the time-dependent ¬eld setting. We de¬ne the vector ¬elds Y1 , Y2 , Y3 by

Y1 = V7 ’ V21 ,

Y2 = V8 ’ V31 , (1.135)

Y3 = V9 ’ V11 ,

6. SYMMETRIES OF THE SELF-DUAL SU (2) YANG“MILLS EQUATIONS 53

i.e., put C 1 , C 2 , and C 3 equal to g ’1 . It results in 36 equations for the

functions Aa :

µ

A2 + A1 ’ x2 A1 + x1 A1 = 0, nonumber

1: (1.136)

1 2 1,1 1,2

’A1 + A2 ’ x2 A2 + x1 A2 = 0,

2: 1 2 1,1 1,2

A3 ’ x2 A3 + x1 A3 = 0,

3: 2 1,1 1,2

’A1 + A2 ’ x2 A1 + x1 A1 = 0,

4: 1 2 2,1 2,2

’A2 ’ A1 ’ x2 A2 + x1 A2 = 0,

5: 1 2 2,1 2,2

’A3 ’ x2 A3 + x1 A3 = 0,

6: 1 2,1 2,2

A2 ’ x2 A1 + x1 A1 = 0,

7: 3 3,1 3,2

’A1 ’ x2 A2 + x1 A2 = 0,

8: 3 3,1 3,2

’x2 A3 + x1 A3 = 0,

9: 3,1 3,2

A2 ’ x2 A1 + x1 A1 = 0,

10 : 4 4,1 4,2

’A1 ’ x2 A2 + x1 A2 = 0,

11 : 4 4,1 4,2

’x2 A3 + x1 A3 = 0,

12 : (1.137)

4,1 4,2

’A3 ’ A1 ’ x1 A1 + x3 A1 = 0,

13 : 1 3 1,3 1,1

’A2 ’ x1 A2 + x3 A2 = 0,

14 : 3 1,3 1,1

A1 ’ A3 ’ x1 A3 + x3 A3 = 0,

15 : 1 3 1,3 1,1

’A3 ’ x1 A1 + x3 A1 = 0,

16 : 2 2,3 2,1

’x1 A2 + x3 A2 = 0,

17 : 2,3 2,1

A1 ’ A3 ’ x1 A3 + x3 A3 = 0,

18 : 2 3 2,3 2,1

A1 ’ x1 A1 + x3 A1 = 0,

19 : 1 3,3 3,1

A2 ’ x1 A2 + x3 A2 = 0,

20 : 1 3,3 3,1

A3 + A1 ’ x1 A3 + x3 A3 = 0,

21 : 1 3 3,3 3,1

’A3 ’ x1 A1 + x3 A1 = 0,

22 : (1.138)

4 4,3 4,1

’x1 A2 + x3 A2 = 0,

23 : 4,3 4,1

A1 ’ x1 A3 + x3 A3 = 0,

24 : (1.139)

4 4,3 4,1

’x2 A1 + x3 A1 = 0,

25 : 1,3 1,2

’A3 ’ x2 A2 + x3 A2 = 0,

26 : 1 1,3 1,2

A2 ’ x2 A3 + x3 A3 = 0,

27 : 1 1,3 1,2

’A1 ’ x2 A1 + x3 A1 = 0,

28 : 3 2,3 2,2

’A3 ’ A2 ’ x2 A2 + x3 A2 = 0,

29 : 2 3 2,3 2,2

A2 ’ A3 ’ x2 A3 + x3 A3 = 0,

30 : 2 3 2,3 2,2

54 1. CLASSICAL SYMMETRIES

A1 ’ x2 A1 + x3 A1 = 0,

31 : 2 3,3 3,2

A2 ’ A3 ’ x2 A2 + x3 A2 = 0.

32 : 2 3 3,3 3,2

A3 + A2 ’ x2 A3 + x3 A3 = 0.

