Proposition 2.10. For any coset [X] ∈ sym(E) there exists a unique

vertical representative and thus

sym(E) = {X ∈ D v (E) | [X, CD(E)] ‚ CD(E)}, (2.16)

where CD(E) is spanned by the ¬elds CY , Y ∈ D(M ).

1

Note that it is the direct sum of F(π)-modules but not of Lie algebras.

2. HIGHER SYMMETRIES AND CONSERVATION LAWS 69

Using this result, we shall identify symmetries of E with vertical vector

¬elds satisfying (2.16).

Lemma 2.11. Let X ∈ sym(π) be a vertical vector ¬eld. Then it is

completely determined by its restriction onto F0 (π) ‚ F(π).

Proof. Let X satisfy the conditions of the lemma and Y ∈ D(M ).

Then for any f ∈ C ∞ (M ) one has

[X, CY ](f ) = X(CY (f )) ’ CY (X(f )) = X(Y (f )) = 0

and hence the commutator [X, CY ] is the vertical vector ¬eld. On the other

hand, [X, CY ] ∈ CD(π), because CD(π) is a Lie algebra ideal. Consequently,

[X, CY ] = 0.

Note now that in special coordinates we have Di (uj ) = uj i for all σ

σ σ+1

and j. From the above said it follows that

X(uj i ) = Di X(uj ) . (2.17)

σ

σ+1

But X is a vertical derivation and thus is determined by its values at the

functions uj .

σ

Let now X0 : F0 (π) ’ F(π) be a derivation. Then equalities (2.17)

allow one to reconstruct locally a vertical derivation X ∈ D(π) satisfying

X F0 (π) = X0 . Obviously, the derivation X thus constructed lies in sym(π)

over the neighborhood under consideration. Consider two neighborhoods

U1 , U2 ‚ J ∞ (π) with the corresponding special coordinates in each of them

and two symmetries X i ∈ sym(π |Ui ), i = 1, 2, arising by the described

procedure. But the restrictions of X 1 and X 2 onto F0 (π |U1 ©U2 ) coincide.

Hence, by Lemma 2.11, the ¬eld X 1 coincide with X 2 over the intersection

U1 © U2 . In other words, the reconstruction procedure X0 ’ X is a global

one. So we have established a one-to-one correspondence between elements

of sym(π) and derivations F0 (π) ’ F(π).

To complete description of sym(π), note that due to vector bundle struc-

ture in π : E ’ M , derivations F0 (π) ’ F(π) are identi¬ed with sections

—

of the pull-back π∞ (π), or with elements of F(π, π).

Theorem 2.12. Let π : E ’ M be a vector bundle. Then:

(i) The F(π)-module sym(π) is in one-to-one correspondence with ele-

ments of the module F(π, π).

(ii) In special coordinates the correspondence F(π, π) ’ sym(π) is ex-

pressed by the formula

‚

def

Dσ (•j ) j ,

•’ • = (2.18)

‚uσ

j,σ

where • = (•1 , . . . , •m ) is the component-wise representation of the

section • ∈ F(π, π).

Proof. The ¬rst part of the theorem has already been proved. To prove

the second one, it su¬ces to use equality (2.17).

70 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

Definition 2.10. Let π : E ’ M be a vector bundle.

(i) The ¬eld • of the form (2.18) is called an evolutionary vector ¬eld

on J ∞ (π).

(ii) The section • ∈ F(π, π) is called the generating section of the ¬eld

•.

Remark 2.6. Let ζ : N ’ M be an arbitrary smooth ¬ber bundle and

ξ : P ’ M be a vector bundle. Then it is easy to show that any ζ-verti-

cal vector ¬eld X on N can be uniquely lifted up to an R-linear mapping

X ξ : “(ζ — (ξ)) ’ “(ζ — (ξ)) such that

f ∈ C ∞ (N ), ψ ∈ “(ζ — (ξ)).

X ξ (f ψ) = X(f )ψ + f X ξ (ψ), (2.19)

In particular, taking π∞ for ζ, for any evolution derivation • we obtain

ξ

the family of mappings • : F(π, ξ) ’ F(π, ξ) satisfying (2.19).

π

Consider the mapping • : F(π, π) ’ F(π, π) and recall that the diag-

onal element ρ0 ∈ F0 (π, π) ‚ F(π, π) is de¬ned (see Example 1.1 on p. 5).

As it can be easily seen, the following identity is valid

π

• (ρ0 ) =• (2.20)

which can be taken for the de¬nition of the generating section.

Let • , ψ be two evolutionary derivations. Then, since sym(π) is a Lie

algebra and by Theorem 2.12, there exists a unique section {•, ψ} satisfying

[ • , ψ ] = {•,ψ} .

Definition 2.11. The section {•, ψ} is called the (higher ) Jacobi

bracket of the sections •, ψ ∈ F(π).

