while the correspondence C : D(M ) ’ D(π) is C ∞ (M )-linear. In other

words, this correspondence is a connection in the bundle π∞ : J ∞ (π) ’ M .

Definition 2.3. The connection C : D(M ) ’ D(π) de¬ned above is

called the Cartan connection in J ∞ (π).

Let now E ‚ J k (π) be a formally integrable equation. Then, due to

the fact that Cθ (E ∞ ) = Cθ (π) at any point θ ∈ E ∞ , we see that the ¬elds

CX are tangent to E ∞ for all vector ¬elds X ∈ D(M ). Thus we obtain the

Cartan connection in the bundle π∞ : E ∞ ’ M which is denoted by the

same symbol C.

Let x1 , . . . , xn , . . . , uj , . . . be special coordinates in J ∞ (π) and X =

σ

X1 ‚/‚x1 + · · · + Xn ‚/‚xn be a vector ¬eld on M represented in this co-

ordinate system. Then the ¬eld CX is to be of the form CX = X + X v ,

where X v = j,σ Xσ ‚/‚uj is a π∞ -vertical ¬eld. The de¬ning conditions

j

σ

j j

iCX ωσ = 0, where ωσ are the Cartan forms on J ∞ (π), imply

«

n n

‚ ‚

j

Xi

CX = + uσ+1i j = Xi Di . (2.5)

‚xi ‚uσ

i=1 j,σ i=1

In particular, we see that C(‚/‚xi ) = Di , i.e., total derivatives are just

liftings to J ∞ (π) of the corresponding partial derivatives by the Cartan

connection.

To obtain a similar expression for the Cartan connection on E ∞ , it needs

only to obtain coordinate representation for total derivatives in internal

coordinates. For example, in the case of equations (1.22) (see p. 16) we have

n n

‚ ‚

C Xi +T = Xi Di + T D t ,

‚xi ‚t

i=1 i=1

where D1 , . . . , Dn , Dt are given by formulas (2.1) and Xi , T ∈ C ∞ (M ) are

the coe¬cients of the ¬eld X ∈ D(M ).

Consider the following construction now. Let V be a vector ¬eld on E ∞

and θ ∈ E ∞ be a point. Then the vector Vθ can be projected parallel to

the Cartan plane Cθ onto the ¬ber of the projection π∞ : E ∞ ’ M passing

through θ. Thus we get a vertical vector ¬eld V v . Hence, for any f ∈ F(E)

a di¬erential one-form UE (f ) ∈ Λ1 (E) is de¬ned by

def

iV (UE (f )) = V v (f ), V ∈ D(E). (2.6)

62 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

The correspondence f ’ UE (f ) is a derivation of the algebra F(E) with the

values in the F(E)-module Λ1 (E), i.e.,

UE (f g) = f UE (g) + gUE (f )

for all f, g ∈ F(E). This correspondence contains all essential data about

the equation E.

Definition 2.4. The derivation UE : F(E) ’ Λ1 (E) is called the struc-

tural element of the equation E.

For the “empty” equation, i.e., in the case E ∞ = J ∞ (π), the structural

element Uπ is locally represented in the form

‚

j

ωσ — j ,

Uπ = (2.7)

‚uσ

j,σ

j

where ωσ are the Cartan forms on J ∞ (π). To obtain the expression in the

general case, one needs to rewrite (2.7) in local coordinates. For example,

in the case of evolution equations we get the same expression with σ =

j

(σ1 , . . . , σn , 0) and the forms ωσ given by (2.2). Contrary to the Cartan

forms, the structural element is independent of local coordinates.

We shall now give a “more algebraic” version of the Cartan connection

de¬nition.

Proposition 2.4. For any vector ¬eld X ∈ D(M ), the equality

j∞ (•)— (CX(f )) = X(j∞ (•)— (f )) (2.8)

takes place, where f ∈ F(π) and • ∈ “loc (π). Equality (2.8) uniquely deter-

mines the Cartan connection in J ∞ (π).

Proof. Both statements follow from the fact that in special coordinates

the right-hand side of (2.8) is of the form

‚ |σ| •j

‚f

X .

j ‚xσ

j,σ ‚uσ j∞ (•)(M )

Corollary 2.5. The Cartan connection in E ∞ is ¬‚at, i.e.,

C[X, Y ] = [CX, CY ]

for any X, Y ∈ D(M ).

