(U ) d dX+b

νµ (0)η(U )U

η X+

g0

k=1

e(0)kw(ν0)kη(U )U η XD−1w

µ

(0)kζ(U )U ζ X,

ˆ

J₍νµ₁₎=gνµ₍₁₎(U ) d dX+b

νµ (1)η(U )U

η X+

g1

k=1

e(1)kw(ν1)kη(U )U η XD−1w

µ

(1)kζ(U )U ζ X,

(1.1)

wheree(0)k, e(1)k= ±1 andD−1=(d/dX)−1are defined in a “skew-symmetric” way

D−1=1 2

X

−∞dX− +∞

X dX

(1.2)

and require that the expression

ˆ

Jλνµ=Jˆ νµ (0)+λJˆ

νµ

(1) (1.3)

defines the Poisson bracket satisfying Jacobi identity for anyλ.

We mention here that the brackets of this kind are the generalization of Dubrovin-Novikov local homogeneous brackets of hydrodynamic type [6,7,8]:

with the Hamiltonian operator

ˆ

JDNνµ =gνµ(U ) d dX+b

νµ

η (U )UXη. (1.5)

Theorem 1.1 (Dubrovin and Novikov). Consider the bracket (1.4) with

non-degenerate tensor gνµ_{(U ). From the Leibnitz identity, it follows that g}νµ_{(U ) and}

Γµ

νη(U )= −gνξ(U )bξµη (U ) (gνξgξµ=δµν)transforms as a metric with upper indices and

the Christoffel symbols under the pointwise coordinate transformationsU˜ν_=U˜ν_{(U ).}

Bracket (1.4) is skew-symmetric if and only ifgνµ _{is symmetric and the connection}

Γµ

νηis compatible with the metric∇ηgνµ≡0.

Bracket (1.4) satisfies the Jacobi identity if and only if the connectionΓνηµ is symmetric and has zero curvatureR_ηξνµ≡0.

It follows from Dubrovin-Novikov theorem that any bracket (1.4) with non-degenerategνµ _{can be written locally in the “constant form”}

nν(X), nµ(Y )=νδνµδ(X−Y ), ν= ±1 (1.6)

in the flat coordinatesnν_=nν_{(U ).}

The functionals

Nν₌ +∞

−∞n

ν_{(X) dX (1.7)}

are Casimirs of bracket (1.4) and the functional

P= +∞

−∞ 1 2

N

ν=1

νnν(X)nν(X) dX (1.8)

is a momentum operator generating the flowUTν=UXν. Form (1.6) can be considered

as the canonical form for the DN-bracket (1.4) with the nondegenerate tensorgνµ_.

It can be also seen that any functional of “hydrodynamic type”

H= +∞

−∞h(U ) dX (1.9)

generates a “hydrodynamic type system”

UTν=Vµν(U )U µ

X (1.10)

according to bracket (1.4).

We also mention that bracket (1.4) with degenerate tensorgνµ_{(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 Mokhov-Ferapontov bracket [29]

Uν(X), Uµ(Y )=gνµ(U )δ(X−Y )+bνµη (U )UXηδ(X−Y )+cUXνν(X−Y )U µ

whereν(X−Y )=1/2 sgn(X−Y ), corresponding to the Hamiltonian operator

ˆ

JDNνµ =gνµ(U ) d dX+b

νµ

η (U )UXη+cUXνD−1U µ

X. (1.12)

Theorem1.2(Mokhov and Ferapontov). Consider bracket (1.11) with nondegener-ate tensorgνµ_{(U ). Then}

(1) bracket (1.11) is skew-symmetric and satisfies Leibnitz identity if and only if the

tensorgνµ_{(U )is a metric with upper indices andΓ}µ

νη= −gνξbξµη are the

connec-tion coefficients compatible withgνµ_{(U );}

(2) bracket (1.11) satisfies the Jacobi identity if and only if the connection Γνηµ is

symmetric and has the constant curvature equal toc, that is,

Rµηντ=c

δνµδτη−δτµδνη

. (1.13)

Bracket (1.11) has a weakly nonlocal form. However, any local translationally invari-ant functional

H=

h(U ) dX (1.14)

generates a local system of hydrodynamic type with respect to (1.11). Indeed, we have

U_Xµ ∂h

∂Uµ ≡∂Xh (1.15)

ifhdoes not depend onXexplicitly; so, the application ofD−1gives the local expres-sion 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 )=νδνµ−cnνnµδ(X−Y )−cnνXnµδ(X−Y )

+cnν

Xν(X−Y )n µ Y,

(1.16)

where nν_=nν_{(U )are the annihilators for bracket (1.11) (on the space of rapidly}

decreasing functionsnν_{(X)atX→ ±∞). Also, the implicit expression for the density}

ofPwas 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 functionsUν_(X)

forX→ ±∞(see [25]) (the conditionUν_{→0,X→ ±∞, in general, is not invariant under}

the pointwise transformations ˜Uν_=U˜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

The general Ferapontov bracket [11,12,13,16] has the form

Uν_{(X), U}µ_{(Y )=g}νµ_{(U )δ(X−Y )+b}νµ

η (U )UXηδ(X−Y )

+

g

k=1

ekwν kη(U )U

η

Xν(X−Y )w µ kζ(U )U

ζ Y,

(1.17)

ek= ±1, which corresponds to the weakly nonlocal Hamiltonian operator

ˆ

JFνµ=gνµ(U ) d dX+b

νµ

η (U )UXη+ g

k=1

ekw_kην (U )UXηD−1w µ kζ(U )U

ζ

X. (1.18)

Theorem 1.3(Ferapontov [11, 16]). Consider bracket (1.17) with nondegenerate tensorgνµ_{(U ). Then,}

(1) bracket (1.17) is skew-symmetric and satisfies Leibnitz identity if and only if

ten-sorgνµ_{(U )is a metric with upper indices andΓ}µ

νη= −gνξbηξµare the connection

coefficients compatible withgνµ_{(U );}

(2) bracket (1.11) satisfies the Jacobi identity if and only if the connection Γµ

νη is

symmetric and the metricgνµ _{and tensorsw}ν

kη(U )satisfy the equations gντw_kτµ =gµτw_kτν , ∇νw_kηµ = ∇ηw_kνµ ,

Rντ µη=

g

k=1

ekwν

kµwkητ −wkµτ wkην

. (1.19)

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 submanifoldᏹN _{with flat normal connection in}

the pseudo-Euclidean spaceEN+g_{. In this consideration, the tensor g}νµ _{is the first}

quadratic form of ᏹN_{, and w}ν

kη are the Weingarten operators corresponding tog

parallel vector fields in the normal bundleNk, such thatNk,Nm =ekδkm. It was also

proved by Ferapontov that these brackets can be constructed as a Dirac restriction of the local DN-bracket

ZI(X), ZJ(Y )=IδIJδ(X−Y ), I, J=1, . . . , N+g, I= ±1 (1.20)

inEN+g _{to the submanifoldᏹ}N _[12,16].