Proposition 2.13. Let •, ψ ∈ F(π, π) be two sections. Then:

π π

(i) {•, ψ} = • (ψ) ’ ψ (•).

(ii) In special coordinates, the Jacobi bracket of • and ψ is expressed by

the formula

‚ψ j j

± ‚•

j ±

{•, ψ} = Dσ (• ) ± ’ Dσ (ψ ) ± , (2.21)

‚uσ ‚uσ

±,σ

where superscript j denotes the j-th component of the corresponding

section.

Proof. To prove (i) let us use (2.20):

π π π π π π π

{•, ψ} = ’ ’

{•,ψ} (ρ0 ) = • ( ψ (ρ0 )) ψ ( • (ρ0 )) = • (ψ) ψ (•).

The second statement follows from the ¬rst one and from equality (2.18).

Consider now a nonlinear di¬erential operator ∆ : “(π) ’ “(ξ) and let

•∆ be the corresponding section. Then for any • ∈ F(π, π) the section

• (•∆ ) ∈ F(π, ξ) is de¬ned and we can set

def

∆ (•) = • (•∆ ). (2.22)

2. HIGHER SYMMETRIES AND CONSERVATION LAWS 71

Definition 2.12. The operator ∆ : F(π, π) ’ F(π, ξ) de¬ned by

(2.22) is called the universal linearization operator of the operator

∆ : “(π) ’ “(ξ).

From the de¬nition and equality (2.18) we obtain that for a scalar dif-

ferential operator

‚ |σ| •j

∆ : • ’ F x1 , . . . , xn , . . . , ,...

‚xσ

= ( 1 , . . . , m ), m = dim π, where

one has ∆ ∆ ∆

‚F

±

= Dσ . (2.23)

∆

‚u± σ

σ

If dim ξ = r > 1 and ∆ = (∆1 , . . . , ∆r ), then

1 2 m

...

∆1 ∆1 ∆1

1 2 m

...

∆2 ∆2 ∆2

= . (2.24)

∆

... ... ... ...

1 2 m

...

∆r ∆r ∆r

In particular, we see that the following statement is valid.

Proposition 2.14. For any di¬erential operator ∆, its universal lin-

earization is a C-di¬erential operator.

Now we can describe the algebra sym(E), E ‚ J k (π) being a formally

integrable equation. Let I(E) ‚ F(π) be the ideal of the equation E (see

Subsection 1.1). Then, by de¬nition, • is a symmetry of E if and only if

‚ I(E).

• (I(E)) (2.25)

Assume now that E is given by a di¬erential operator ∆ : “(π) ’ “(ξ) and

locally is described by the system of equations

F 1 = 0, . . . , F r = 0, F j ∈ F(π).

Then the functions F 1 , . . . , F r are di¬erential generators of the ideal I(E)

and condition (2.25) may be rewritten as

j

a± Dσ (F ± ), a± ∈ F(π).

• (F )= j = 1, . . . , m, (2.26)

σ σ

±,σ

With the use of (2.22), the last equation acquires the form2

a± Dσ (F ± ), a± ∈ F(π).

F j (•) = j = 1, . . . , m, (2.27)

σ σ

±,σ

But by Proposition 2.14, the universal linearization is a C-di¬erential op-

erator and consequently can be restricted onto E ∞ (see Corollary 2.8). It

means that we can rewrite equation (2.27) as

F j |E ∞ (• |E ∞ ) = 0, j = 1, . . . , m. (2.28)

2

Below we use the notation F , F ∈ F(π, ξ), as a synonym for where ∆ : “(π) ’

∆,

“(ξ) is the operator corresponding to the section F .

72 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

Combining these equations with (2.23) and (2.24), we obtain the following

fundamental result:

Theorem 2.15. Let E ‚ J k (π) be a formally integrable equation and

∆ = ∆E : “(π) ’ “(ξ) be the operator corresponding to E. Then an evolu-

tionary derivation • , • ∈ F(π, π), is a symmetry of E if and only if

E (•)

¯ = 0, (2.29)

and • on E ∞ respectively. In other

and • denote restrictions of

¯

where E ∆

words,

sym(E) = ker E. (2.30)

Remark 2.7. From the result obtained it follows that higher symmetries

of E can be identi¬ed with elements of F(E, π) satisfying equation (2.29).

Below we shall say that a section • ∈ F(E, π) is a symmetry of E keeping

in mind this identi¬cation. Note that due to the fact that sym(E) is a

Lie algebra, for any two symmetries •, ψ ∈ F(E, π) their Jacobi bracket

{•, ψ}E = {•, ψ} ∈ F(E, π) is well de¬ned and is a symmetry as well.

