Compatible Dubrovin-Novikov Hamiltonian operators, Lie derivative and integrable systems of hydrodynamic type

11 Pages·0.14 MB·English

by  O. I. Mokhov

Compatible Dubrovin–Novikov
Hamiltonian operators, Lie derivative and
integrable systems of hydrodynamic type1
2
0
0
2
O. I. Mokhov

n
a
J

9
2
1 Introduction

]
G
In the present paper, we prove that a local Hamiltonian operator of hydrodynamic type
D Kij (Dubrovin–NovikovHamiltonianoperator[1])iscompatiblewithanondegeneratelocal
1
. Hamiltonian operator of hydrodynamic type Kij if and only if the operator Kij is locally
h 2 1
t the Lie derivative of the operator K2ij along a vector field in the corresponding domain of
a
localcoordinates. Thisresultgives,firstofall,aconvenientgeneralinvariantcriterionofthe
m
compatibilityfortheDubrovin–NovikovHamiltonianoperatorsand,inaddition,thisgivesa
[
naturalinvariantdefinitionoftheclassofspecialflatmanifoldscorrespondingtoalltheclass

1 ofcompatibleDubrovin–NovikovHamiltonianoperators(theFrobenius–Dubrovinmanifolds
v naturally belong to this class of flat manifolds). There is an integrable bi-Hamiltonian hi-
1
erarchy corresponding to every flat manifold of this class. The integrable systems are also
8
studiedinthepresentpaper. Thisclassofintegrablesystemsisexplicitlygivenbysolutions
2
1 of the nonlinear system of equations, which is integrated by the method of inverse scatter-
0 ing problem. The corresponding results on compatible nonlocal Hamiltonian operators of
2 hydrodynamictype andotherHamiltonianandsymplectic differential-geometrictype oper-
0
ators related by the Lie derivative and the results on the corresponding to them integrable
/
h bi-Hamiltonian systems will be published in other our works.
t Recall that an operator Kij[u(x)] is called Hamiltonian if it defines a Poisson bracket
a
m (skew-symmetric and satisfying the Jacobi identity)
:
v δI δJ
{I,J}= Kij[u(x)] dx (1.1)
i δui(x) δuj(x)
X
Z
r for arbitrary functionals I[u(x)] and J[u(x)] on the space of functions (fields) u(x) =
a
{ui(x), 1 ≤ i ≤ N}, where u1,...,uN are local coordinates on a certain given smooth
N-dimensionalmanifoldM. It is obviousthat anyHamiltonianoperatoralwaysbehavesas
1This work was supported by the Alexander von Humboldt Foundation (Germany), the RussianFoun-
dationforBasicResearch(grantNo. 99–01–00010) andtheINTAS(grantNo. 99–1782). Theworkwillbe
published in Proceedings of the International Conference “Nonlinear Evolution Equations and Dynamical
Systems,”Cambridge(England),July24–30, 2001.
1
a contravariant two-valent operator tensor field with respect to arbitrary local changes of
coordinates ui =ui(v1,...,vN), 1≤i≤N, on the manifold M:
∂vs ∂vr
Ksr[v(x)]= Kij[u(v(x))]◦ , (1.2)
∂ui ∂uj
where the symbol ◦ meansethe operator multiplication (it is important to note that in
contrast to the usual tensor fields the components of operators in (1.2) multiply only in
the indicated order). Accordingly, the classical tensor constructions, in particular, the Lie
derivative along a vector field, are applied to Hamiltonian operators (see, for example, the
monograph [2] and also section 3 below).