As far as we know, the cases of brackets (1.11), (1.17) with the degenerate tensors

gνµ_{(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 diagonalgνµ_{(U )) satisfies Tsarëv “semi-Hamiltonian” property, and}

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 manifoldᏹN _{has a holonomic net of lines of}

curvature, the metricgνµ_{(U )and all the operatorsw}ν

kη can be written in the

diago-nal form in the corresponding coordinatesrν_=rν_{(U ). Here, we do not impose this}

requirement and consider any brackets of Ferapontov type.

We will assume that the flowsw_kην(U )UXηin the nonlocal part of (1.17) are linearly

independent (with constant coefficients). (The nonlocal part in (1.17) actually repre-sents the nondegenerate quadratic form on the linear space generated bywν

kη(U )U η X, k=1, . . . , gwritten in the canonical form withek= ±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 functionalHis a conservation law for any of the flowsw_kην (U )UXηsuch that the

ex-pressions

wkην (U )U η X

∂h

∂Uν (1.22)

represent the total derivatives with respect toXof some functionsQk(U )for anyk. This fact is also true for more general weakly nonlocal Poisson brackets having the form

ϕi_{(x), ϕj(y)=} G

k=1

B_kijϕ, ϕx, . . .δ(k)_(x−y)

+

g

k=1

ekS_kiϕ, ϕx, . . .ν(x−y)S_kjϕ, ϕy, . . .,

(1.23)

whereδ(k)_{(x−y)=(d/dx)}k_{δ(x−y),ek= ±1, and the set{S}i

k(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)= −ϕxν(x−y)ϕy (1.24)

for the Krichever-Novikov equation

ϕt=ϕxxx−3₂ϕ2xx ϕx +

h(ϕ)

whereh(ϕ)=c3ϕ3+c2ϕ2+c1ϕ+c0, with the Hamiltonian function

H= ₁

2

ϕ2

xx ϕ2x +

1 3

h(ϕ) ϕ2x

dx. (1.26)

As established in [23,24], the flowsS_ki(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

H=

hϕ, ϕx, . . .dx. (1.27)

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)

and the Magri bracket

ϕ(x), ϕ(y)= −δ(x−y)+4ϕ(x)δ(x−y)+2ϕxδ(x−y) (1.29)

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 evolu-tion system connected with the Whitham method for slow modulaevolu-tions 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 sim-ilar facts for the case of the nondegenerate pencils (i.e., detg₍νµ₀₎≠0) with also some nondegeneracy conditions for nonlocal part of ˆJ(0)+λJ(ˆ1).

We also mention that the wide classes of local pencils of hydrodynamic type (DN-brackets) 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.

Consider bracket (1.17) with nondegenerate tensorgνµ_{(U ). According to}

Ferapon-tov results, we can represent it as a Dirac restriction of DN-bracket inRN+g _{to the}

submanifoldᏹN_⊂RN+g _{with flat normal connection. We fix the pointz∈ᏹ}N _and

introduce the corresponding loop space in the vicinity ofz

LᏹN, z=γ:R1 _→ᏹN_{:γ(−∞)=γ(+∞)=z∈ᏹ}N_{. (2.1)}

So, we fix the boundary conditions for the functionsUν_{(X)and require that these}

functions rapidly approach their boundary values atX→ ±∞.

Theorem 2.1(Maltsev and Novikov [25]). Consider the bracket (1.17) with

non-degenerate gνµ_{(U ) defined on the loop spaceL(ᏹ}N_{, z). Consider the corresponding}

embeddingᏹN_⊂RN+g_{with flat normal connection. Consider the flat orthogonal}

coor-dinatesZI_inRN+g_{such that}

(1) ZI_{(z)=0,I=1, . . . , N+g, and the corresponding DN-bracket inR}N+g _{has the} form

ZI(X), ZJ(Y )=EIδIJδ(X−Y ), I, J=1, . . . , N+g, EI= ±1; (2.2)

(2) the firstNcoordinatesZ1_{, . . . , Z}N _{are tangential toᏹ}N _{at the pointz;}

(3) the lastgcoordinatesZN+1_{, . . . , Z}N+g_{are orthogonal toᏹ}N _{at the pointz.} Then,

(1) bracket (1.17) has exactlyNlocal Casimirs given by the functionals

Nν₌ +∞

−∞n

ν_{(X)dX, (2.3)}

wherenν_{(U )are the restrictions of the firstN(tangential toᏹ}N_{atz) coordinates} Z1_{, . . . , Z}N _onᏹN_,

(2) all the flows

Uν tk=w

ν kη(U )U

η

X (2.4)

are Hamiltonian with respect to (1.17) with local Hamiltonian functionals

Hk= _+∞

−∞hk(X) dX, k=1, . . . , g, (2.5)

wherehk(U )are the restriction of the lastgcoordinatesZN+k_{,k=1, . . . , gon} ᏹN _andwν

kη(U )are the Weingarten operators corresponding to parallel vector

fieldsNk(U )in the normal bundle with the normalization

Nk(z)=0, . . . ,0,−EN+k,0, . . . ,0T, (2.6)

where −EN+k _{stays at the position N+k, in the coordinatesZ}1_{, . . . , Z}N+g_{, so}

