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

these properties is the following result.

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

section corresponds to some conservation law.3 Let — (•) = 0 and the equa-

E

— (•) = (F ) for some C-

tion E be given by the operator ∆ = ∆F . Then ∆

di¬erential operator .

Proposition 2.21. A solution • of the equation — (•) = 0 corresponds

E

to a conservation law of the -normal equation E, if there exists a C-di¬er-

such that — = and the equality

ential operator

—

—¦

•+ = ∆

E ∞.

is valid being restricted onto

Let us describe the action of symmetries on the space of generating

sections. Assume, as above, that E is given by equations F = 0.

Proposition 2.22. Let ω be a conservation law of an -normal equation

E and ψω be the corresponding generating section. Then, if • ∈ sym(E) is a

symmetry, then the generating section

—

• (πω ) + (ψω )

• (ω),

corresponds to the conservation law where the operator is such

that • (F ) = (F ).

We ¬nish this subsection with a discussion of Euler“Lagrange equations

and N¨ther symmetries.

o

Definition 2.18. Let π : E ’ M , dim M = n, be a vector bundle and

L = [ω] ∈ Hh (π), ω ∈ Λn (π), be a Lagrangian. The equation EL = {E(L) =

n

h

0} is called the Euler“Lagrange equation corresponding to the Lagrangian

L, where E is the Euler“Lagrange operator (2.38).

We say that an evolutionary vector ¬eld • is a N¨ther symmetry of L,

o

if • (L) = 0 and denote the Lie algebra of such symmetries by sym(L). It

easy to show that sym(L) ‚ sym(EL ).

∈

Proposition 2.23 (N¨ther theorem). To any N¨ther symmetry

o o •

sym(L) there corresponds a conservation law of the equation EL .

Proof. In fact, since • ∈ sym(L), one has • (ω) = dh ρ for some

ρ ∈ Λn’1 (π). Then, by Green™s formula (2.35), one has

h

—

’ dh (ρ) = = ω(•) ’ dh (ρ) = + dh θ(•) ’ dh (ρ)

• (ω) ω (1)(•)

= E(L)(•) + dh (θ(•) ’ ρ) = 0.

Hence, the form dh (θ(•) ’ ρ) vanishes on E ∞ L and · = θ(•) ’ ρ |E ∞ L is a

desired conserved density.

We illustrate relations between symmetries and conserved densities by

explicit computations for the nonlinear Dirac equations (see Section 5 of

Chapter 1).

2,n’1

3

If E1 (E) = 0, then, as it follows from Theorem 2.20(ii), there is a one-to-one

correspondence between conservation laws and their generating sections.

2. HIGHER SYMMETRIES AND CONSERVATION LAWS 77

Example 2.4 (Conservation laws of the Dirac equations). Let us con-

sider the nonlinear Dirac equations with nonvanishing rest mass (case 4

in Section 5 of Chapter 1). Among the symmetries of this equation there

are the following ones:

‚ ‚ ‚ ‚

V1 = X19 = u4 ’ u3 2 ’ u2 3 + u1 4

‚u1 ‚u ‚u ‚u

‚ ‚ ‚ ‚

’ v4 1 + v3 2 + v2 3 ’ v1 4 ,

‚v ‚v ‚v ‚v

‚ ‚ ‚ ‚

V2 = X20 = v 1 1 + v 2 2 + v 3 3 + v 4 4

‚u ‚u ‚u ‚u

‚ ‚ ‚ ‚

’ u1 1 ’ u2 2 ’ u3 3 ’ u4 4 ,

‚v ‚v ‚v ‚v

‚ ‚ ‚ ‚

V3 = X23 = v 4 1 ’ v 3 2 ’ v 2 3 + v 1 4

‚u ‚u ‚u ‚u

‚ ‚ ‚ ‚

+ u4 1 ’ u3 2 ’ u2 3 + u1 4 . (2.42)

‚v ‚v ‚v ‚v

The generators V1 , V2 , V3 are vertical vector ¬elds on the space J 0 (π) =

π

R8 — R4 ’’ R4 with coordinates x1 , . . . , x4 in the base and u1 , . . . , v 4 along

’

the ¬ber. The ¬elds under consideration are generated by ‚/‚u1 , ‚/‚u2 ,

‚/‚u3 , ‚/‚u4 , ‚/‚v 1 , ‚/‚v 2 , ‚/‚v 3 , ‚/‚v 4 , i.e.,

π— Vj = 0, j = 1, . . . , 3.

(1) (1) (1)

In fact, we need the prolonged vector ¬elds V1 , V2 , V3 to J 1 (π) which

can be calculated from (2.42) using formulas (1.34) on p. 26.

Let L(u, v, uj , v j ) be the Lagrangian de¬ned on J 1 (π) by

L = ’u4 v1 + v 4 u1 ’ u3 v1 + v 3 u2 ’ u2 v1 + v 2 u3 ’ u1 v1 + v 1 u4

1 2 3 4

1 1 1 1

’ v 4 v2 ’ u 4 u 1 + v 3 v2 + u 3 u 2 ’ v 2 v2 ’ u 2 u 3 + v 1 v2 + u 1 u 4

1 2 3 4

2 2 2 2

’ u 3 v3 + v 3 u 1 + u 4 v3 ’ v 4 u 2 ’ u 1 v3 + v 1 u 3 + u 2 v3 + v 2 u 3

1 2 3 4

3 3 3 4

’ u 1 v4 + v 1 u 1 ’ u 2 v4 + v 2 u 2 ’ u 3 v4 + v 3 u 3 ’ u 4 v4 + v 4 u 4

1 2 3 4

4 4 4 4

1

’ K(1 + »3 K), (2.43)

2

where

(x, u, v, uj , v j ) = (x1 , . . . , x4 , u1 , . . . , v 4 , u1 , . . . , u1 , . . . , v1 , . . . , v4 )

4 4

(2.44)

1 4

are local coordinates on J 1 (π) = R44 . An easy calculation shows that the

Euler“Lagrange equations associated to (2.43), i.e.,

‚ ‚L ‚L

’ A =0 (2.45)

A

‚xa ‚za ‚z

are just nonlinear the Dirac equations (1.88), see p. 39. In (2.45) we used the

notation z A , A = 1, . . . , 8, instead of u1 , . . . , u4 , v 1 , . . . , v 4 and summation

convention over A = 1, . . . , 8, a = 1, . . . , 4, if an index occurs twice.