Hamiltonian operators are called compatible if any their linear combination is also a
Hamiltonian operator (Magri, [3]; see also [2]). In the theory of Hamiltonian operators,
thereisthefollowinggeneralfact(see,forexample,[2]),whichisimportantforapplications:
if the secondcohomologygroupofthe correspondingcomplexofformalvariationalcalculus
istrivial,thenfromthecompatibilityoftwoHamiltonianoperatorsKij andKij,whereKij
1 2 2
isaninvertibleoperator,itfollowsthatthereexistsaformalvectorfieldX[u(x)]depending,
generallyspeaking,inanarbitraryandnonlocalway,onthefieldsu(x)andtheirderivatives
and such that K1 =LXK2, where LXK is the Lie derivative of a Hamiltonian operator K
along a formal vector field X. Moreover,if an operator K is Hamiltonian and, in addition,
the operator L K, where X is a certain arbitrary formal vector field, is also Hamiltonian,
X
then the Hamiltonian operators K and L K are always compatible. If an operator K is
X
Hamiltonian and, besides, L2 K = 0, where X is a certain vector field, then the operator
X
L K is always Hamiltonian, and consequently the Hamiltonian operators K and L K
X X
are compatible in this case. This beautiful construction plays the very important role in
applications and the corresponding class of compatible Hamiltonian operators (the pairs of
operators of the form Kij, (L K)ij, such that (L2 K)ij =0) deserves a separate study.
X X
An important special partial case of this construction arose afterwards in the Dubrovin
theoryofFrobeniusmanifolds(thequasihomogeneouscompatiblenondegenerateDubrovin–
NovikovHamiltonianoperators)[4]–[6]. ThestudyofgeneralcompatibleDubrovin–Novikov
Hamiltonian operators of this class, that is, of the form Bij and (L B)ij and such that
X
L2 B =0, was started by the present author and Fordy in [7], where the first classification
X
results were obtained.
2 Compatible Dubrovin–Novikov
Hamiltonian operators
Recall that a Hamiltonian operator given by an arbitrary matrix homogeneous first-order
ordinary differential operator, that is, a Hamiltonian operator of the form
d
Pij[u(x)]=gij(u(x)) +bij(u(x))uk, (2.1)
dx k x
2
is called a local Hamiltonian operator of hydrodynamic type or Dubrovin–Novikov Hamilto-
nianoperator[1]. Thisdefinitiondoesnotdependonthechoiceoflocalcoordinatesu1,...,uN
on the manifold M, since it follows from (1.2) that the form of operator (2.1) is invariant
with respect to local changes of coordinates on M. Operator (2.1) is called nondegenerate
if det(gij(u)) 6≡ 0. If det(gij(u)) 6≡ 0, then operator (2.1) is Hamiltonian if and only if 1)
gij(u) is an arbitrary contravariant flat pseudo-Riemannian metric (a metric of zero Rie-
mannian curvature), 2) bij(u)=−gis(u)Γj (u), where Γj (u) is the Levi-Civita connection
k sk sk
generatedbythemetricgij(u)(theDubrovin–Novikovtheorem[1]). Inparticular,itfollows
fromtheDubrovin–NovikovtheoremthatforanynondegeneratelocalHamiltonianoperator
of hydroodynamic type there always exist local coordinates v1,...,vN (flat coordinates of
the metric gij(u)) in which all the coefficients of the operator are constant:
gij(v)=ηij = const, Γi (v)=0, bij(v)=0, (2.2)
jk k
that is the corresponding Poissonbracket has the form
e e e
δI d δJ
{I,J}= ηij dx, (2.3)
δvi(x) dxδvj(x)
Z
where (ηij) is a nondegenerate symmetric constant matrix:
ηij =ηji, ηij =const, det(ηij)6=0. (2.4)
Moreover, it immediately follows from the Dubrovin–Novikov theorem that any two
nondegenerateDubrovin–NovikovHamiltonianoperatorsPij[u(x)]andPij[u(x)]generated
1 2
by flat contravariant metrics gij(u) and gij(u) respectively are compatible if and only if 1)
1 2
any linear combination of these flat metrics
gij(u)=λ1g1ij(u)+λ2g2ij(u), (2.5)
where λ1 and λ2 are arbitrary constants for which det(gij(u)) 6≡ 0, is also a flat metric, 2)
the coefficients of the corresponding Levi-Civita connections are related by the same linear
formula:
Γikj(u)=λ1Γi1j,k(u)+λ2Γi2j,k(u). (2.6)
The derived purely differential-geometric conditions (2.5) and (2.6) on flat metrics gij(u)
1
and gij(u) define a flat pencil of metrics [4]. In this case, we shall also say that the flat
2
metrics gij(u) and gij(u) are compatible (see [8]).