Nk(U ),Nl(U ) =EN+k_{δkl (there is an arithmetic mistake in the journal}

vari-ant of [25]where the generated flows are written asEN+k_wν

(3) bracket (1.17) has in coordinatesnν_{(U )the “canonical form” corresponding to} the spaceL(ᏹN_{, z), that is,}

nν(X), nµ(Y )=

νδνµ−

g

k=1

ekf_kν(n)f_kµ(n)

δ(X−Y )

−

g

k=1

ekfkν(n)

Xf µ

k(n)δ(X−Y )

+

g

k=1

ekfν k(n)

Xν(X−Y )

f_kµ(n)_Y,

(2.7)

whereν_=Eν_{,ν=1, . . . , N, ek=E}N+k_{,k=1, . . . , g,n}ν_(z)=0,fν

k(z)=0(i.e., fkν(0)=0). (Actually, we have the equalityfkν(U )≡Nkν(U ), where Nkν(U )are the firstNcomponents of the vectorsNk(U )in the coordinatesZ1_{, . . . , Z}N+g_.)

We note here that the Casimirs and “canonical functionals,” as well as the canonical forms, depend on the phase spaceL(ᏹN_{, z). So, if we do not fix the loop space, it is}

better to speak just about theN+gcanonical functions (the restrictions of the flat co-ordinates fromRN+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 differentf_kν(U )in this case.)

For the case of Mokhov-Ferapontov bracket, the canonical form will be (1.16) for any fixed spaceL(ᏹN_{, z)(although the coordinatesn}ν_{(U )will be different for}

differ-ent loop spaces). The explicit form of canonical functional, which is the momdiffer-entum operator in this case, can then be written as (see [25])

P=1_c +∞

−∞   1−

1−c

N

ν=1

ν_nν_nν  

dX. (2.8)

Using the restriction of the symplectic form of DN-bracket onᏹN_{, it is also}

possi-ble to get the symplectic formΩνµ(X, Y )for theF-bracket (1.17) with nondegenerate

gνµ_{(U )[25]. The symplectic form appears to be also weakly nonlocal and can be}

writ-ten as

Ωνµ(X, Y )=

N+g

I=1

EI∂Z I_{(U )}

∂Uν (X)ν(X−Y ) ∂ZI_{(U )}

∂Uµ (Y )

=

N

τ=1

τ∂n τ

∂Uν(X)ν(X−Y ) ∂nτ ∂Uµ(Y )+

g

k=1

ek∂hk

∂Uν(X)ν(X−Y ) ∂hk ∂Uµ(Y )

(2.9)

onᏹN_.

We can also write the corresponding symplectic operator ˆΩνµ onL(ᏹN_{, z)as}

ˆ

Ωνµ=

N

τ=1

τ∂n τ

∂UνD−

1∂nτ

∂Uµ+ g

k=1

ek∂hk ∂UνD−

1∂hk

∂Uµ (2.10)

We introduce the spaceᐂ0(z)of vector fieldsξν(X)rapidly decreasing atX→ ±∞ onL(ᏹN_{, z). It is easy to see that all the hydrodynamic type systems (1.10) satisfy}

this requirement as the vector fields onL(ᏹN_{, z)and so belong toᐂ}

0(z). We can then naturally define the action of symplectic operator ˆΩνµon the spaceᐂ0(z).

We will also need the spaceᏲ0(z)of 1-formsων(X)rapidly decreasing atX→ ±∞ onL(ᏹN_{, z). We note here that ˆΩνµξ}µ_(X)∉Ᏺ

0(z)in the general case.

We will now prove by the direct calculation that the symplectic form ˆΩνµ is the inverse of the Hamiltonian operator ˆJFνµon the appropriate functional spaces.

Theorem2.2. (I)On the functional spaceᐂ0(z), the relation

ˆ

JFνξΩξµˆ =ˆI (2.11)

is true, whereˆIis the identity operator on the spaceᐂ0(z). (II)On the functional spaceᏲ0(z), the relation

ˆ

Ωνξ_Jˆξµ

F =ˆI (2.12)

is true, whereˆIis the identity operator onᏲ0(z). Proof. (I) We have

ˆ

J_Fνξ_Ωξµˆ ₌ 

gνξ_{(U )} d dX+b

νξ

η (U )UXη+ g

k=1

ekwν kη(U )U

η XD−1w

ξ kζ(U )U

ζ X  

×  N

τ=1

τ∂n τ

∂UξD−

1∂nτ

∂Uµ+ g

l=1

el∂h l

∂UξD−

1∂hl

∂Uµ  ,

(2.13)

where the operatorswkην (U )correspond to the vector fieldsN(k)(U )such that

N(k)(z)=0, . . . ,0,−ek,0, . . . ,0T (2.14)

(−ekstays at the positionN+k), and we have identicallyN(k)(U ),N(l)(U ) =ekδkl. Since the integrals ofnτ_{(U ), andh}l_{(U )generate the local flows according to ˆJ}νµ

F ,

we have

wkζξ (U )U ζ X

∂nτ ∂Uξ ≡

qkτ(U )

X, w

ξ kζ(U )U

ζ X

∂hl ∂Uξ ≡

plk(U )

X (2.15)

for some functionsqkτ(U )andpkl(U ). Moreover, considerNtangential vectors toᏹN

inRN+g_{corresponding to the coordinatesU}ν

e(ξ)(U )= ∂n_∂U1ξ, . . . , ∂nN ∂Uξ,

∂h1

∂Uξ, . . . , ∂hg ∂Uξ

T

(2.16)

Since by the following definition:

d

dXN(k)(U )=w ξ kζU

ζ

Xe(ξ)(U ), (2.17)

we have from (2.15)

q_kτX=

N_(k)τ X,

p_klX=

N_(k)N+lX, τ=1, . . . , N, l=1, . . . g (2.18)

for the components ofN(k)(U ). We normalize the functionsqτ

k(U )andplk(U )such that

qτ

k(z)=0, plk(z)=0. (2.19)

We then have

qτ

k(U )=N(k)τ (U ), p l

k(U )=N N+l

(k) (U )+ekδ l

k. (2.20)