78 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

Let us introduce the form ˜ by

˜ = Lω + (‚A )θA § ωa ,

a

(2.46)

where

ω = dx1 § dx2 § dx3 § dx4 ,

‚ ‚ ‚

a

‚a = , ‚A = A , ‚A = A ,

‚xa ‚z ‚za

ωa = ‚a ω,

θA = dz A ’ za dxa ,

A

(2.47)

and za refers to either uj or va . From (2.45) we derive

j

A

a

˜ = Lω + (‚A L)(dz A ) § ωa ’ (‚A L)za ω

a a A

= L ’ (‚A L)za ω + (‚A L)(dz A ) § ωa . (2.48)

a A a

A

Since L de¬ned by (2.43) is linear with respect to za we derive

1

L ’ (‚A L)za = ’K(1 + »3 K).

a A

(2.49)

2

We now want to compute the Lie derivatives

(1)

Vi ˜,

(1)

i.e., the Lie derivatives of the form ˜ with respect to the vector ¬eld Vi ,

i = 1, 2, 3. We prove the following

Lemma 2.24. The form ˜ is Vi -invariant, i.e.,

(1)

Vi ˜ = 0, i = 1, 2, 3.

Proof. The proof splits in two parts:

1

(1)

1 : Vi K(1 + »3 K)ω = 0, i = 1, 2, 3, (2.50)

2

(1) a

2 : Vi (‚A L) dz A § ω = 0, i = 1, 2, 3, a = 1, . . . , 4. (2.51)

Proof of 1. One has

1

(1) (1)

Vi K(1 + »3 K)ω = Vi (’1 ’ »3 K)dK § ω

2

and due to the de¬nition of K (1.89) on p. 39, dK = 2(u1 du1 + u2 du2 ’

u3 du3 ’ u4 du4 + v 1 dv 1 + v 2 dv 2 ’ v 3 dv 3 ’ v 4 dv 4 ) an easy calculation leads to

(1)

Vi dK = 0, i = 1, 2, 3, (2.52)

which completes the proof of part 1.

Proof of 2. In order to prove (2.51), we introduce four 1-forms

V1— = (‚A L)dz A = v 4 du1 + v 3 du2

1

+ v 2 du3 + v 1 du4 ’ u4 dv 1 ’ u3 dv 2 ’ u2 dv 3 ’ u1 dv 4 ,

V2— = (‚A L)dz A = ’u4 du1 + u3 du2

2

2. HIGHER SYMMETRIES AND CONSERVATION LAWS 79

’ u2 du3 + u1 du4 ’ v 4 dv 1 + v 3 dv 2 ’ v 2 dv 3 + v 1 dv 4 ,

V3— = (‚A L)dz A = v 3 du1 ’ v 4 du2

3

+ v 1 du3 ’ v 2 du4 ’ u3 dv 1 + u4 dv 2 ’ u1 dv 3 + u2 dv 4 ,

V4— = (‚A L)dz A = v 1 du1 + v 2 du2

4

+ v 3 du3 + v 4 du4 ’ u1 dv 1 ’ u2 dv 2 ’ u3 dv 3 ’ u4 dv 4 ,

from which we obtain

dV1— = ’2(du1 § dv 4 + du2 § dv 3 + du3 § dv 2 + du4 d § v 1 ),

dV2— = 2(du1 § du4 ’ du2 § du3 + dv 1 § dv 4 ’ dv 2 § dv 3 ),

dV3— = 2(’du1 § dv 3 + du2 § dv 4 ’ du3 § dv 1 + du4 § dv 2 ),

dV4— = ’2(du1 § dv 1 + du2 § dv 2 + du3 § dv 3 + du4 § dv 4 ). (2.53)

Using (2.42) and (2.53), a somewhat lengthy calculation leads to the follow-

ing result

(1)

Vi (Vj— ) = 0, i = 1, 2, , 3, j = 1, . . . , 4. (2.54)

This completes the proof of the lemma.

Now due to the relation

(1) (1) (1)

(Vi )˜ = (Vi ) d˜ + d(Vi ˜) = 0, i = 1, 2, 3, (2.55)

and

(1)

(Vi ) d˜ = 0, i = 1, 2, 3, (2.56)

on the “equation manifold”, [95], we arrive at

(1)

d(Vi ˜) = 0, i = 1, 2, 3 (2.57)

(1)

on the “equation manifold”. This means that Vi ˜ are conserved currents,

i=1,2,3. Combination of (2.42), (2.48), and (2.54) leads to

(1) (1)