1 2
SotheproblemofdescriptionforcompatiblenondegeneratelocalHamiltonianoperators
of hydrodynamic type is the purely differential-geometric problem of description of general
flat pencils of metrics (see [4]). In [4], [5] Dubrovin considered all the tensor relations for
the general flat pencils of metrics.
First of all, let us introduce the necessary notation. Let ∇1 and ∇2 be the operators of
covariantdifferentiationgivenbythe Levi-CivitaconnectionsΓij (u)andΓij (u)generated
1,k 2,k
3
by the metrics gij(u) and gij(u) respectively. The indices of the covariant differentials
1 2
are raised and lowered by the corresponding metrics: ∇i1 = g1is(u)∇1,s, ∇i2 = g2is(u)∇2,s.
Consider the tensor
∆ijk(u)=gis(u)gjp(u) Γk (u)−Γk (u) , (2.7)
1 2 2,ps 1,ps
(cid:0) (cid:1)
introduced by Dubrovin in [4], [5].
Theorem 2.1 (Dubrovin [4], [5]) If metrics gij(u) and gij(u) form a flat pencil, then
1 2
there exists a vector field fi(u) such that the tensor ∆ijk(u) and the metric gij(u) have the
1
form
∆ijk(u)=∇i∇jfk(u), (2.8)
2 2
gij(u)=∇ifj(u)+∇jfi(u)+cgij(u), (2.9)
1 2 2 2
where c is a certain constant, and the vector field fi(u) satisfies the equations
∆ij(u)∆sk(u)=∆ik(u)∆sj(u), (2.10)
s l s l
where
∆ikj(u)=g2,ks(u)∆sij(u)=∇2,k∇i2fj(u), (2.11)
and
(g1is(u)g2jp(u)−g2is(u)g1jp(u))∇2,s∇2,pfk(u)=0. (2.12)
Conversely, for the flat metric gij(u) and the vector field fi(u) that is a solution of the
2
system of equations (2.10) and (2.12), the metrics gij(u) and (2.9) form a flat pencil.
2
In[6]DubrovinprovedthatthetheoryofFrobeniusmanifoldsconstructedhimin[4](the
Frobenius manifolds correspond to two-dimensional topological field theories) is equivalent
to the theory of quasihomogeneous compatible nondegenerate Dubrovin–Novikov Hamilto-
nian operators or, in other words, quasihomogeneous flat pencils of metrics.
A flat pencil of metrics generated by flat metrics gij(u) and gij(u) is called quasihomo-
1 2
geneous of degree d if there exists a function τ(u) such that for the vector fields
∂τ ∂τ
E =∇1τ, Ei =g1is(u)∂us, e=∇2τ, ei =g2is(u)∂us, (2.13)
the following conditions are satisfied:
1) [e,E]=e,
2) LEg1 =(d−1)g1,
3) Leg1 =g2,
4) Leg2 =0,
where L g is the Lie derivative of a metric g along a vector field X (Dubrovin, [6]). Note
X
that in the quasihomogeneous case we always have LeQ1 = Q2, LeQ2 = 0, L2eQ1 = 0, for
4
the corresponding quasihomogeneous compatible nondegenerate Dubrovin–Novikov Hamil-
tonianoperatorsQij[u(x)]andQij[u(x)](moreindetailaboutthisimportantclassofquasi-
1 2
homogeneous compatible nondegenerate Dubrovin–NovikovHamiltonian operatorssee [7]).
In the present author’s work [9], a necessary for further, explicit and simple ctiterion of
compatibility for two local Poisson brackets of hydrodynamic type was stated in a suitable
forus way,thatis, itis shownwhatanexplicitformissufficientandnecessaryfortwolocal
Hamiltonian operators of hydrodynamic type to be compatible (see also theorem 2.1).