Now, using the equalities

qτk

X=

d dXq

τ k−qkτ

d dX,

plk

X=

d dXp

l k−pkl

d

dX (2.21)

onL(ᏹN_{, z), we can write}

ˆ

J_Fνξ_Ωξµˆ _|ᐂ 0(z)=

N

τ=1

τ

gνξ ∂nτ ∂Uξ

X+ bνξη UXη

∂nτ ∂Uξ+

g

k=1

ekwν kηU

η XD−1

d dXq

τ k

D−1∂n τ

∂Uµ

+

g

l=1

el

gνξ ∂hl ∂Uξ

X+ bνξη UXη

∂hl ∂Uξ+

g

k=1

ekwν kηU

η XD−1

d dXp

l k

D−1∂h l

∂Uµ

+gνξ  N

τ=1

τ∂nτ ∂Uξ

d dXD

−1∂nτ

∂Uµ+ g

l=1

el∂h l

∂Uξ d dXD

−1∂hl

∂Uµ  

−

g

k=1

N

τ=1

ekτ_wν kηU

η XD−1qτk

d dXD

−1∂nτ

∂Uµ

−

g

k=1

g

l=1

ekelwν kηU

η XD−1plk

d dXD

−1∂hl

∂Uµ.

(2.22)

We can here replace(d/dX)D−1_{by identity, and we have}

D−1fX=f (X)−1₂f (−∞)+f (+∞) (2.23)

for any functionf (U )onL(ᏹN_{, z)according to the definition ofD}−1_{. So, according to} normalization (2.19), we can replace the operatorsD−1_(d/dX)qτ

k andD−1(d/dX)p l k

According to the same normalization, the expressions within the brackets in the first two terms are equal to

ˆ

JνξF ∂nτ ∂Uξ

L(ᏹN,z)=

0,

ˆ

JFνξ ∂hl ∂Uξ

L(ᏹN,z)= wν

lηU η

X. (2.24)

So, we have

ˆ

J_Fνξ_Ωξµˆ _|ᐂ 0(z)=

g l=1 elwν lηU η xD−1∂h

l

∂Uµ+g νξ

 N

τ=1

τ∂nτ ∂Uξ

∂nτ ∂Uµ+

g l=1 el∂h l ∂Uξ ∂hl ∂Uµ   − g

k=1

ekw_kην UXηD−1  N

τ=1

τq_kτ∂n τ

∂Uµ+ g

l=1

elpl_k∂h l

∂Uµ  .

(2.25)

Using (2.20) and (2.16), we now get

N

τ=1

τqτ_k∂n τ

∂Uµ+ g

l=1

elpl_k∂h l

∂Uµ =

N(k),e(µ)

+_∂U∂hkµ = ∂hk

∂Uµ. (2.26)

Also, using the evident relation

N τ=1 τ∂n τ ∂Uξ ∂nτ ∂Uµ+

g l=1 el∂h l ∂Uξ ∂hl

∂Uµ≡gξµ(U ), (2.27)

we get the statement (I) of the theorem. (II) We have

ˆ

ΩνξJˆFξµ=  N

τ=1

τ∂nτ ∂UνD

−1∂nτ

∂Uξ+ g

l=1

el∂h l

∂UνD

−1∂hl

∂Uξ  

× gξµ d dX+b

ξµ η UXη+

g

k=1

ekwkηξ U η XD−1w

µ kζU ζ X = N τ=1

τ∂nτ ∂UνD

−1 d

dX ∂nτ ∂Uξg

ξµ₊ g

l=1

∂hl ∂UνD

−1 d

dX ∂hl ∂Uξg

ξµ − N τ=1 τ∂n τ

∂UνD−

1 ∂nτ

∂Uξg ξµ X− g l=1 ∂hl ∂UνD−

1 ∂hl

∂Uξg ξµ

X

+

N

τ=1

τ∂nτ ∂UνD

−1∂nτ

∂Uξb ξµ η UXη+

g

l=1

el∂h l

∂UνD

−1∂hl

∂Uξb ξµ η UXη

+ N τ=1 g k=1

τek∂n τ

∂UνD−

1 d

dXq τ k−q

τ k

d dX

D−1w_kζµ UXζ

+ g l=1 g k=1 elek∂h l

∂UνD−

1 d

dXp l k−pkl

d dX

D−1wkζµ U ζ X.

(2.28)

have(∂hl_/∂Uξ_{)(z)=0; so, we also putD}−1_(d/dX)(∂hl_/∂Uξ_)=(∂hl_/∂Uξ_{). Now, using}

the same arguments, We get

ˆ

ΩνξJˆ_Fξµ=ˆI+

N

τ=1

τ∂nτ ∂Uν

D−1 d dX−ˆI

_∂nτ

∂Uξg ξµ

−

N

τ=1

τ∂nτ ∂UνD

−1_Jˆµξ F

∂nτ ∂Uξ

L(ᏹN,z)− g

l=1

el∂h l

∂UνD

−1_Jˆµξ F

∂hl ∂Uξ

L(ᏹN,z)

+

g

k=1  N

τ=1

τ∂nτ ∂Uνq

τ kD−1w

µ kζU

ζ X+

g

l=1

el∂h l

∂Uνp l kD−1w

µ kζU

ζ X  

(2.29)

(we used the skew-symmetry of the operator ˆJFµξfor the action from the right).

Again, using (2.24) and (2.26), we get

ˆ

Ωνξ_Jˆξµ F =ˆI+

N

τ=1

τ∂n τ

∂Uν

D−1 d dX−ˆI

_∂nτ

∂Uξg

ξµ_{, (2.30)}

that is,

ˆ

ΩνξJˆ_Fξµfµ(X)=fµ(X)−

N

τ=1

τ∂nτ ∂Uν(X)

∂nτ ∂Uξ(z)g

ξµ_{(z)fµ(z) (2.31)}

for anyfµ(X). From the definition ofᏲ0(z), we now obtain the part (II) of the theorem.

Remark2.3. We can also say that the operator ˆJFνξΩξµˆ is identity if it is acting from

the left onᏲ0(z)and from the right onᐂ0(z). This interpretation will be convenient later for the consideration of the recursion operator.