Va— )ωa ,

Vi θ = (Vi (2.58)

i.e., the conserved currents associated to V1 , V2 , V3 are given by

1 : 2 u4 v 4 ’ u3 v 3 ’ u2 v 2 + u1 v 1 dx2 § x3 § dx4

’ (u1 )2 + (u2 )2 ’ (u3 )2 ’ (u4 )2 ’ (v 1 )2 ’ (v 2 )2

+ (v 3 )2 + (v 4 )2 dx1 § dx3 § dx4

+ 2 u4 v 3 + u3 v 4 ’ u2 v 1 ’ u1 v 2 dx1 § dx2 § dx4

’ 2 u4 v 1 ’ u3 v 2 ’ u2 v 3 + u1 v 4 dx1 § dx2 § dx3 ,

2 : 2 v 1 v 4 + v 2 v 3 + u1 u4 + u2 u3 dx2 § dx3 § dx4

80 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

’ 2 ’ u4 v 1 + u3 v 2 ’ u2 v 3 + u1 v 4 dx1 § dx3 § dx4

+ 2 v 1 v 3 ’ v 2 v 4 + u1 u3 ’ u2 u4 dx1 § dx2 § dx4

’ (u1 )2 + (u2 )2 + (u3 )2 + (u4 )2 + (v 1 )2 + (v 2 )2

+ (v 3 )2 + (v 4 )2 dx1 § dx2 § dx3 ,

’ (u1 )2 + (u2 )2 + (u3 )2 ’ (u4 )2 + (v 1 )2 ’ (v 2 )2

3:

’ (v 3 )2 + (v 4 )2 dx2 § dx3 § dx4

’ 2 u4 v 4 ’ u3 v 3 + u2 v 2 + u1 v 1 dx1 § dx3 § dx4

+ 2 v 3 v 4 ’ v 2 v 1 ’ u3 u4 + u1 u2 dx1 § dx2 § dx4

’ 2 v 1 v 4 ’ v 3 v 2 ’ u1 u4 + u2 u3 dx1 § dx2 § dx3 .

Remark 2.10. It is possible to derive the conservation laws obtained

above by the N¨ther theorem 2.23, but we preferred here the explicit way.

o

3. The Burgers equation

Consider the Burgers equation E

ut = uxx + uux (2.59)

and choose internal coordinates on E ∞ by setting uk = u(k,0) . Below we

compute the complete algebra of higher symmetries for (2.59) using the

method described in [60] and ¬rst published in [105].

3.1. De¬ning equations. Let us rewrite restrictions onto E ∞ of all

basic concepts in this coordinate system.

For the total derivatives we obviously obtain

∞

‚ ‚

Dx = + ui+1 , (2.60)

‚x ‚ui

k=0

∞

‚ ‚

i

Dt = + Dx (u2 + u0 u1 ) . (2.61)

‚t ‚ui

k=0

The operator of universal linearization for E is then of the form

2

= D t ’ u 1 ’ u 0 Dx ’ D x , (2.62)

E

and, as it follows from Theorem 2.15 on p. 72, an evolutionary vector ¬eld

∞

‚

i

= Dx (•) (2.63)

•

‚ui

i=1

3. THE BURGERS EQUATION 81

is a symmetry for E if and only if the function • = •(x, t, u0 , . . . , uk ) satis¬es

the equation

2

Dt • = u1 • + u0 Dx • + Dx •, (2.64)

2

where Dt , Dx are given by (2.60), (2.61). Computing Dx • we obtain

k k k

‚2• ‚2• ‚2• ‚•

2

Dx • = +2 ui+1 + ui+1 uj+1 + ui+2 ,

‚x2 ‚x‚ui ‚ui ‚uj ‚ui

i=1 i,j=0 i=0

while

i

i

i

Dx (u0 u1 + u3 ) = u± ui’±+1 + ui+3 .

±

±=0

Hence, (2.64) transforms to

k i

‚• ‚ 2 •

‚• i ‚•

+ u± ui’±+1 = u1 • + u0 +

‚x2

‚t ± ‚ui ‚x

i=1 ±=1

k k

‚2• ‚2•

+2 ui+1 + ui+1 uj+1 . (2.65)

‚x‚ui ‚ui ‚uj

i=1 i,j=0

3.2. Higher order terms. Note now that the left-hand side of (2.65)

is independent of uk+1 while the right-hand one is quadratic in this variable

and is of the form

k’1

‚2• ‚2• ‚2•

2

uk+1 2 + 2uk+1 + ui+1 .

‚x‚uk ‚ui ‚uk

‚uk i=0

It means that

• = Auk + ψ, (2.66)

where A = A(t) and ψ = ψ(t, x, u0 , . . . , uk’1 ). Substituting (2.66) into

equation (2.65) one obtains

k’1 i k

‚ψ i ‚ψ k

™

Auk + + u± ui’±+1 + ui uk’i+1 A

‚t ± ‚ui i

i=1 ±=1 i=1

k’1

‚ψ ‚ 2 ψ ‚2ψ

= u1 (Auk + ψ) + u0 + +2 ui+1

‚x2

‚x ‚x‚ui

i=1

k’1

‚2ψ

+ ui+1 uj+1 ,

‚ui ‚uj

i,j=0

82 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

def

™

where A = dA/dt. Here again everything is at most quadratic in uk , and

equating coe¬cients at u2 and uk we get

k

k’2

‚2ψ ‚2ψ ‚2ψ ™

= 0, 2 ui+1 + = ku1 A + A.

‚u2 ‚ui ‚uk’1 ‚x‚uk’1

k’1 i=0

Hence,

1 ™

ψ = (ku0 A + Ax + a)uk’1 + O[k ’ 2],

™

2

where a = a(t) and O[l] denotes a function independent of ui , i > l. Thus

1 ™

• = Auk + (ku0 A + Ax + a)uk’1 + O[k ’ 2]

™ (2.67)

2

which gives the “upper estimate” for solutions of (2.64).