Proof. Consider the case E ∞ = J ∞ (π). Then from Proposition 2.4 we

have

j∞ (•)— (C[X, Y ](f )) = [X, Y ](j∞ (•)— (f ))

= X(Y (j∞ (•)— (f ))) ’ Y (X(j∞ (•)— (f )))

for any • ∈ “loc (π), f ∈ F(π). On the other hand,

1. BASIC STRUCTURES 63

j∞ (•)— ([CX, CY ](f )) = j∞ (•)— (CX(CY (f )) ’ CY (CX(f )))

= X(j∞ (•)— (Y (f ))) ’ Y (j∞ (•)— (CX(f )))

= X(Y (j∞ (•)— (f ))) ’ Y (X(j∞ (•)— (f )))

To prove the statement for an arbitrary formally integrable equation E, it

su¬ces to note that the Cartan connection in E ∞ is obtained by restricting

the ¬elds CX onto in¬nite prolongation of E.

The construction of Proposition 2.4 can be generalized.

1.4. C-di¬erential operators. Let π : E ’ M be a vector bundle and

ξ1 : E1 ’ M , ξ2 : E2 ’ M be another two vector bundles.

Definition 2.5. Let ∆ : “(ξ1 ) ’ “(ξ2 ) be a linear di¬erential operator.

The lifting C∆ : F(π, ξ1 ) ’ F(π, ξ2 ) of the operator ∆ is de¬ned by

j∞ (•)— (C∆(f )) = ∆(j∞ (•)— (f )), (2.9)

where • ∈ “loc (π) and f ∈ F(π, ξ1 ) are arbitrary sections.

Immediately from the de¬nition, we obtain the following properties of

operators C∆:

Proposition 2.6. Let π : E ’ M , ξi : Ei ’ M , i = 1, 2, 3, be vector

bundles. Then

(i) For any C ∞ (M )-linear di¬erential operator ∆ : “(ξ1 ) ’ “(ξ2 ), the

operator C∆ is an F(π)-linear di¬erential operator of the same order.

(ii) For any ∆, : “(ξ1 ) ’ “(ξ2 ) and f, g ∈ F(π), one has

C(f ∆ + g ) = f C∆ + gC .

(iii) For ∆1 : “(ξ1 ) ’ “(ξ2 ), ∆2 : “(ξ2 ) ’ “(ξ3 ), one has

C(∆2 —¦ ∆1 ) = C∆2 —¦ C∆1 .

From this proposition and from Proposition 2.4 it follows that if ∆ is a

scalar di¬erential operator C ∞ (M ) ’ C ∞ (M ) locally represented as ∆ =

|σ| ∞

σ aσ ‚ /‚xσ , aσ ∈ C (M ), then

C∆ = a σ Dσ (2.10)

σ

in the corresponding special coordinates. If ∆ = ∆ij is a matrix operator,

then C∆ = C∆ij .

From Proposition 2.6 it follows that C∆ may be understood as a dif-

ferential operator acting from sections of the bundle π to linear di¬erential

operators from “(ξ1 ) to “(ξ2 ). This observation is generalized as follows.

Definition 2.6. An F(π)-linear di¬erential operator ∆ : F(π, ξ1 ) ’

F(π, ξ2 ) is called a C-di¬erential operator, if it admits restriction onto graphs

of in¬nite jets, i.e., if for any section • ∈ “(π) there exists an operator

∆• : “(ξ1 ) ’ “(ξ2 ) such that

j∞ (•)— (∆(f )) = ∆• (j∞ (•)— (f )) (2.11)

64 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

for all f ∈ F(π, ξ1 ).

Thus, C-di¬erential operators are nonlinear di¬erential operators taking

their values in C ∞ (M )-modules of linear di¬erential operators. The follow-

ing proposition gives a complete description of such operators.

Proposition 2.7. Let π, ξ1 , ξ2 be vector bundles over M . Then any C-

di¬erential operator ∆ : F(π, ξ1 ) ’ F(π, ξ2 ) can be presented in the form

a± C∆± , a± ∈ F(π),

∆=

±

where ∆± are linear di¬erential operators acting from “(ξ1 ) to “(ξ2 ).

Proof. Recall ¬rst that we consider the ¬ltered theory; in particular,

there exists an integer l such that ∆(Fk (π, ξ1 )) ‚ Fk+l (π, ξ2 ) for all k.