We also introduce the momentum functionalPgenerating the flow

Uν

T =UXν (2.32)

with respect to the general bracket (1.17) (with nondegenerategνµ_{(U )).}

Lemma2.4. AnyF-bracket (1.17) with nondegenerate tensorgνµ_{(U )has the local}

momentum operatorPgenerating the flow (2.32) on the spaceL(ᏹN_{, z). The functional}

Pcan be written in the form

P= +∞

−∞p(U ) dX= 1 2

+∞

−∞  N

τ=1

τnτnτ+

g

k=1

ekhkhk 

dX, (2.33)

where the functionsnτ_andhk_{correspond to the loop spaceL(ᏹ}N_{, z).}

Proof. We should here just prove the relation

∂p

∂Uν=ΩνξUˆ ξ

onL(ᏹN_{, z)according to part (I) ofTheorem 2.2. So, we have}

N

τ=1

τ∂n τ

∂UνD−

1∂nτ

∂UξU ξ X+

g

k=1

ek∂h k

∂UνD−

1∂hk

∂UξU ξ X

=

N

τ=1

τ∂nτ ∂Uνn

τ₊ g

k=1

ek∂h k

∂Uνh k₌ ∂

∂Uνp.

(2.35)

3. Theλ-pencils and the integrable hierarchies. We now consider the operator

ˆ

J_λνµ=Jˆ₍νµ₀₎+λJˆ₍νµ₁₎= gνµ₍₀₎+λgνµ₍₁₎! d dX+ b

νµ (0)η+λb

νµ (1)η

! UXη

+

g0

k=1

e(0)kw(ν0)kηU η XD−1w

µ (0)kζU

ζ X+λ

g1

k=1

e(1)kw(ν1)kηU η XD−1w

µ (1)kζU

ζ X

(3.1)

for ˆJ₍νµ₀₎and ˆJ₍νµ₁₎having form (1.18). We will call theλ-pencil (3.1) nondegenerate, for smallλ, if detgνµ₍₀₎(U )≠0.

We will admit here that the linear spacesᐃ(0)andᐃ(1), generated by the sets

wν (0)1ηU

η

X, . . . , w(ν0)g0ηU

η X

, wν (1)1ηU

η

X, . . . , w(ν1)g1ηU

η X

, (3.2)

can have a nontrivial intersectionᐂ. We introduce the basis inᐂ

ˆ

vν

1ηU η

X, . . . ,vˆdην U η X

, (3.3)

whered=dimᐂ, and consider the linear spaceᐃ, generated by all the flows from ᐃ(0)andᐃ(1)(dimᐃ=g0+g1−d) with basis

Ꮾ=wˆν (0)1ηU

η

X, . . . ,wˆ(ν0)(g0−d)ηU η X,vˆ1νηU

η

X, . . . ,vˆdην U η

X,wˆ(ν1)1ηU η

X, . . . ,wˆ(ν1)(g1−d)ηU η X

=w˜1νη(U )U η

X, . . . ,w˜gν0+g1−d,η(U )U η X

, (3.4)

where ˆwν

(0)kη and ˆw(ν1)sη are some linear combinations of operatorsw(0) and w(1),

respectively, and the flows corresponding to ˆw(0)k, ˆvm, and ˆw(1)sare all linearly

inde-pendent. The flows

ˆ

w₍ν_0)1ηUXη, . . . ,wˆ(ν0)(g0−d)ηU η X,vˆ1νηU

η

X, . . . ,vˆdην U η X

,

ˆ

v1νηU η

X, . . . ,vˆdην U η

X,wˆ(ν1)1ηU η

X, . . . ,wˆ(ν1)(g1−d)ηU η X

(3.5)

will then give bases inᐃ(0)andᐃ(1), respectively.

The nonlocal part of the bracket ˆJνµ_λ

g0

k=1

e(0)kw(ν0)kηU η XD−1w

µ (0)kζU

ζ X+λ

g1

s=1

e(1)sw(ν1)sηU η XD−1w

µ (1)sζU

ζ

X (3.6)

will correspond in our case to some quadratic formQks_λ (linear inλ),k, s=1, . . . , g0+

In our further consideration, the question ifQks_λ is nondegenerate onᐃforλ≠0 or not will be important, and we will mainly consider the pencils (3.1) such thatQksλ

is nondegenerate onᐃ.

We now formulate the properties of nondegenerate pencils (3.1) also satisfying the requirement of nondegeneracy of the formQks_λ forλ≠0 connected with the “canoni-cal” integrable hierarchies.

Theorem3.1. Consider the nondegenerate pencil (3.1) (detg₍νµ₀₎≠0) such that the

formQks_λ is also nondegenerate onᐃforλ≠0(small enough). Then,

(I) it is possible to introduce the local functionals

Nν(λ)= +∞

−∞n

ν_{(U , λ) dX, ν=1, . . . , N,}

P (λ)= +∞

−∞p(U , λ) dX,

Hk (0)(λ)=

+∞

−∞h

k

(0)(U , λ) dX, k=1, . . . , g0,

H₍s₁₎(λ)= _+∞

−∞h

s

(1)(U , λ) dX, s=1, . . . , g1,

(3.7)

which are the Casimirs, momentum operator and the Hamiltonian functions for

the flowswν

(0)kη(U )U η

Xandw(ν1)sη(U )U η

Xfor the bracketJˆ νµ

λ , respectively;

(II) all the functionsnν_{(U , λ),p(U , λ),h}k

(0)(U , λ), andh s

(1)(U , λ)are regular atλ→

0and can be represented as regular series

nν(U , λ)=

+∞

q=0

nνq(U )λq, p(U , λ)=

+∞

q=0

pq(U )λq,

hk₍₀₎(U , λ)= +∞

q=0

hk_(0),q(U )λq_{, h}s

(1)(U , λ)=

+∞

q=0

hs

(1),q(U )λq.