3.3. Estimating Jacobi brackets. Let

• = •(t, x, u0 , . . . , uk ), ψ = ψ(t, x, u0 , . . . , ul )

be two symmetries of E. Then their Jacobi bracket restricted onto E ∞ looks

as

l k

‚ψ ‚•

i j

{•, ψ} = ’

Dx (•) Dx (ψ) . (2.68)

‚ui ‚uj

i=0 i=0

Suppose that the function • is of the form (2.67) and similarly

1 ™

™

ψ = Bul + (lu0 B + Bx + b)ul’1 + O[l ’ 2]

2

and let us compute (2.68) for these functions temporary denoting ku0 A +

™ ™ ¯ ¯

A + a and lu0 B + B + b by A and B respectively. Then we have:

1¯ 1 l’1 1¯ ¯

l

{•, ψ} = Dx (Auk + Auk’1 )B + Dx (Auk + Auk’1 )B

2 2 2

1¯ 1 k’1 1¯ ¯

k

’ Dx (Bul + Bul’1 )A ’ Dx (Buk + Bul’1 )A + O[k + l ’ 1]

2 2 2

1 ¯1 1¯

¯ ¯

= (lDx (A)uk+l’2 + Auk+l’1 )B + (Auk+l’1 + Auk+l’2 )B

2 2 2

1 ¯1 1¯

¯ ¯

’ (kDx (B)uk+l’2 + Buk+l’1 )A ’ (Buk+l’1 + Buk+l’2 )A +

2 2 2

O[k + l ’ 3],

or in short,

1™ ™

{•, ψ} = (lAB ’ K BA)uk+l’2 + O[k + l ’ 3]. (2.69)

2

3. THE BURGERS EQUATION 83

3.4. Low order symmetries. These computations were done already

in Section 3 of Chapter 1 (see equation (1.61)). They can also be done

independently taking k = 2 and solving equation (2.64) directly. Then one

obtains ¬ve independent solutions which are

•0 = u 1 ,

1

•1 = tu1 + 1,

1

•0 = u 2 + u 0 u 1 ,

2

1 1

•1 = tu2 + (tu0 + x)u1 + u0 ,

2

2 2

2 2 2

•2 = t u2 + (t u0 + tx)u1 + tu0 + x. (2.70)

3.5. Action of low order symmetries. Let us compute the action

def

Tij = {•j , •} = ’

•j •j

i i i

of symmetries •j on other symmetries of the equation E.

i

0 one has

For •1

‚ ‚

0

’ ’ Dx = ’ .

T1 = = ui+1

u1 u1

‚ui ‚x

i≥0

Hence, if • = Auk + O[k ’ 1] is a function of the form (2.67), then we obtain

1™

0

T1 • = ’ Auk’1 + O[k ’ 2].

2

Consequently, if • is a symmetry, then, since sym(E) is closed under the

Jacobi bracket,

k’1

dk’1 A

1

(T1 )k’1 •

0

’

= u1 + O[0]

dtk’1

2

is a symmetry as well. But from (2.70) one sees that ¬rst-order symmetries

are linear in t. Thus, we have the following result:

Proposition 2.25. If • = Auk + O[k ’ 1] is a symmetry of the Burgers

equation, then A is a k-th degree polynomial in t.

3.6. Final description. Note that direct computations show that the

equation E possesses a third-order symmetry of the form

3 3 3

•0 = u 3 + u 0 u 2 + u 2 + u 2 u 1 .

3

20 40

2

2 0

Using the actions T2 and T3 , one can see that

k’1

k!(k ’ 1)!

3

((T2 )i

2 0

T2 )k’1 )u1

2

—¦ —¦ ’ uk + O[k ’ 1]

(T3 = (2.71)

(k ’ i)!

2

is a symmetry, since u1 is the one.

84 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

Theorem 2.26. The symmetry algebra sym(E) for the Burgers equation

E = {ut = uux + uxx }, as a vector space, is generated by elements of the

form

•i = ti uk + O[k ’ 1], k ≥ 1, i = 0, . . . , k,

k

which are polynomial in all variables. For the Jacobi bracket one has

1

{•i , •j } = (li ’ kj)•i+j’1 + O[k + l ’ 3]. (2.72)

k l k+l’2

2

The Lie algebra sym(E) is simple and has •0 , •2 , and •0 as its generators.

1 2 3

Proof. It only remains to prove that all •i are polynomials and that

k

sym(E) is a simple Lie algebra. The ¬rst fact follows from (2.71) and from

2 0

the obvious observation that coe¬cients of both T2 and T3 are polynomials.

Let us prove that sym(E) is a simple Lie algebra. To do this, let us

introduce an order in the set {•i } de¬ning

k

def

¦ k(k+1) +i = •i .

k

2

Then any symmetry may be represented as s »± ¦± , » ∈ R.

±=1

Let I ‚ sym(E) be an ideal and ¦ = ¦s + s’1 »± ¦± be its element.

±=1

i for some k ≥ 1 and i ¤ k.

Assume that ¦s = •k

Note now that

‚ ‚ ‚

1 ±

’ tDx = ’t

T1 = Dx (tu1 + 1)

‚u± ‚u0 ‚x

±≥0

and

‚ ‚

0 ± 2

’ D x ’ u 0 Dx ’ u 1 = ’ .

T2 = Dx (u2 + u0 u1 )

‚u± ‚t

±≥0

Therefore,

((T1 )k’1 —¦ (T2 )i )¦ = c•0 ,

1 0

1

where the coe¬cient c does not vanish. Hence, I contains the function •0 . 1

But due to (2.71) the latter, together with the functions •2 and •0 , generates

2 3

the whole algebra.