Consequently, since “(ξ1 ) is embedded into F0 (π, ξ1 ), we have ∆(“(ξ1 )) ‚

¯

Fl (π, ξ2 ) and the restriction ∆ = ∆ “(ξ1 ) is a C ∞ (M )-di¬erential operator

taking its values in Fl (π, ξ2 ). Then one can easily see that the equality

¯

∆• = j∞ (•)— —¦ ∆ holds, where • ∈ “loc (π) and ∆• is the operator from

(2.11). It means that any C-di¬erential ∆ operator is completely determined

¯

by its restriction ∆.

¯ ¯

On the other hand, the operator ∆ is represented in the form ∆ =

∞

± a± ∆± , a± ∈ Fl (π) and ∆± : “(ξ1 ) ’ “(ξ2 ) being C (M )-linear dif-

¯ def

ferential operators. Let us de¬ne C ∆ = ± a± C∆± . Then the di¬erence

¯

∆ ’ C ∆ is a C-di¬erential operator such that its restriction onto “(ξ1 ) van-

¯

ishes. Therefore ∆ = C ∆.

Remark 2.1. From the result obtained it follows that C-di¬erential op-

erators are operators “in total derivatives”. By this reason, they are called

total di¬erential operators sometimes.

Corollary 2.8. C-di¬erential operators admit restrictions onto in¬nite

prolongations: if ∆ : F(π, ξ1 ) ’ F(π, ξ2 ) is a C-di¬erential operator and

E ‚ J k (π) is a k-th order equation, then there exists a linear di¬erential

operator ∆E : F(E, ξ1 ) ’ F(E, ξ2 ) such that

µ— —¦ ∆ = ∆ E —¦ µ — ,

where µ : E ∞ ’ J ∞ (π) is the natural embedding.

Proof. The result immediately follows from Example 2.1 and Proposi-

tion 2.7.

We shall now consider an example which will play a very important role

in the sequel.

Example 2.3. Let ξ1 = „i— , ξ2 = „i+1 , where „p : p T — M ’ M (see

— —

Example 1.2 on p. 6), and ∆ = d : Λi (M ) ’ Λi+1 (M ) be the de Rham

def

di¬erential. Then we obtain the ¬rst-order operator dh = Cd : Λi (π) ’

h

p

i+1 — ). Due Corollary 2.8, the

Λh (π), where Λh (π) denotes the module F(π, „p

operators d : Λi (E) ’ Λi+1 (E) are also de¬ned, where Λp (E) = F(E, „p ).

—

h h h

1. BASIC STRUCTURES 65

Definition 2.7. Let E ‚ J k (π) be an equation.

(i) Elements of the module Λi (E) are called horizontal i-forms on E ∞ .

h

i (E) ’ Λi+1 (E) is called the horizontal de Rham

(ii) The operator dh : Λh h

di¬erential on E ∞ .

(iii) The sequence

d d

0 ’ F(E) ’ Λ1 (E) ’ · · · ’ Λi (E) ’ Λi+1 (E) ’ · · ·

’h ’h

h

is called the horizontal de Rham sequence of the equation E.

From Proposition 2.6 (iii) it follows that d—¦d = 0 and hence the de Rham

sequence is a complex. It cohomologies are called the horizontal de Rham

— i

cohomologies of E and are denoted by Hh (E) = i≥0 Hh (E).

In local coordinates, horizontal forms of degree p on E ∞ are represented

as ω = i1 <···<ip ai1 ...ip dxi1 § · · · § dxip , where ai1 ...ip ∈ F(E), while the

horizontal de Rham di¬erential acts as

n

Di (ai1 ...ip ) dxi § dxi1 § · · · § dxip .

dh (ω) = (2.12)

i=1 i1 <···<ip

In particular, we see that both Λi (E) and Hh (E) vanish for i > dim M .

i

h

Remark 2.2. In fact, the above introduced cohomologies are horizontal

cohomologies with trivial coe¬cients. The case of more general coe¬cients

will be considered in Chapter 4 (see also [98, 52]). Below we make the ¬rst

step to deal with a nontrivial case.