(3.8)

Moreover, we can choose these functionals in such a way that

∂pq

∂Uµ(z)=0,

∂hk_(0),q ∂Uµ (z)=0,

∂hs_(1),q

∂Uµ (z)=0, q≥0, ∂nν

q

∂Uµ(z)=0, q≥1,

∂nν0

∂Uµ(z)=e ν µ,

(3.9)

wheredeteν µ≠0;

(III) the integralsNν_{(0),P (0), H}k

(0)(0), and H s

(1)(0)are the Casimirs, momentum

operator, and the Hamiltonian functions for the flows w₍ν_0)kη(U )UXη and

wν

(1)sη(U )U η

X with respect toJˆ νµ

(0), while the flows generated by the functionals

Fq= _+∞

−∞fq(U ) dX (3.10)

are connected by the relation

ˆ

J₍νξ₀₎∂f(q+1) ∂Uξ = −Jˆ

νξ (1)

∂fq

for any functionalF (λ)from the set (3.7). All the functionalsFq given by the

expansions (3.8) generate the local flows and commute with each other with

respect to both bracketsJ(ˆ0)andJ(ˆ1).

Proof. We now putλ >0 and write ˆJ_λνµin the form

ˆ

J_λνµ= gνµ₍₀₎+λgνµ₍₁₎! d dX+ b

νµ (0)η+λb

νµ (1)η

! UXη

+

g0

k=1

e(0)kw(ν0)kηU η XD−1w

µ (0)kζU

ζ X

+

g1

k=1

e(1)k "

λwν (1)kηU

η XD−1

"

λw₍µ_1)kζUXζ.

(3.12)

In the case of the nondegenerate formQks_λ onᐃ, we can consider the corresponding embedding ofᏹN _toRN+g0+g1 _{depending onλ. Indeed, all the flows ˜ws from the set} Ꮾwill in this case satisfy conditions

g_λνξw˜_sξµ =gµξ_λ w˜ν

sξ, ∇νw˜sηµ = ∇ηw˜sν,µ R_ηζνµ=

k,s

˜

w_kηνQks_λw˜_sζµ −w˜_kηµQks_λw˜_sζν (3.13)

for nondegenerategλµξaccording to Ferapontov theorem.

Since the operatorswν

(0)kη(U )andw(ν1)sη(U )are just linear combinations of ˜wnην (U )

(with constant coefficients) and the formQks

λ coincides with the nonlocal part of (3.12),

we will have the corresponding Gauss and Petersson-Codazzi equations for these flows and the curvature tensorRνµηζ. So, for nondegenerateg

νµ

λ , we will get the local

embed-ding ofᏹN_toRN+g0+g1_{(actually, to some subspace}RN+g0+g1−d⊂RN+g0+g1_{) depending} onλ.

Since the embedding is defined up to the Poincaré transformation inRN+g0+g1_{, we} can choose, at allλ, the coordinatesZI_{,I=1, . . . , N+g}

0+g1in such a way that (1) allZI_{=0 at the pointz∈ᏹ}N_{, and the metricG}IJ_inRN+g0+g1_{has the form}

GIJ=EIδIJ, (3.14)

whereEN+k_=e(

0)k,k=1, . . . , g0,EN+g0+s=e(1)s,s=1, . . . , g1;

(2) the firstNcoordinatesZν_{,ν=1, . . . , Nare tangential toᏹ}N_{at the pointz∈ᏹ}N

at allλ;

(3) the lastg0+g1coordinates are orthogonal toᏹN at the pointz, and the Wein-garten operators

w(ν0)1η(U ), . . . , w(ν0)g0η(U ), "

λw(ν1)1η(U ), . . . , "

λw(ν1)g1η(U ) (3.15)

correspond to the parallel vector fieldsN(0)(k)(U ),N(1)(s)(U )in the normal bun-dle such that

(−EN+k_{stays at the positionN+k)k=1, . . . , g0,}

N(1)(k)(z)=

0, . . . ,0,−EN+g0+s_{,0, . . . ,0}T

(3.17)

(−EN+g0+s _{stays at the positionN+g}

0+s)s=1, . . . , g1.

So, according to [25], the restriction of the firstNcoordinatesZ1_{, . . . , Z}N _{gives the}

Casimirs of bracket (3.12) onL(ᏹN_{, z)}

˜

Nν(λ)= +∞

−∞n˜

ν_{(U , λ) dX=} +∞

−∞Z

ν_|

ᏹN(U , λ) dX, ν=1, . . . , N, (3.18)

while the restrictions of the lastg0+g1coordinates give the Hamiltonian functions for the flowsw₍ν_0)kηUXη

˜

Hk (0)(λ)=

+∞

−∞ ˜

hk

(0)(U , λ) dX= +∞

−∞Z

N+k_|

ᏹN(U , λ) dX, k=1, . . . , g0 (3.19)

and√λwν (1)sηU

η X

˜

H(s1)(λ)= +∞

−∞ ˜

hs(1)(U , λ) dX= +∞

−∞Z

N+g0+s_|

ᏹN(U , λ) dX, s=1, . . . , g1. (3.20)

Remark3.2. Forλ <0, the signature ofRN+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 theλ-dependence of the functions ˜nν_{(U , λ), ˜h}k

(0)(U , λ), and ˜h s (1)(U , λ)

atλ→0.

ConsiderNtangential vectors toᏹN_{, corresponding to the coordinate system{U}ν_},

that is,

e(ν)=

∂Z1

∂Uν, . . . ,

∂ZN+g0+g1

∂Uν 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), andN(1)(s)(U , λ)onᏹN:

de(ν)=Γνη(U , λ)µ e(µ)dUη− g0

k=1

EN+k_{gνµ(U , λ)w}µ

(0)kη(U , λ)N(0)(k)dUη

−"λ g1

s=1

EN+g0+sgνµ(U , λ)w(µ1)sη(U , λ)N(1)(s)dUη,

dN(0)(k)=w(ν0)kη(U , λ)e(ν)dU η_,

dN(1)(s)= "

λw(ν1)sη(U , λ)e(ν)dUη,

(3.22)

So, for any curveγ(t)onᏹN_{(γ(0)=z), we have the evolution system fore(ν)(t),}

N(0)(k)(t), andN(1)(s)(t)having the general form

d dt

   

e(ν)(t)

N(0)(k)(t) N(1)(s)(t)    =

   

∗(t, λ) ∗(t, λ) √λ∗(t, λ)

∗(t, λ) 0 0

√

λ∗(t, λ) 0 0

       

e(ν)(t)

N(0)(k)(t) N(1)(s)(t)   

, (3.23)

where all∗(t, λ)are regular atλ→0 (matrix-) functions ofλ.