2.2. Conservation laws. This subsection contains a brief review of

the main de¬nitions and facts concerning the theory of conservation laws for

nonlinear di¬erential equations. We con¬ne ourselves with main de¬nition

and results referring the reader to [102] and [52] for motivations and proofs.

Definition 2.13. Let E ‚ J k (π), π : E ’ M being a vector bundle, be

a di¬erential equation and n be the dimension of the manifold M .

(i) A horizontal (n ’ 1)-form ρ ∈ Λn’1 (E) on E ∞ is called a conserved

h

density on E, if dh ρ = 0.

(ii) A conserved density ρ is called trivial, if ρ = dh ρ for some ρ ∈

Λn’2 (E).

h

n’1

(iii) The horizontal cohomology class [ρ] ∈ Hh (E) of a conserved density

ρ is called a conservation law on E.

We shall always assume below that the manifold M of independent vari-

ables is cohomologically trivial which means triviality of all de Rham coho-

mology groups H i (M ) except for the group H 0 (M ).

0,n’1 0,n’1

n’1

Note now that the group Hh (E) is the term E1 = E1 (E) of

the spectral sequence associated to the bicomplex (dh , dC ) (see Subsection

1.4 and Remark 2.3 in particular). This fact is not accidental and to clarify

it we shall need more information about this spectral sequence. Let us start

with the “trivial” case and ¬rst introduce preliminary notions and notations.

For any equation E, we shall denote by κ = κ(E) the module F(E, π).

In particular, κ(π) denotes the module κ in the case E ∞ = J ∞ (π). Let ξ

and ζ be two vector bundles over M and P = F(E, ξ), Q = F(E, ζ). Denote

by C Diff alt (P, Q) the F(E)-module of R-linear mappings

l

∆: P — · · · — P ’ Q

l times

2. HIGHER SYMMETRIES AND CONSERVATION LAWS 73

such that:

(i) ∆ is skew-symmetric,

(ii) for any p1 , . . . , pl’1 ∈ P , the mapping

∆p1 ,...,pl’1 : P ’ Q, p ’ ∆(p1 , . . . , pl’1 , p),

is a C-di¬erential operator.

In particular, C Diff(P, Q) denotes the module of all C-di¬erential operators

acting from P to Q.

De¬ne the complex

dP

C Diff(P, Λ1 (E)) ’ · · · ’ C Diff(P, Λq (E))

0 h

0’C Diff(P, Λh (E)) ’’ h h

dP

’’ C Diff(P, Λq+1 (E)) ’ · · · ’ C Diff(P, Λn (E)) ’ 0 (2.31)

h

h

h

def

by setting dP (∆) = dh —¦ ∆.

h

Lemma 2.16. The above introduced complex (2.31) is acyclic at all terms

except for the last one. The cohomology group at the n-th term equals the

def

module P = homF (E) (P, Λn (E)).

h

Let ∆ : P ’ Q be a C-di¬erential operator. Then it generates the

cochain mapping

∆ : (C Diff(Q, Λ— (E)), dQ ) ’ (C Diff(P, Λ— (E)), dP )

h h h

h

and consequently the mapping of cohomology groups

∆— : Q = homF (E) (Q, Λn (E)) ’ P = homF (E) (P, Λn (E)). (2.32)

h h

Definition 2.14. The above introduced mapping ∆— is called the ad-

joint operator to the operator ∆.

In the case E ∞ = J ∞ (π), the local coordinate representation of the

adjoint operator is as follows. For the scalar operator ∆ = σ aσ Dσ one

has

∆— = (’1)|σ| Dσ —¦ aσ . (2.33)

σ

In the multi-dimensional case, ∆ = ∆ij , the components of the adjoint

operator are expressed by

(∆— )ij = ∆— , (2.34)

ji

where ∆— are given by (2.33).

ji

Relation between the action of an C-di¬erential ∆ : P ’ Q and its ad-

joint ∆— : Q ’ P is given by

Proposition 2.17 (Green™s formula). For any elements p ∈ P and q ∈

Q there exists an n ’ 1-form ω ∈ Λn’1 (E) such that

h

p, ∆— (q) ’ ∆(p), q = dh ω, (2.35)

where R, R ’ Λn (E) denotes the natural pairing.

h

74 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

Finally, let us de¬ne F(E)-submodules Kl (P ) ‚ C Diff alt (P, P ), l > 0,

l’1

by setting

def

Kl (P ) =

{∆ ∈ C Diff alt (P, P ) | ∆—1 ,...,pl’2 = ’∆p1 ,...,pl’2 , ∀p1 , . . . , pl’2 ∈ P }.

p

l’1

Theorem 2.18 (one-line theorem). Let π : E ’ M be a vector bundle

over a cohomologically trivial manifold M , dim M = n. Then:

0,n n

(i) E1 (π) = Hh (E).

p,n

(ii) E1 (π) = Kp (κ(π)), p > 0.