33 : 2 3 3,3 3,2

’x2 A1 + x3 A1 = 0.

34 : 4,3 4,2

’A3 ’ x2 A2 + x3 A2 = 0.

35 : 4 4,3 4,2

A2 ’ x2 A3 + x3 A3 = 0.

36 : (1.140)

4 4,3 4,2

We shall now indicate in more detail how to solve (1.140).

Note, that due to (1.137)

A3 = F 1 (r1,2 , x3 ), (1.141)

4

where

1

r1,2 = (x2 + x2 ) 2 , (1.142)

1 2

and due to (1.139)

‚F 1 (r1,2 , x3 ) x3 ‚F 1 (r1,2 , x3 )

A1 ’

= x1 . (1.143)

4

‚x3 r1,2 ‚r1,2

Now let

‚F 1 (r1,2 , x3 ) x3 ‚F 1 (r1,2 , x3 ) def

’ = H(r1,2 , x3 ). (1.144)

‚x3 r1,2 ‚r1,2

Substitution of (1.141) and (1.144) into (1.138) results in

x2 ‚H(r1,2 , x3 ) ‚H(r1,2 , x3 )

1

’ x2

F (r1,2 , x3 ) = x3 H(r1,2 , x3 ) + 1 , (1.145)

1

r1,2 ‚r1,2 ‚x3

or

1 ‚H(r1,2 , x3 ) ‚H(r1,2 , x3 )

F 1 (r1,2 , x3 ) ’ x3 H(r1,2 , x3 ) = x2 ’ .

1

r1,2 ‚r1,2 ‚x3

(1.146)

Di¬erentiation of (1.146) with respect to x1 , x2 yields

1 ‚H(r1,2 , x3 ) ‚H(r1,2 , x3 )

’ = 0,

r1,2 ‚r1,2 ‚x3

F 1 (r1,2 , x3 ) = x3 H(r1,2 , x3 ). (1.147)

From the second equation in (1.147) and equation (1.144) we obtain

H(r1,2 , x3 ) = l(r), (1.148)

where

1

r = (x2 + x2 + x2 ) 2 , (1.149)

1 2 3

and ¬nally, due to (1.147) and (1.123), one has

A2 = x1 l(r), A2 = x2 l(r), A3 = x3 l(r). (1.150)

4 4 4

6. SYMMETRIES OF THE SELF-DUAL SU (2) YANG“MILLS EQUATIONS 55

Handling the remaining system in a similar way, a straightforward but te-

dious computation leads to the general solution of (1.123), i.e.,

1 1

A1 = x2 f (r) + k(r), A1 = x1 x2 f (r) ’ x3 u(r),

1

21 2

2

1 1

A1 = x1 x3 f (r) + x2 u(r), A2 = x1 x2 l(r) + x3 u(r),

3 1

2 2

1 1

A2 = x2 f (r) + k(r), A2 = x2 x3 f (r) ’ x1 u(r),

2

22 3

2

1 1

A3 = x1 x3 f (r) ’ x2 u(r), A3 = x2 x3 f (r) + x1 u(r),

1 2

2 2

1

A3 = x2 f (r) + k(r), A2 = x1 l(r),

3 3 4

2

A2 = x2 l(r), A3 = x3 l(r), (1.151)

4 4

where u, l, k, f are functions of r.

Substitution of (1.151) into (1.95) and (1.95) yields a system of three

ordinary di¬erential equations for the functions u, l, k, f :

1

l + u ’ gru2 ’ grul + grf k = 0,

2

1

r2 u + 2ru ’ rl ’ gr 3 ul + grk 2 + gr3 f k = 0,

2

1 1

k ’ rf ’ grku ’ grlk ’ gr3 f u = 0. (1.152)

2 2

If we choose

h(r) a(r)

, u(r) = ’

f (r) = k(r) = 0, l(r) = , (1.153)

r r

we are led by (1.151), (1.153) to the monopole solution obtained by Prasad

and Sommerfeld [84] by imposing the ansatz (1.151), (1.153).