Consider the algebra Λ— (E) of all di¬erential forms on E ∞ and let us note

that one has the embedding Λ— (E) ’ Λ— (E). Let us extend the horizontal

h

de Rham di¬erential onto this algebra as follows:

(i) dh (dω) = ’d(dh (ω)),

(ii) dh (ω § θ) = dh (ω) § θ + (’1)ω ω § dh (θ).

Obviously, conditions (i), (ii) de¬ne the di¬erential dh : Λi (E) ’ Λi+1 (E) and

its restriction onto Λ— (E) coincides with the horizontal de Rham di¬erential.

h

def

Let us also set dC = d ’ dh : Λ— (E) ’ Λ— (E). Then, by de¬nition,

h h

dh —¦ dh = dC —¦ dC = 0, dC —¦ dh + dh —¦ dC = 0.

d = d h + dC ,

In other words, the pair (dh , dC ) forms a bicomplex in Λ— (E) with the total

di¬erential d. Hence, the corresponding spectral sequence converges to the

de Rham cohomology of E ∞ .

Remark 2.3. We shall rede¬ne this bicomplex in a more general alge-

braic situation in Chapter 4. On the other hand, it should be noted that

the above mentioned spectral sequence (in the case, when dh is taken for the

¬rst di¬erential and dC for the second one) is a particular case of the Vino-

gradov C-spectral sequence (or the so-called variational bicomplex) which is

essential to the theory of conservation laws and Lagrangian formalism with

constraints; cf. Subsection 2.2 below.

66 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

To conclude this section, let us write down the coordinate representation

for the di¬erential dC and the extended dh . First note that by de¬nition and

due to (2.12), one has

n

uj i dxi ,

dC (uj ) d(uj ) dh (uj ) duj

’ ’

= =

σ σ σ σ σ+1

i=1

i.e., dC takes coordinate functions uj to the corresponding Cartan forms.

σ

Since obviously dC (xi ) = 0 for any coordinate function on the base, we

obtain

‚f j

f ∈ F(π).

dC (f ) = ωσ , (2.13)

j

j,σ ‚uσ

The same representation, written in internal coordinates, is valid on E ∞ .

Therefore, the image of dC spans the Cartan submodule CΛ1 (E) in Λ1 (E).

By this reason, we call dC the Cartan di¬erential on E ∞ . From the equality

d = dh + dC it follows that the direct sum decomposition

Λ1 (E) = Λ1 (E) • CΛ1 (E)

h

takes place which extends to the decomposition

Λq (E) — C p Λ(E).

Λi (E) = (2.14)

h

p+q=i

Here the notation

def

C p Λ(E) = CΛ1 (E) § · · · § CΛ1 (E)

p times

j

is used. Consequently, to ¬nish computations, it su¬ces to compute dh (ωσ ).

But we have

dh (ωσ ) = dh dC (uj ) = ’dC dh (uj )

j

σ σ

and thus

n

j

j

dh (ωσ ) = ’ ωσ+1i § dxi . (2.15)

i=1

Note that from the results obtained it follows, that

dh (Λq (E) — C p Λ(E)) ‚ Λq+1 (E) — C p Λ(E),

h h

dC (Λq (E) — C p Λ(E)) ‚ Λq (E) — C p+1 Λ(E).

h h

Remark 2.4. Note that the sequence dh : Λq (E) — C — (E) ’ Λq+1 (E) —

h h

— (E) can be considered as the horizontal de Rham complex with coe¬cients

C

in Cartan forms

2. HIGHER SYMMETRIES AND CONSERVATION LAWS 67

Remark 2.5. From (2.14) it follows that to any form ω ∈ Λ— (E) we

can put into correspondence its “purely horizontal” component ωh ∈ Λ— (E).

h

k (π), then, due to the equality duj =

Moreover, if the form ω “lives” on J σ

j

n — k+1 (π)). This correspon-

i=1 uσ+1i dxi + ωσ , the form ωh belongs to Λ (J

dence coincides with the one used in Example 1.7 on p. 14 to construct

Monge“Ampere equations.

2. Higher symmetries and conservation laws

In this section, we brie¬‚y expose the theory of higher (or Lie“B¨cklund)

a

symmetries and conservation laws for nonlinear partial di¬erential equations

(for more details and examples see [60, 12]).

2.1. Symmetries. Let π : E ’ M be a vector bundle and E ‚ J k (π)

be a di¬erential equation. We shall still assume E to be formally integrable,

though is it not restrictive in this context.