Further details on the structure of sym(E) one can ¬nd in [60].

4. The Hilbert“Cartan equation

We compute here classical and higher symmetries of the Hilbert“Cartan

equation [2]. Since higher symmetries happen to depend on arbitrary func-

tions, we consider some special choices of these functions [38].

4. THE HILBERT“CARTAN EQUATION 85

4.1. Classical symmetries. The Hilbert“Cartan equation is in e¬ect

an underdetermined system of ordinary di¬erential equations in the sense of

De¬nition 1.10 of Subsection 2.1 in Chapter 1. The number of independent

variables, n, is one while the number of dependent variables, m, is two. Local

coordinates are given by x, u, v in J 0 (π), while the order of the equations is

two, i.e.,

2

ux = vxx (2.73)

The representative morphism (see De¬nition 1.6 on p. 6) ¦ is given by

2

¦∆ (x, u, v, ux , vx , uxx , vxx ) = ux ’ vxx . (2.74)

The total derivative operator Dx is given by the formula

‚ ‚ ‚ ‚ ‚

+ ···

D = Dx = + ux + vx + uxx + vxx (2.75)

‚x ‚u ‚v ‚ux ‚vx

To construct classical symmetries for (2.73), we start from the vector ¬eld

X, given by

‚ ‚ ‚

X = X(x, u, v) + U0 (x, u, v) + V0 (x, u, v)

‚x ‚u ‚v

‚ ‚

+ U1 (x, u, x, ux , vx ) + V1 (x, u, x, ux , vx )

‚ux ‚vx

‚ ‚

+ U2 (x, u, x, ux , vx , uxx , vxx ) + V2 (x, u, x, ux , vx , uxx , vxx ) .

‚uxx ‚vxx

The de¬ning relations (1.34) (see p. 26) for U1 , V1 , U2 , V2 are

U1 = D(U0 ) ’ ux D(X) = D(U0 ’ ux X) + uxx X,

V1 = D(V0 ) ’ vx D(X) = D(V0 ’ vx X) + vxx X,

U2 = D(U1 ) ’ uxx D(X) = D 2 (U0 ’ ux X) + uxxx X,

V2 = D(V1 ) ’ vxx D(X) = D 2 (V0 ’ vx X) + vxxx X. (2.76)

From (2.76) we derive the following explicit expressions for U1 , V1 , U2 , V2 :

U1 = U0,x + U0,u ux + U0,v vx ’ ux (X0,x + X0,u ux + X0,v vx ),

V1 = V0,x + V0,u ux + V0,v vx ’ ux (X0,x + X0,u ux + X0,v vx ),

U2 = U0,xx + 2U0,xu ux + 2U0,xv vx + U0,uu u2 + 2U0,uv ux vx + U0,u uxx

x

2

+ U0,vv vx + U0,v vxx ’ 2uxx (X0,x + X0,u ux + X0,v vx )

’ ux (X0,xx + 2X0,xu ux + 2X0,xv vx

+ X0,uu u2 + 2X0,uv ux vx + X0,u uxx + X0,vv vx + X0,v vxx ),

2

x

V2 = V0,xx + 2V0,xu ux + 2V0,xv vx + V0,uu u2 + 2V0,uv ux vx + V0,u uxx

x

2

+ V0,vv vx + V0,v vxx ’ 2uxx (X0,x + X0,u ux + X0,v vx )

’ vx (X0,xx + 2X0,xu ux + 2X0,xv vx + X0,uu u2

x

2

+ 2X0,uv ux vx + X0,u uxx + X0,vv vx + X0,v vxx ). (2.77)

86 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

Now the symmetry-condition X(¦∆ ) |E = 0 results in

2

U1 ’ 2vxx V2 = »(ux ’ vxx ) (2.78)

which is equivalent to

1

U1 ’ 2(ux ) 2 V2 = 0 mod ¦∆ = 0, (2.79)

which results in

U0,x + U0,u ux + U0,v vx ’ ux (X0,x + X0,u ux + X0,v vx )

’ V0,xx + 2V0,xu ux + 2V0,xv vx + V0,uu u2 + 2V0,uv ux vx + V0,u uxx

x

2

+ V0,vv vx + V0,v vxx ’ 2uxx (X0,x + X0,u ux + X0,v vx )

’ vx (X0,xx + 2X0,xu ux + 2X0,xv vx + X0,uu u2 + 2X0,uv ux vx

x

+ X0,u uxx + X0,vv vx + X0,v vxx ) · 2(ux )1/2 = 0.

2

(2.80)

Equation (2.80) is a polynomial in the “variables” (ux )1/2 , vx , uxx , the

coe¬cients of which should vanish. From this observation we obtain the

following system of equations:

1: U0,x = 0,

u1/2 : ’2V0,xx = 0,

x

u1/2 vx : ’4V0,xv + 2X0,xx = 0,

x

u1/2 uxx : ’2V0,u = 0,

x

u1/2 uxx vx : 2X0,u = 0,

x

u1/2 vx :

2

’2V0,vv + 4X0,xv = 0,

x

u1/2 vx :

3

2X0,vv = 0,

x

U0,u ’ X0,x ’ 2V0,v + 4X0,x = 0,

ux :

’X0,v + 4X0,v + 2X0,v = 0,

u x vx :

u2 : ’X0,u + 4X0,u = 0,

x

u3/2 : ’4v0,xu = 0,

x

u3/2 vx : ’4V0,uv + 4X0,xu = 0,

x

u3/2 vx :

2

4X0,uv = 0,

x

u5/2 : ’2V0,uu = 0,

x

u5/2 vx : 2X0,uu = 0,

x

vx : U0,v = 0. (2.81)