56 1. CLASSICAL SYMMETRIES

CHAPTER 2

Higher symmetries and conservation laws

In this chapter, we specify general constructions described for in¬nite jets

to in¬nitely prolonged di¬erential equations. We describe basic structures

existing on these objects, give an outline of di¬erential calculus over them

and introduce the notions of a higher symmetry and of a conservation law.

We also compute higher symmetries and conservation laws for some

equations of mathematical physics.

1. Basic structures

Now we introduce the main object of our interest:

Definition 2.1. The inverse limit proj liml’∞ E l with respect to pro-

jections πl+1,l is called the in¬nite prolongation of the equation E and is

denoted by E ∞ ‚ J ∞ (π).

In the sequel, we shall mostly deal with formally integrable equations

E ‚ J k (π) (see De¬nition 1.20 on p. 30), which means that all E l are smooth

manifolds while the mappings πk+l+1,k+l : E l+1 ’ E l are smooth locally

trivial ¬ber bundles.

In¬nite prolongations are objects of the category M∞ (see Example 1.5

on p. 10). Hence, general approach exposed in Subsection 1.3 of Chapter 1

can be applied to them just in the same manner as it was done for manifolds

of in¬nite jets. In this section, we give a brief outline of calculus over E ∞ and

describe essential structures speci¬c for in¬nite prolongations of di¬erential

equations.

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

a k-th order di¬erential equation. Then we have the embeddings µl : E l ‚

J k+l (π) for all l ≥ 0. We de¬ne a smooth function on E l as the restriction

f |E l of a smooth function f ∈ Fk+l (π). The set Fl (E) of all functions

on E l forms an R-algebra in a natural way and µ— : Fk+l (π) ’ Fl (E) is a

l

homomorphism. In the case of formally integrable equations, the algebra

def

Fl (E) coincides with C ∞ (E l ). Let Il = ker µ— .

l

57

58 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

Due to commutativity of the diagram

µ—

l

Fk+l (π) ’ Fl (E)

— —

πk+l+1,k+l πk+l+1,k+l

“ “

µ—

l+1

Fk+l+1 (π) ’ Fl+1 (E)

one has Il (E) ‚ Il+1 (E). Then I(E) = l≥0 Il (E) is an ideal in F(π) which

is called the ideal of the equation E. The function algebra on E ∞ is the quo-

tient F(E) = F(π)/I(E) and coincides with inj liml’∞ Fl (E) with respect

—

to the system of homomorphisms πk+l+1,k+l . For all l ≥ 0, we have the

homomorphisms µ— : Fl (E) ’ F(E). When E is formally integrable, they

l

are monomorphic, but in any case the algebra F(E) is ¬ltered by the images

of µ— .

l

Now, to construct di¬erential calculus on E ∞ , one needs the general

algebraic scheme exposed in Chapter 4 and applied to the ¬ltered algebra

F(E). However, in the case of formally integrable equations, due to the

fact that all E l are smooth manifolds, this scheme may be simpli¬ed and

combined with a purely geometrical approach (cf. similar constructions of

Subsection 1.3 of Chapter 1).

Namely, di¬erential forms in this case are de¬ned as elements of the

def

module Λ— (E) = l≥0 Λ— (E l ), where Λ— (E l ) is considered to be embedded

into Λ— (E l+1 ) by πk+l+1,k+l . A vector ¬eld on E ∞ is a derivation X : F(E) ’

—

F(E) agreed with ¬ltration, i.e., such that X(Fl (E)) ‚ Fl+± (E) for some

integer ± = ±(X) ∈ Z. Just like in the case J ∞ (π), we de¬ne the de Rham

complex over E ∞ and obtain “usual” relations between standard operations

(contractions, de Rham di¬erential and Lie derivatives).