0,0

(iii) E1 (π) = R.

p,q

(iv) E1 (π) = 0 in all other cases.

Moreover, the following result is valid.

Theorem 2.19. The sequence

1,n

1,n d1

d dh E 2,n

Λ0 (π) ’h Λn (π) ’

’ ... ’’ ’ E1 ’ ’’ E1 ’ · · · (2.36)

h h

where the operator E is the composition

0,1

d1

0,n 1,n

Λn (π) n

’ E1 (π) ’ ’

’

E: Hh (π) = E1 (π), (2.37)

h

the ¬rst arrow being the natural projection, is exact.

Definition 2.15. Let π : E ’ M be a vector bundle, dim M = n.

(i) The sequence (2.36) is called the variational complex of the bundle π.

(ii) The operator E de¬ned by (2.37) is called the Euler“Lagrange opera-

tor.

It can be shown that for any ω ∈ Λn (π) one has

h

—

E(ω) = ω (1), (2.38)

from where an explicit formula in local coordinates for E is obtained:

‚

(’1)|σ| Dσ —¦ j .

Ej = (2.39)

‚uσ

σ

The di¬erentials dp,n can also be computed explicitly. In particular, we have

1

d1,n (•) = 1,n

—

’ • ∈ E1 (π) = κ(π).

•, (2.40)

•

1

p,q

Let us now describe the term E1 (E) for a nontrivial equation E. We

shall do it for a broad class of equations which is introduced below.

Note ¬rst that a well-de¬ned action of C-di¬erential operators ∆ ∈

C Diff(F(E, E) on Cartan forms ω ∈ CΛ1 (E) exists. Namely, for a zero-

def

order operator (i.e., for a function on E ∞ ) we set ∆(ω) = ∆ · ω. If now

∆ = σ Xσ , where Xσ = CXi1 —¦ · · · —¦ CXis , X± ∈ D(M ), then

def

∆(ω) = LXi1 (. . . (LXis (ω)) . . . ).

σ=(i1 ...is )

2. HIGHER SYMMETRIES AND CONSERVATION LAWS 75

In general, such a action is not well de¬ned because of the identity

LaY (ω) = aLY (ω) + d(a) § iY (ω).

But if Y = CX and ω ∈ CΛ1 (E), the second summand vanishes and we

obtain the action we seek for.

Let now ∆ ∈ C Diff(κ, F(E)) and ∆1 , . . . , ∆m be the components of this

operator. Then we can de¬ne the form

def

ω∆ = ∆1 (ω 1 ) + · · · + ∆m (ω m ),

j

where ω j = ω(0,...,0) are the Cartan forms. Thus we obtain the mapping

C Diff(κ, F(E)) ’ CΛ1 (E), ∆ ’ ω∆ . On the other hand, assume that

the equation E is determined by the operator ∆ : “(π) ’ “(ξ) and let

P = F(E, ξ). Then to any operator ∈ C Diff(P, F(E)) we can put into cor-

respondence the operator —¦ E ∈ C Diff(κ, F(E)), where E is the restriction

of ∆ onto E ∞ . It gives us the mapping C Diff(P, F(E)) ’ C Diff(κ, F(E)).

In Chapter 5 it will be shown that the forms ω —¦ E vanish which means that

the sequence

0 ’ C Diff(P, F(E)) ’ C Diff(κ, F(E)) ’ CΛ1 (E) ’ 0 (2.41)

is a complex.

Definition 2.16. We say that equation E is -normal if (2.41) is an

exact sequence.

Theorem 2.20 (two-line theorem). Let E ‚ J k (π) be a formally inte-

grable -normal equation in a vector bundle π : E ’ M over a cohomologi-

cally trivial manifold M , dim M = n. Then:

p,q

(i) E1 (E) = 0, if p ≥ 1 and q = n ’ 1, n.

0,n’1 1,n’1

(ii) The di¬erential d0,n’1 : E1 (E) ’ E1 (E) is a monomorphism

1,n’1 ).

and its image coincides with ker(d

1,n’1

(E) coincides with ker( — ).

(iii) The group E1 E

Remark 2.8. The theorem has a stronger version, see [98], but the one

given above is su¬cient for our purposes.

Remark 2.9. The number of nontrivial lines at the top part of the term

E1 relates to the length of the so-called compatibility complex for the opera-

tor E (see [98, 52]). For example, for the Yang“Mill equations (see Section

6 of Chapter 1 one has the three-line theorem, [21].

1,n’1

(E) = ker( — ) are called gen-

Definition 2.17. The elements of E1 E

erating sections of conservation laws.

Theorem 2.20(iii) gives an e¬cient method to compute generating sec-

tions of conservation laws. The following result shows when a generating

76 2. HIGHER SYMMETRIES AND CONSERVATION LAWS