From system (2.81) we ¬rst derive:

X0,u = X0,v = 0, V0,uu = V0,uv = V0,vv = 0 = V0,u = V0,xx ,

4. THE HILBERT“CARTAN EQUATION 87

[Ai , Aj ] A1 A2 A3 A4 A5 A6

A1 0 0 0 0 A1 A3

0 2A2 ’3A2

A2 0 0

A3 0 A3 0 0

’A6

A4 0 0

A5 0 A6

A6 0

Figure 2.1. Commutator table for classical symmetries of

the Hilbert“Cartan equation

which results in the equality X(x, u, v) = H(x) and in the fact that V0 is

independent of u, being of degree 1 in v and of degree 1 in x, i.e.,

X(x, u, v) = H(x), V0 = a0 + a1 x + a2 v + a3 xv.

1/2

Now from the equation labeled by ux vx in (2.81) we derive

H(x) = a3 x2 + a4 x + a5 . (2.82)

From the equations U0,v = 0 and U0,u + 3X0,x ’ 2V0,v = 0 we get

U0 = ’(4a3 x + 2a2 ’ 3a5 )u + G(x). (2.83)

Finally from U0,x = 0 we arrive at a3 = 0, G(x) = a6 , from which the general

solution is obtained as

U0 = (2a2 ’ 3a4 )u + a6 ,

X = a4 x + a5 , V0 = a0 + a1 x + a2 v.

This results in a 6-dimensional Lie algebra, the generators of which are given

by

‚

A1 = ,

‚x

‚

A2 = ,

‚u

‚

A3 = ,

‚v

‚ ‚

A4 = 2u +v ,

‚u ‚v

‚ ‚

’ 3u ,

A5 = x

‚x ‚u

‚

A6 = x ,

‚v

while the commutator table is given on Fig. 2.1.

4.2. Higher symmetries. As a very interesting and completely com-

putable application of the theory of higher symmetries developed in Subsec-

tion 2.1, we construct in this section the algebra of higher symmetries for

88 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

the Hilbert“Cartan equation E

2

ux ’ vxx = 0. (2.84)

First of all, note that E ∞ is given by the system of equations:

Di (ux ’ vxx ) = 0,

2

i = 0, 1, . . . (2.85)

where D is de¬ned by

∞ ∞

‚ ‚ ‚

D= + uk+1 + vk+1 , (2.86)

‚x ‚uk ‚vk

k=0 k=0

and uk = ux . . . x . So from (2.84) we have

k times

D1 F = u2 ’ 2v2 v3 = 0,

D2 F = u3 ’ 2v3 ’ 2v2 v4 = 0,

2

i

i

i

D F = u1+i ’ v2+l v2+i’l = 0,

l

l=0

2

i = 3, . . . , with F (x, u, v, u1 , v1 , u2 , v2 ) = u1 ’ v2 = 0.

In order to construct higher symmetries of (2.84), we introduce internal

coordinates on E ∞ which are

x, u, v, v1 , v2 , v3 , · · · (2.87)

The total derivative operator restricted to E ∞ , again denoted by D, is given

by the following expression

‚ 2‚ ‚ ‚

D= + v2 + v1 + vi+1 ,

‚x ‚u ‚v ‚vi

i>0

n

‚ 2‚ ‚ ‚

D(n) = + v2 + v1 + vi+1 . (2.88)

‚x ‚u ‚v ‚vi

i>0

Suppose that a vertical vector ¬eld V = with the generating function ¦,

¦

¦ = f u (x, u, v, v1 , . . . , vn ), f (x, u, v, v1 , . . . , vn ) ,

v

(2.89)

is a higher symmetry of E. We introduce the notation

f [vk ] = f (x, u, v, v1 , . . . , vk ). (2.90)

Since the vertical vector ¬eld V is formally given by

‚ ‚ ‚ ‚

V = f u [vn ] + f v [vn ] + f v1 [vn+1 ] + f v2 [vn+2 ] + ..., (2.91)

‚u ‚v ‚v1 ‚v2

we derive the following symmetry conditions from (2.84)

D(n) f u [vn ] ’ 2v2 f v2 [vn+2 ] = 0,

D(n) f v [vn ] ’ f v1 [vn+1 ] = 0,

D(n+1) f v1 [vn+1 ] ’ f v2 [vn+2 ] = 0. (2.92)

4. THE HILBERT“CARTAN EQUATION 89

In e¬ect, the second and third equation of (2.92) are just the de¬nitions

of f v1 [vn+1 ] and f v2 [vn+2 ], due to the evolutionary property of ¦ . We now

want to construct the general solution of system (2.92). In order to do so,

we ¬rst solve the third equation in (2.92) for f v2 [vn+2 ],

f v2 [vn+2 ] = D(n+1) f v1 [vn+1 ], (2.93)

and the system reduces to

D(n) f u [vn ] ’ 2v2 D(n+1) f v1 [vn+1 ] = 0,

D(n) f v [vn ] ’ f v1 [vn+1 ] = 0. (2.94)

Remark 2.11. At this stage it would be possible to solve the last equa-

tion for f v1 [vn+1 ], but we prefer not to do so.

Now (2.94) is a polynomial in vn+2 of degree 1 and (2.94) reduces to

‚f v1 [vn+1 ]

’2v2

vn+2 : = 0,

‚vn+1

D(n) f u [vn ] ’ 2v2 D(n) f v1 [vn ] = 0,

1:

D(n) f v [vn ] ’ f v1 [vn ] = 0.