The formal solution of (3.23) can be written as the chronological exponent

Texp t

0 ˆ

H(t)dt

=ˆ_I++∞

n=1 1

n! t1

0 ··· tn

0 T _ˆ

Ht1

···_Hˆ_{tndt1···dtn (3.24)}

applied to the initial datae(ν)(0),N(0)(k)(0), andN(1)(s)(0), where ˆH(t)is the matrix of system (3.23).

It is now easy to verify that, for anyn≥1, we have

THˆt1

···_Hˆ_tn=    

∗(t, λ) ∗(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 atλ→0.

For the densities of Casimirs (3.18) and Hamiltonian functions (3.19) and (3.20), we can now write the equations

dn˜ν_{(t, λ)}

dt =

U_tµe(µ)(t, λ)ν=ν_Uµ

te(µ)(t, λ),eν(0)

dh˜k₍₀₎(t, λ) dt = −

Utµe(µ)(t, λ),N(0)(k)(0)

dh˜s (1)(t, λ)

dt = −

U_tµe(µ)(t, λ),N(1)(s)(0)

(3.26)

along the same curveγ(t)onᏹN_{. Sinceγ(t)is just the arbitrary curve, we have}

˜

nν_{(U , λ)= ∗(U , λ), h˜}k

(0)(U , λ)= ∗(U , λ), h˜ s

(1)(U , λ)= "

λ∗(U , λ), (3.27)

where∗(U , λ)are regular atλ→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)=h˜k(0)(U , λ), hs(1)(U , λ)=

1 √

λ

˜

hs

(1)(U , λ) (3.28)

According to the geometric construction, we have

∂hk (0)(U , λ)

∂Uµ ##

##_z≡0, ∂h s (1)(U , λ)

∂Uµ ##

##_z≡0 (3.29)

and also

∂p(U , λ) ∂Uµ

##

##_z≡0. (3.30)

Since the functions ˜nν_{(U , λ)locally give the coordinate system onᏹ}N _{at everyλ,}

we have

det ∂n˜

ν

∂Uµ(z, λ)

≠0, (3.31)

and we can put

nν_{(U , λ)=}∂n˜ν ∂Uξ(z,0)

∂Uξ ∂n˜η(z, λ)n˜

η_{(U , λ) (3.32)}

such that

∂nν_{(U , λ)} ∂Uµ

## ##_z≡∂nν

∂Uµ(z,0)=

e(µ)ν, ν=1, . . . , N, (3.33)

wheree(µ)are the tangent vectors at the pointzforλ=0,nν(U ,0)=n˜ν(U ,0). So, we

get parts (I) and (II) of the theorem.

For part (III), we first prove here the following lemma.

Lemma3.3. Under the conditions formulated inTheorem 3.1the functionals

Nν p=

+∞

−∞n

ν

p(U ) dX, Pq= +∞

−∞pq(U ) dX,

H₍k_0)l= _+∞

−∞h

k

(0)l(U ) dX, H s (1)t=

_+∞

−∞h

s

(1)t(U ) dX,

(3.34)

p, q, l, t=0,1, . . . ,generate the local flows with respect to bothJ(ˆ0) andJ(ˆ1)and com-mute with the functionalsN₀µ,P0,H(m0)0, andHn(1)0with respect to the bracketJ(ˆ0).

Proof. Since the functionalsN₀µ are the annihilators of ˆJ(0), they commute with

all other functionals with respect to ˆJ(0) on L(ᏹN, z). Also, from the translational

invariance of all functionalsNν

p,Pq,H(k0)l, andH s

(1)t, we get that they also commute

with the momentum operatorP0of ˆJ(0)with respect to ˆJ(0).

We then know that the functionalsNν_{(λ),P (λ),H}k

(0)(λ), andH(s1)(λ)generate the

local flows with respect to ˆJλ. In the case of nondegenerate formQks_λ (onᐃ), this means that they are the conservation laws for all the flows ˜wν

nη(U )U η

X introduced in (3.4).

So, they are the conservation laws for the flows w(ν0)kη(U )U η

X and w(ν1)sη(U )U η X and

generate the local flows with respect to ˆJ(0)and ˆJ(1). Now, since the flowsw(ν0)mη(U )U η X

andw(ν1)nη(U )U η

Xare generated by the functionalsHm(0)0andH(n1)0with respect to ˆJ(0),

we get that all Nν

For the commutativity of allNν

p,Pq,H(k0)l, andH s

(1)twith respect to both brackets, we

can now use just the common approach for the bi-Hamiltonian systems [20] writing

δFqJ(ˆ0)δGk= −δFq−1J(ˆ1)δGk=δFq−1J(ˆ0)δGk+1= ··· =δF0J(ˆ0)δGk+q=0 (3.35)

for any two functionalsFqandGkfrom the set (3.34) (the same for the bracket ˆJ(1)).

Remark3.4. We point out here that the requirement of nondegeneracy of the form

Qks_λ onᐃis important, and, in general,Theorem 3.1is not true without it. As example, we consider here the Poisson pencil ˆJλ=_J(ˆ_0)+λJ(ˆ_1)where

ˆ

J(0)=

1 0

0 −1

d dX+

U1 _U2 −U2 _−U1

U1

X UX2

D−1 1 0 0 1 U1 X UX2 + 1 0 0 1 UX1 U2

X

D−1

U1 _U2 −U2 _−U1

UX1 U2

X

,

ˆ

J(1)= 0 1 1 0 d dX+ 0 0

2U1 ₀

UX1 U2 X D−1 1 0 0 1 UX1 U2 X + 1 0 0 1 U1 X UX2

D−1

0 0

2U1 ₀

U1

X UX2

.

(3.36)

We have here the three-dimensional spaceᐃgenerated by the flows

˜

w1νηU η X=

U1 _U2 −U2 _−U1

U1

X UX2

, w˜2νηU η X=

U1

X UX2

,

˜

wν

3ηU η X=

0 0

2U1 ₀

UX1 U2

X

.