Lemma 2.1 ([9]) (explicit criterion of compatibility for local Poisson brackets of
hydrodynamictype)AnylocalPoissonbracketofhydrodynamictype{I,J}2iscompatible
with the constant nondegenerate Poisson bracket (2.3) if and only if it has the form
δI ∂hj ∂hi d ∂2hj δJ
{I,J}2 = δvi(x) ηis∂vs +ηjs∂vs dx +ηis∂vs∂vkvxk δvj(x)dx, (2.14)
Z (cid:18)(cid:18) (cid:19) (cid:19)
where hi(v), 1 ≤ i ≤ N, are smooth functions defined in a certain domain of local coordi-
nates.
We do not require in lemma 2.1 that the Poisson bracketof hydrodynamic type {I,J}2
is nondegenerate. Besides, it is important to note that this statement is local.
3 Lie derivative and Dubrovin–Novikov
Hamiltonian operators
Let Kij[u(x)] be an arbitrary Hamiltonian operator, ξ(u) = {ξi(u), 1 ≤ i ≤ N} is an
arbitrary smooth vector field on the manifold M. The Lie derivative of the Hamiltonian
operator Kij[u(x)] (just as any contravarianttwo-valentoperator tensor field of type (1.2))
along the vector field ξ(u) is the operator
d ∂ξi ∂ξj
(L K)ij[u(x)]= δi −t Ksr[u(x)+tξ(u(x))]◦ δj −t , (3.1)
ξ dt s ∂us r ∂ur
(cid:18) (cid:20)(cid:18) (cid:19) (cid:18) (cid:19)(cid:21)(cid:19)(cid:12)t=0
(cid:12)
which also always behaves as contravariant two-valent operator tensor field o(cid:12)f type (1.2)
(cid:12)
with respect to arbitrary local changes of coordinates on the manifold M.
For the Lie derivative of a local Hamiltonian operator of hydrodynamic type Pij[u(x)]
along the vector field ξ(u) we get
∂gij ∂ξi ∂ξj d
(L P)ij[u(x)]= ξs −gsj −gis
ξ ∂us ∂us ∂us dx
(cid:18) (cid:19)
∂bij ∂ξi ∂ξj ∂ξs ∂2ξj
+ ξs k −bsj −bis +bij −gis uk. (3.2)
∂us k ∂us k ∂us s ∂uk ∂us∂uk x
!
5
Theorem 3.1 Any Dubrovin–Novikov Hamiltonian operator Pij is compatible with a non-
1
degenerate Dubrovin–Novikov Hamiltonian operator Pij if and only if there locally exists a
2
vector field ξ(u) such that
P1ij =(LξP2)ij. (3.3)
In a different form, which is not connected with the Lie derivative, the general com-
patibility conditions for the Dubrovin–Novikov Hamiltonian operators were shown in [4],
[8]–[12] (see also section 2 above).
Question. If there always globally exists such a smooth vector field on every flat
manifold M, on the loop space of which are globally defined compatible Dubrovin–Novikov
Hamiltonian operators Pij and Pij? If not always, then it is necessary to investigate the
1 2
corresponding differential-geometric and topology obstructions and also to single out and
study all the flat manifolds on which there globally exists a vector field such that the Lie
derivativeoftheDubrovin–NovikovHamiltonianoperatorgeneratedbytheflatmetricofthe
manifoldalongthevectorfieldisalsoaHamiltonianoperator. Inparticular,itisinteresting
tostudyallsuchvectorfieldsinRN orRN ,andalsoindomainsofthesespaces. Locally,
k,N−k
any such vector field is a solution of the nonlinear system of equations integrable by the
methodofinversescatteringproblem(see[10]–[12])andgeneratesintegrablebi-Hamiltonian
systems of hydrodynamic type (we shall consider them in the next section).