: (2.95)

In (2.95) and below, “vn+2 :” refers to the coe¬cient at vn+2 in a particular

equation. From (2.95) we arrive, due to the fact that second and third

equation are polynomial in vn+1 , at

‚f u [vn ] ‚f u1 [vn ]

’ 2v2

vn+1 : = 0,

‚vn ‚vn

D(n’1) f u [vn ] ’ 2v2 D(n’1) f v1 [vn ] = 0,

1:

‚f v [vn ]

vn+1 : = 0,

‚vn

D(n’1) f v [vn ] ’ f v1 [vn ] = 0.

1: (2.96)

To solve system (2.96), we ¬rst note that

f v [vn ] = f v [vn’1 ]. (2.97)

By di¬erentiation of the fourth equation in (2.96) twice with respect to v n ,

we obtain

‚ 2 f v1 [vn ]

= 0. (2.98)

2

‚vn

By consequence, f v1 is linear with respect to vn , i.e.,

f v1 [vn ] = H 1 [vn’1 ] + vn H 2 [vn’1 ]. (2.99)

Now, substitution of (2.97) and (2.99) into (2.96) yields the following

system of equations

‚f u [vn ]

’ 2v2 H 2 [vn’1 ] = 0,

‚vn

90 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

D(n’1) f u [vn ] ’ 2v2 D(n’1) H 1 [vn’1 ] ’ 2v2 vn D(n’1) H 2 [vn’1 ] = 0,

D(n’1) f v [vn’1 ] ’ H 1 [vn’1 ] ’ vn H 2 [vn’1 ] = 0. (2.100)

We solve the ¬rst equation in (2.100) for f u [vn ], i.e.,

f u [vn ] = 2v2 vn H 2 [vn’1 ] + H 3 [vn’1 ], (2.101)

and from the second and third equation in (2.100) we arrive at

2v3 vn H 2 [vn’1 ] + 2v2 vn D(n’1) H 2 [vn’1 ] + D(n’1) H 3 [vn’1 ]

’ 2v2 D(n’1) H 1 [vn’1 ] ’ 2v2 vn D(n’1) H 2 [vn’1 ] = 0,

D(n’1) f v [vn’1 ] ’ H 1 [vn’1 ] ’ vn H 2 [vn’1 ] = 0. (2.102)

Due to cancellation of second and ¬fth term in the ¬rst equation of (2.102)

and its polynomial structure with respect to vn , we obtain a resulting system

of four equations:

‚H 3 [vn’1 ] ‚H 1 [vn’1 ]

2

’ 2v2

vn : 2v3 H [vn’1 ] + = 0,

‚vn’1 ‚vn’1

D(n’2) H 3 [vn’1 ] ’ 2v2 D(n’2) H 1 [vn’1 ] = 0,

1:

‚f v [vn’1 ]

’ H 2 [vn’1 ] = 0,

vn :

‚vn’1

D(n’2) f v [vn’1 ] ’ H 1 [vn’1 ] = 0.

1: (2.103)

From (2.103) we solve the third equation for H 2 [vn’1 ],

‚f v [vn’1 ]

2

H [vn’1 ] = , (2.104)

‚vn’1

and integrate the ¬rst one in (2.103):

‚f v [vn’1 ] ‚H 3 [vn’1 ] ‚H 1 [vn’1 ]

’ 2v2

2v3 + = 0, (2.105)

‚vn’1 ‚vn’1 ‚vn’1

which leads to

H 3 [vn’1 ] = 2v2 H 1 [vn’1 ] ’ 2v3 f v [vn’1 ] + H 4 [vn’2 ]. (2.106)

By obtaining (2.106), we have to put in the requirement n ’ 1 > 3 and we

shall return to this case in the next subsection.

We now proceed by substituting the results (2.104) and (2.106) into

(2.103), which leads to

2v3 H 1 [vn’1 ] + 2v2 D(n’2) H 1 [vn’1 ] ’ 2v4 f v [vn’1 ] ’ 2v3 D(n’2) f v [vn’1 ]

+ D(n’2) H 4 [vn’2 ] ’ 2v2 D(n’2) H 1 [vn’1 ] = 0,

D(n’2) f v [vn’1 ] ’ H 1 [vn’1 ] = 0. (2.107)

By cancellation of the second and sixth term in the ¬rst equation of (2.107),

we ¬nally arrive at

D(n’2) f v [vn’1 ] ’ H 1 [vn’1 ] = 0,

4. THE HILBERT“CARTAN EQUATION 91

D(n’2) H 4 [vn’2 ] ’ 2v4 f v [vn’1 ] = 0, (2.108)

where the ¬rst equation in (2.108) can be considered as de¬ning relation

for H 1 [vn’1 ], while the second equation determines f v [vn’1 ] in terms of an

arbitrary function H 4 [vn’2 ]. The ¬nal result can now be obtained by (2.104)

and (2.106):

‚f v [vn’1 ]

2

H [vn’1 ] = ,

‚vn’1

H 3 [vn’1 ] = 2v2 H 1 [vn’1 ] ’ 2v3 f v [vn’1 ] + H 4 [vn’2 ], (2.109)

together with (2.108) and (2.101):

‚f v [vn’1 ]

u

+ 2v2 H 1 [vn’1 ] ’ 2v3 f v [vn’1 ] + H 4 [vn’2 ],

f [vn ] = 2v2 vn

‚vn’1

f v [vn ] = f v [vn’1 ], (2.110)

whereas in (2.110) f v [vn’1 ], H 1 [vn’1 ] are de¬ned by (2.108) in terms of an

arbitrary function H 4 [vn’2 ]! The general result of this section can now be

formulated in the following

Theorem 2.27. Let H be an arbitrary function of the variables x, u,

v, . . . , vn’2 , i.e.,

H = H[vn’2 ], (2.111)

and let us de¬ne

1 (n’2)

f v [vn’1 ] = D H[vn’2 ],

2v4

f u [vn ] = 2v2 D(n’1) f v [vn’1 ] ’ 2v3 f v [vn’1 ] + H[vn’2 ]. (2.112)

Then the vector ¬eld

‚ ‚

V = f u [vn ] + f v [vn’1 ] (2.113)

‚u ‚v

is a higher symmetry of (2.85).