Consider a symmetry F of the equation E and let θk+1 be a point of E 1

such that πk+1,k (θk+1 ) = θk ∈ E. Then the R-plane Lθk+1 is taken to an

R-plane F— (Lθk+1 ) by F , since F is a Lie transformation, and F— (Lθk+1 ) ‚

TF (θk ) , since F is a symmetry. Consequently, the lifting F (1) : J k+1 (π) ’

J k+1 (π) is a symmetry of E 1 . By the same reasons, F (l) is a symmetry of the

l-th prolongation of E. From here it also follows that for any in¬nitesimal

symmetry X of the equation E, its l-th lifting is a symmetry of E l as well.

In fact, the following result is valid:

Proposition 2.9. Symmetries of a formally integrable equation E ‚

J k (π)

coincide with symmetries of any prolongation of this equation. The

same is valid for in¬nitesimal symmetries.

Proof. We have shown already that to any (in¬nitesimal) symmetry

of E there corresponds an (in¬nitesimal) symmetry of E l . Consider now an

(in¬nitesimal) symmetry of E l . Then, due to Theorems 1.12 and 1.13 (see

pp. 24 and 26), it is πk+l,k -¬berwise and therefore generates an (in¬nitesimal)

symmetry of E.

The result proved means that a symmetry of E generates a symmetry of

E∞ which preserves every prolongation up to ¬nite order. A natural step to

generalize the concept of symmetry is to consider “all symmetries” of E ∞ .

Let us clarify such a generalization.

First of all note that only in¬nitesimal point of view may be e¬cient

in the setting under consideration. Otherwise we would have to deal with

di¬eomorphisms of in¬nite-dimensional manifolds with all natural di¬cul-

ties arising as a consequence. Keeping this in mind, we proceed with the

following de¬nition. Recall the notation

def

CD(π) = {X ∈ D(π) | X lies in C(π)},

cf. (1.31) on p. 25.

68 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

Definition 2.8. Let π be a vector bundle. A vector ¬eld X ∈ D(π) is

called a symmetry of the Cartan distribution C(π) on J ∞ (π), if [X, CD(π)] ‚

CD(π).

Thus, the set of symmetries coincides with DC (π) (see (1.32) on p. 25)

and forms a Lie algebra over R and a module over F(π). Note that since the

Cartan distribution on J ∞ (π) is integrable, one has CD(π) ‚ DC (π) and,

moreover, CD(π) is an ideal in the Lie algebra DC (π).

Note also that symmetries belonging to CD(π) are of a special type:

they are tangent to any integral manifold of the Cartan distribution. By

this reason, we call such symmetries trivial. Respectively, the elements of

the quotient Lie algebra

def

sym(π) = DC (π)/CD(π)

are called nontrivial symmetries of the Cartan distribution on J ∞ (π).

Let now E ∞ be the in¬nite prolongation of an equation E ‚ J k (π).

Then, since CD(π) is spanned by the ¬elds of the form CY , Y ∈ D(M ) (see

Example 2.1), any vector ¬eld from CD(π) is tangent to E ∞ . Consequently,

either all elements of the coset [X] = X mod CD(π), X ∈ D(π), are tangent

to E ∞ or neither of them is. In the ¬rst case we say that the coset [X] is

tangent to E ∞ .

Definition 2.9. An element [X] = X mod CD(π), X ∈ D(π), is called

a higher symmetry of E, if it is tangent to E ∞ .

The set of all higher symmetries forms a Lie algebra over R and is de-

noted by sym(E). We shall usually omit the adjective higher in the sequel.

Let us describe the algebra sym(E) in e¬cient terms. We start with

describing sym(π) as the ¬rst step. To do this, note the following. Consider

a vector ¬eld X ∈ D(π). Then, substituting X into the structural element

Uπ (see (2.7)), we obtain a ¬eld X v ∈ D(π). The correspondence Uπ : X ’

X v = X Uπ possesses the following properties:

(i) The ¬eld X v is vertical, i.e., X v (C ∞ (M )) = 0.

(ii) X v = X for any vertical ¬eld.

(iii) X v = 0 if and only if the ¬eld X lies in CD(π).

Therefore, we obtain the direct sum decomposition1

D(π) = D v (π) • CD(π),

where D v (π) denotes the Lie algebra of vertical ¬elds. A direct corollary of