Here let us prove theorem 3.1. Let Dubrovin–Novikov Hamiltonian operators Pij[u(x)]
1
andPij[u(x)]becompatibleandtheoperatorPij[u(x)]benondegenerate. Considerthelocal
2 2
coordinates v = (v1,...,vN) in which the nondegenerate Dubrovin–Novikov Hamiltonian
operator Pij[u(x)] is reduced to the constant form (2.3):
2
d
Pij[v(x)]=ηij , (3.4)
2
dx
where (ηij) is an arbitrary nondegenerate constant symmetric matrix: det(ηij) 6= 0, ηij =
const, ηij = ηji (there always exist such local coordinates by the Dubrovin–Novikov theo-
rem). Inthesecoordinates,accordingtoformula(3.2),forthe Liederivativeofthe operator
Pij[v(x)] along an arbitrary vector field ξ(v)=(ξ1(v),...,ξN(v)) we get
2
∂ξi ∂ξj d ∂2ξj
(LξP2)ij[v(x)]= −ηsj∂vs −ηis∂vs dx −ηis∂vs∂vkvxk. (3.5)
(cid:18) (cid:19)
Accordingto lemma 2.1 any Dubrovin–NovikovHamiltonian operatorPij[v(x)] compatible
1
with the Hamiltonian operator (3.4) must have namely such the form (3.5) in the local
coordinates v = (v1,...,vN) (in formula (2.14) hi(v) = −ξi(v), 1 ≤ i ≤ N). Thus there
exists a vector field ξi(v), 1≤i≤N, such that
P1ij[v(x)]=(LξP2)ij[v(x)]. (3.6)
Then by virtue of tensor invariance of the Lie derivative formula (3.3) is valid in arbitrary
local coordinates.
6
Conversely, let two Dubrovin–Novikov Hamiltonian operators Pij[u(x)] and Pij[u(x)]
1 2
arerelatedbyformula(3.3)andtheoperatorPij[u(x)]isnondegenerate. Then,inthelocal
2
coordinates v = (v1,...,vN) in which the nondegenerate Dubrovin–Novikov Hamiltonian
operatorPij[u(x)]isreducedtotheconstantform(3.4),theHamiltonianoperatorPij[v(x)]
2 1
has the form (3.5). According to lemma 2.1 a pair of Hamiltonian operators Pij[v(x)] and
1
Pij[v(x)] of such the form are necessarily compatible. Thus theorem 3.1 is proved.
2
Now the special class of flat manifolds which corresponds to the class of all com-
patible Dubrovin–Novikov Hamiltonian operators and generalizes the class of Frobenius–
Dubrovin manifolds is naturally singled out. Theorem 3.1 gives the following invariant
definition of these manifolds. We consider the manifolds (M,g,ξ), where M is a flat mani-
foldwithaflatmetricg equippedalsowithavectorfieldξ suchthat,forthenondegenerate
Dubrovin–Novikov Hamiltonian operator Pij[u(x)] generated by the flat metric g, the op-
erator (L P)ij[u(x)] is also Hamiltonian (that is, the operator defines a Poissonbracket on
ξ
the corresponding loop space of the manifold M). Generally speaking, for the description
of all compatible Dubrovin–Novikov Hamiltonian operators it is necessary to consider the
followingmore weakcondition: anexistence ofsuchvectorfield locally,in a neighbourhood
of every point of the manifold. Locally, such vector fields are described by the nonlinear
system integrable by the method of inverse scattering problem (see [10]–[12]).
4 Class of integrable bi-Hamiltonian systems
of hydrodynamic type
AnarbitrarypairofcompatibleDubrovin–NovikovHamiltonianoperatorsPij andPij,one
1 2
of which(let us assume Pij) is nondegenerate,can be reduced to the following specialform
2
by a local change of coordinates:
d
Pij[v(x)]=ηij , (4.1)
2
dx
∂hj ∂hi d ∂2hj
Pij[v(x)]= ηis +ηjs +ηis vk, (4.2)
1 ∂vs ∂vs dx ∂vs∂vk x
(cid:18) (cid:19)
where (ηij) is an arbitrary nondegenerate constant symmetric matrix: det(ηij) 6= 0, ηij =
const, ηij = ηji; hi(v), 1 ≤ i ≤ N, are smooth functions given in a certain domain of local
coordinatessuchthatoperator(4.2)isHamiltonian(lemma2.1,seealsotheorem2.1above).
An operator of form (4.2) is Hamiltonian if and only if
∂2hj ∂2hk ∂2hk ∂2hj
ηsr =ηsr , (4.3)
∂vs∂vi∂vl∂vr ∂vs∂vi∂vl∂vr
∂hs ∂hi ∂2hr ∂hs ∂hj ∂2hr
ηip +ηsp ηjl = ηjp +ηsp ηil . (4.4)
∂vp ∂vp ∂vl∂vs ∂vp ∂vp ∂vl∂vs
(cid:18) (cid:19) (cid:18) (cid:19)
7
([13], see also [9] and theorem 2.1).