Conversely, given a higher symmetry of (2.85), then there exists a func-

tion H, such that the components f u , f v of V are de¬ned by (2.112).

4.3. Special cases. Due to the restriction n > 4 the result (2.109) and

(2.110) holds for

n = 5, . . . (2.114)

meaning that H 4 [vn’2 ] is a free function of x, u, v, . . . , vn’2 and f v [vn’1 ] is

obtained by (2.109)

1 (n’2) 4

f v [vn’1 ] = D H [vn’2 ]. (2.115)

2v4

92 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

From (2.115) and (2.109) it is clear that f v [vn’1 ] is linear with respect

to the variable vn’1 and

vn’1 ‚H 4 [vn’2 ]

v

+ f v [vn’2 ].

f [vn’1 ] = (2.116)

2v4 ‚vn’2

Moreover, the requirement that f v [vn’1 ] is independent of vn’1 reduces to

H 4 [vn’2 ] to be independent of vn’2 , i.e.,

‚f v [vn’1 ]

= 0 ’ H 4 [vn’2 ] = H 4 [vn’3 ]. (2.117)

‚vn’1

The result (2.117) holds for all n > 5.

The results for higher symmetries, or Lie“B¨cklund transformations, for

a

n < 6 are obtained by imposing additional conditions on the coe¬cient f v

of the evolutionary vector ¬eld.

The case n = 5.

1 ‚H 4 [v3 ] 4 ‚H 4 [v3 ]

2 ‚H [v3 ]

v

f [v4 ] = ( + v2 + v1

2v4 ‚x ‚u ‚v

‚H 4 [v3 ] ‚H 4 [v3 ] ‚H 4 [v3 ]

+ v2 + v3 + v4 ).

‚v1 ‚v2 ‚v3

The requirement that f v [v4 ] is independent of v4 now leads to a genuine ¬rst

order partial di¬erential equation, i.e.,

‚H 4 4 ‚H 4 ‚H 4 ‚H 4

2 ‚H

+ v2 + v1 + v2 + v3 = 0, (2.118)

‚x ‚u ‚v ‚v1 ‚v2

and the general solution is given in terms of the invariants of the correspond-

ing vector ¬eld

‚ 2‚ ‚ ‚ ‚

U= + v2 + v1 + v2 + v3 , (2.119)

‚x ‚u ‚v ‚v1 ‚v2

where the set of invariants is given by

z1 = v 3 ,

z2 = v2 ’ v3 x,

z3 = 2v1 ’ 2v2 x + v3 x2 ,

z4 = 6v ’ 6v1 x + 3v2 x2 ’ v3 x3 ,

z5 = 3u ’ 3v2 x + 3v2 v3 x2 ’ v3 x3 .

2 2

(2.120)

So H 4 is given by

H 4 = H 4 (z1 , z2 , z3 , z4 , z5 ), (2.121)

whereas the formulas for f v and f u reduce to

‚H 4 ‚H 4 ‚2H 4

u 4

f = H ’ v2 ’ v3 + v 2 v4 2,

‚v2 ‚v3 ‚v3

1 ‚H 4

v

f= . (2.122)

2 ‚v3

5. THE CLASSICAL BOUSSINESQ EQUATION 93

The requirement the function f v is independent of

The case n = 4.

v3 reduces to

‚2H 4

2 = 0, (2.123)

‚v3

and (2.118)

‚H 4 4 ‚H 4 ‚H 4 ‚H 4

2 ‚H

+ v2 + v1 + v2 + v3 = 0. (2.124)

‚x ‚u ‚v ‚v1 ‚v2

Substitution of (2.123) into (2.124) immediately leads to the condition

‚‚

H 4 = 0, (2.125)

‚v2 ‚v3

i.e.,

f v = f v (x, u, v, v1 ), (2.126)

and the result completely reduces to the second order higher symmetries

obtained by Anderson [3] and [2] leading to the 14-dimensional Lie algebra

G2 .

5. The classical Boussinesq equation

The classical Boussinesq equation is written as the following system

of partial di¬erential equations in J 3 (π), where π : R2 — R2 ’ R2 with

independent variables x, t and u, v for dependent ones:

ut = (uv + ±vxx )x = ux v + uvx + ±vxxx ,

1

vt = (u + v 2 )x = ux + vvx . (2.127)

2

So in this application u = (u, v) and (x1 , x2 ) = (x, t). In order to construct

higher symmetries of (2.127), we have to construct solutions of the symmetry

condition which are discussed in Section 2. For evolution equations it is

custom to choose internal coordinates as x, t, u, v, u1 , v1 , u2 , v2 , . . . , where

‚iu ‚iv

ui = , vi = . (2.128)

‚xi ‚xi

The partial derivative operators Dx and Dt are de¬ned on E ∞ by

‚ ‚ ‚

Dx = + ui+1 + vi+1 ,

‚x ‚ui ‚vi

i>0 i>0

‚ ‚ ‚

Dt = + uit + vit , (2.129)

‚t ‚ui ‚vi

i>0 i>0

while expressions for uit and vit are derived from (2.127) by