(3.37)

Operator ˆJλcan be written as

ˆ

Jλ=

1 λ

λ −1

d dX+

U1 _U2

−U2_+2λU1 _−U1

U1

X UX2

D−1

U1

X UX2

+

UX1 U2

X

D−1

U1 _U2

−U2_+2λU1 _−U1

UX1 U2

X

,

(3.38)

and the form

Qλ=    

0 1 0 1 0 λ

0 λ 0   

 (3.39)

is degenerate onᐃ. The metric

gνµ_λ (U )=

1 λ

λ −1

here is just the flat metric onR2_{, detg}νµ

λ ≠0, and it is easy to check the equations

gνξλ W µ

1ξ(λ)=g µξ

λ W1νξ(λ), g νξ λ W

µ

2ξ=g µξ

λ W2νξ, (3.41)

where

W₁µ_ξ(λ)=

U1 _U2

−U2_+2λU1 _−U1

, W₂µ_ξ(λ)=

1 0 0 1

. (3.42)

Also,

∂νW₁ξ_µ(λ)=∂µW₁ξ_ν(λ), ∂νW₂ξ_µ=∂µW₂ξ_ν (3.43)

in the flat coordinatesU1_,U2_{, and}

R1212=W111(λ)W222+W211W122(λ)−W112(λ)W221 −W212W121(λ) =W1

11(λ)+W122(λ)≡0;

(3.44)

so, ˆJλrepresents a Poisson bracket of Ferapontov type for allλ.

Bothgνµ₍₀₎andg₍νµ₁₎are nondegenerate in this case. However, the flow ˜w1νηU η X is not

Hamiltonian with respect to ˆJ(1), and ˜w3νηU η

X is not Hamiltonian with respect to ˆJ(0)

as follows from

g₍νξ₀₎w˜₃µ_ξ≠gµξ₍₀₎w˜₃ν_ξ, g₍νξ₁₎w˜₁µ_ξ≠g₍µξ₁₎w˜₁ν_ξ. (3.45)

It is also easy to check that the flows ˜wν

1ηU η

Xand ˜w3νηU η

Xdo 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 ˆJ(0)and consider the recursion

operator ˆRν

µ=Jˆ(ντ1)Ω(ˆ0)τµunder the assumptions ofTheorem 3.1.

We put vτ_{(U , λ) =n}τ_{(U , λ), τ = 1, . . . , N, v}N+k_{(U , λ) =h}k

(0)(U , λ), k =1, . . . , g0, v0s(U )≡vs(U ,0),E(τ0)=τ,τ=1, . . . , N, andE

N+k

(0) =e(0)k,k=1, . . . , g0(wherenτ(U , λ) and hk

(0)(U , λ) are the functions fromTheorem 3.1) and write the symplectic form

ˆ

Ω(0)τµas

ˆ

Ω(0)τµ= N+g0

k=1

E₍k₀₎∂v k

0

∂UτD−

1∂v0k

∂Uµ. (4.1)

We can also write the operator ˆJ₍νµ₀₎as

ˆ

Jνµ₍₀₎=gνµ₍₀₎ d dX+b

νµ (0)ηU

η X+

N+g0

k=1

E₍k₀₎

ˆ

J₍ντ₀₎∂v k

0

∂Uτ

D−1

ˆ

Jµσ₍₀₎ ∂v k

0

∂Uσ

For ˆRν

µ, we have the expression

ˆ

Rµν=Jˆ(ντ1)Ω(ˆ 0)τµ= N+g0

k=1

Ek(0)gντ(1) ∂vk

0

∂Uτ

X D−1∂v

k

0

∂Uµ

+

N+g0

k=1

Ek (0)gντ(1)

∂v0k

∂Uτ ∂v0k

∂Uµ

+

N+g0

k=1

Ek₍₀₎bντ_(1)ηUXη ∂vk

0

∂UτD−

1∂v0k

∂Uµ

+

g1

s=1

N+g0

k=1

e(1)sE(k0)w(ν1)sηU η

XD−1w(τ1)sζU ζ X

∂v0k

∂UτD

−1∂v0k

∂Uµ,

(4.3)

where

w₍τ_1)sζUXζ ∂vk

0

∂Uτ ≡

Qks

X= d dXQ

k s−Qks

d

dX (4.4)

for someQk

s(U )according toTheorem 3.1.

We normalize the functionsQk

s(U )such thatQks(z)=0, and we have

···D−1 d dXQ

k

s··· = ···Qks··· (4.5)

onL(ᏹN_{, z).}

Also,d/dX D−1≡ˆIonL(ᏹN_{, z), and}

N+g0

k=1

Ek₍₀₎gντ (1)

∂v0k

∂Uτ

X D−1∂v

k

0

∂Uµ+ N+g0

k=1

E₍k₀₎bντ (1)ηU

η X

∂v0k

∂UτD

−1∂v0k

∂Uµ

+

g1

s=1

N+g0

k=1

e(1)sEk(0)w(ν1)sηU η XQskD−1

∂vk

0

∂Uµ

=

N+g0

k=1

Ek (0)

ˆ

Jντ (1)

∂v0k

∂Uτ

D−1∂v k

0

∂Uµ.

(4.6)

Also, using the relation

N+g0

k=1

Ek (0)

∂v0k

∂UµQ k

s =Ω(ˆ 0)µτw(τ1)sηU η X=

∂hs(1)(U ,0) ∂Uµ =

∂hs(1)0(U )

∂Uµ (4.7)

onL(ᏹN_{, z)(since∂h}s

(1)q/∂Uµ(z)=0,q≥0), we can write

ˆ

Rν

µ=gντ(1)g(0)τµ+ N+g0

k=1

Ek (0)

ˆ

Jντ (1)

∂v0k

∂Uτ

D−1∂v k

0

∂Uµ− g1

s=1

e(1)sw(ν1)sηU η XD−1

∂hs (1)0

∂Uµ

=Vµν− N+g0

k=1

E₍k₀₎

ˆ

J₍ντ₀₎∂v k

1

∂Uτ

D−1∂v k

0

∂Uµ− g1

s=1

e(1)s

ˆ

J₍ντ₀₎∂h s (1)0

∂Uτ

D−1∂h s (1)0

∂Uµ ,