Thesystemofnonlinearequations(4.3),(4.4),aswasconjecturedinthepresentauthor’s
work[13](therewasstatedthecorrespondingconjecturein[13]),isintegrablebythemethod
of inverse scattering problem. The procedure of integrating this system was presented in
theauthor’swork[10],[11]. Inthework[12]theLaxpairforthissystemwasdemonstrated.
Note that the associativity equations of two-dimensional topological field theory (see [4])
are a natural reduction of equations (4.3) for “potential” vector fields hi(v), 1 ≤ i ≤ N, of
the special form
∂Φ
hi(v)=ηij , (4.5)
∂vj
where Φ(v) is a certain smooth function (“ the potential”) (see also [9], [13]–[15]).
Considertherecursionoperatorgeneratedbythe “canonical”compatibleDubrovin–No-
vikov Hamiltonian operators (4.1), (4.2):
Rli = P1[v(x)](P2[v(x)])−1 il = ηis∂∂hvsj +ηjs∂∂vhsi ddx +ηis∂∂vs2∂hvjkvxk ηjl ddx −1,
h i (cid:18)(cid:18) (cid:19) (cid:19) (cid:18) (cid:19)(4.6)
where (η ) is the matrix which is inverse to the matrix (ηij): ηisη =δi (see [2], [16]–[20]
ij sj j
about recursion operators generated by pairs of compatible Hamiltonian operators).
Letusapply thederivedrecursionoperator(4.6)tothe systemoftranslationsinx, that
is, the system of hydrodynamic type
vi =vi, (4.7)
t x
which is, obviously, Hamiltonian with the Hamiltonian operator (4.1):
δH 1
vi =vi ≡Pij , H = η vj(x)vl(x)dx. (4.8)
t x 2 δvj(x) 2 jl
Z
Any system from the hierarchy
vi =(Rn)i vj, n∈Z, (4.9)
tn j x
is a multi-Hamiltonian integrable system.
In particular, any system of the form
vi =Rivj, (4.10)
t1 j x
that is, the system of hydrodynamic type
∂hj ∂hi d ∂2hj
vi = ηis +ηjs +ηis vk η vl
t1 ∂vs ∂vs dx ∂vs∂vk x jl
(cid:18)(cid:18) (cid:19) (cid:19)
∂hj ∂hi ∂2hj ∂hj
≡ ηis η + +ηisη vl vk ≡ hi(v)+ηis η vl , (4.11)
∂vs jk ∂vk jl∂vs∂vk x ∂vs jl
(cid:18) (cid:19) (cid:18) (cid:19)x
8
where hi(v), 1 ≤ i ≤ N, is an arbitrary solution of the integrable system (4.3), (4.4), is
integrable.
This system of hydrodynamic type is bi-Hamiltonian with the pair of “canonical” Dub-
rovin–NovikovHamiltonian operators (4.1), (4.2):
vti1 = ηis∂∂hvsj +ηjs∂∂vhsi ddx +ηis∂∂vs2∂hvjkvxk δvδHj(x1), H1 = 12 ηjlvj(x)vl(x)dx,
(cid:18)(cid:18) (cid:19) (cid:19) Z
(4.12)
vti1 =ηijddxδvδHj(x2), H2 = ηjkhk(v(x))vj(x)dx. (4.13)
Z
The next system in the hierarchy (4.9) is the integrable system of hydrodynamic type
∂hj ∂hi d ∂2hj ∂hr
vi = ηis +ηjs +ηis vk η hl(v)+ηlp η vq
t2 ∂vs ∂vs dx ∂vs∂vk x jl ∂vp rq
(cid:18)(cid:18) (cid:19) (cid:19) (cid:18) (cid:19)
∂hj ∂hi ∂hl ∂hr ∂2hr
≡ ηis +ηjs η +η +η vq
∂vs ∂vs jl∂vk rk∂vj rq ∂vj∂vk
(cid:18)(cid:18) (cid:19)(cid:18) (cid:19)
∂2hj ∂hr
+ηis η hl(v)+η vq vk. (4.14)
∂vs∂vk jl rq ∂vj x
(cid:18) (cid:19)(cid:19)
The hierarchy of integrable systems (4.9) is “canonical” for all bi-Hamiltonian systems
of hydrodynamic type possessing pairs of compatible local Hamiltonian operatorsof hydro-
dynamic type.
We have also realized a completely similar explicit construction of the corresponding
class of integrable bi-Hamiltonian systems of hydrodynamic type in the case of compatible
nonlocal Hamiltonian operators of hydrodynamic type (see [21]–[26] about the nonlocal
Hamiltonian operators of hydrodynamic type). These results will be published somewhere
else.
References
[1] Dubrovin B.A., Novikov S.P. The Hamiltonian formalism of one-dimensional systems
of hydrodynamic type and the Bogolyubov–Whithamaveragingmethod. Soviet Math.
Dokl. 1983. Vol. 27. P. 665–669.
[2] Dorfman I. Dirac structures and integrability of nonlinear evolution equations. Chich-
ester, England: John Wiley & Sons, 1993.
[3] MagriF. Asimple model ofthe integrableHamiltonianequation.J.Math. Phys.1978.
Vol. 19. No. 5. P. 1156–1162.
9
[4] Dubrovin B. Geometry of 2D topological field theories. Lect. Notes Math. 1996. Vol.
1620. P. 120–348;hep-th/9407018.
[5] Dubrovin B. Differential geometry of the space of orbits of a Coxeter group. Preprint
SISSA-29/93/FM; hep-th/9303152.
[6] Dubrovin B. Flat pencils of metrics and Frobenius manifolds. Preprint SISSA
25/98/FM;math.DG/9803106.
[7] Fordy A.P., Mokhov O.I. On a special class of compatible Poissonstructures of hydro-
dynamic type. Physica D. 2001. Vol. 152–153.Nos. 1–4. P. 475–490.
[8] Mokhov O.I. Compatible and almost compatible pseudo-Riemannian metrics. Func-
tional Anal. Appl. 2001.Vol. 35. No. 2; E-print. arXiv: math.DG/0005051.
[9] Mokhov O.I. On compatible Poisson structures of hydrodynamic type. Russian Math.
Surveys. 1997. Vol. 52. No. 6. P. 1310–1311.
[10] Mokhov O.I. On integrability of the equations for nonsingular pairs of compatible flat
metrics. arXiv: math.DG/0005081.
[11] Mokhov O.I. Flat pencils of metrics and integrable reductions of the Lam´e equations.
Russian Math. Surveys. 2001. Vol. 56. No. 2.
[12] Ferapontov E.V. Compatible Poisson brackets of hydrodynamic type. arXiv:
math.DG/0005221.
[13] Mokhov O.I. Compatible Poisson structures of hydrodynamic type and associativity
equations. Proceedings of the Steklov Institute of Mathematics (Moscow). 1999. Vol.
225. P. 269–284.
[14] Mokhov O.I. Compatible Poisson structures of hydrodynamic type and the equations
of associativity in two-dimensional topological field theory. Reports on Mathematical
Physics. 1999. Vol. 43, No. 1-2. P. 247–256.
[15] Mokhov O.I. On compatible potential deformations of Frobenius algebrasand associa-
tivity equations. Russian Math. Surveys. 1998. Vol. 53. No. 2. P. 396–397.
[16] GelfandI.M.,DorfmanI.Ya.Hamiltonianoperatorsandalgebraicstructuresassociated
with them. Functional Anal. Appl. 1979. Vol. 13. No. 4. P. 246–262.
[17] FuchssteinerB.Applicationofhereditarysymmetriesto nonlinearevolutionequations.
Nonlinear Analysis. Theory, Methods and Applications. 1979.Vol. 3. P. 849–862.
[18] Fokas A.S., Fuchssteiner B. On the structure of symplectic operators and hereditary
symmetries. Lettere al Nuovo Cimento. 1980. Vol. 28. No. 8. P. 299–303.
10