In special coordinates the in¬nite prolongation of the equation E is de-

termined by the system similar to (1.41) on p. 29 with the only di¬erence

that |σ| is unlimited now. Thus, the ideal I(E) is generated by the functions

Dσ F j , |σ| ≥ 0, j = 1, . . . , m. From these remarks we obtain the following

important fact.

Example 2.1. Let E be a formally integrable equation. Then from the

above said it follows that the ideal I(E) is stable with respect to the action

of the total derivatives Di , i = 1, . . . , n = dim M . Consequently, the action

E def E

Di = Di F (E) : F(E) ’ F(E) is well de¬ned and Di are ¬ltered deriva-

tions. We can reformulate it in other words by saying that the vector ¬elds

Di are tangent to any in¬nite prolongation and thus determine vector ¬elds

on E ∞ . We shall often skip the superscript E in the notation of the above

de¬ned restrictions.

The fact established in the last example plays a crucial role in the theory

of in¬nite prolongations. We continue to discuss it in the next section.

1. BASIC STRUCTURES 59

To ¬nish this one, let us make a remark concerning local coordinates.

Let E be locally represented with equations (1.41). Assume that the latter

is resolved in the form

±

uσj = f j (x1 , . . . , xn , . . . , uβ , . . . ),

j

j = 1, . . . , r,

σ

in such a way that

(i) the set of functions u±1 , . . . , u±r has at the left-hand side the empty

1 r

σ

σ

intersection with the set of functions uβ at the left-hand side and

σ

±j

±i

(ii) uσi +„ = uσj +„ for no „, „ unless i = j.

In this case, all coordinate functions in the system under consideration may

±j

be partitioned into two parts: those of the form uσj +„ , |„ | ≥ 0, j = 1, . . . , r,

and all others. We call the latter ones internal coordinates on E ∞ . Note

that all constructions of di¬erential calculus over E ∞ can be expressed in

terms of internal coordinates.

Example 2.2. Consider a system of evolution equations of the form

(1.22) (see p. 16). Then the functions x1 , . . . , xn , t, . . . , uj 1 ,...,σn ,0 , σi ≥ 0,

σ

j = 1, . . . , m, where t = xn+1 , may be taken for internal coordinates on E ∞ .

The total derivatives restricted onto E ∞ are expressed as

n

‚ ‚

uj i

Di = + , i = 1, . . . , n,

σ+1

‚xi ‚uσ

j=1 |σ|≥0

n

‚ ‚

Dσ (f j )

Dt = + (2.1)

‚t ‚uσ

j=1 |σ|≥0

in these coordinates, while the Cartan forms are written down as

n

uj i dxi ’ Dσ (f j ) dt,

j

duj ’

ωσ = (2.2)

σ σ+1

i=1

where all multi-indices σ are of the form σ = (σ1 , . . . , σn , 0).

1.2. Cartan distribution. Let π : E ’ M be a vector bundle and

E ‚ J k (π) be a formally integrable equation.

Definition 2.2. Let θ ∈ J ∞ (π). Then

(i) The Cartan plane Cθ = Cθ (π) ‚ Tθ J ∞ (π) at θ is the linear envelope of

tangent planes to all manifolds j∞ (•)(M ), • ∈ “(π), passing through

the point θ.

def

(ii) If θ ∈ E ∞ , the intersection Cθ (E) = Cθ (π) © Tθ E ∞ is called Cartan

plane of E ∞ at θ.

The correspondence θ ’ Cθ (π), θ ∈ J ∞ (π) (respectively, θ ’ Cθ (E ∞ ),

θ ∈ E ∞ ) is called the Cartan distribution on J ∞ (π) (respectively, on E ∞ ).

The following result shows the crucial di¬erence between the Cartan

distributions on ¬nite and in¬nite jets (or between those on ¬nite and in¬nite

prolongations).

60 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

Proposition 2.1. For any vector bundle π : E ’ M and a formally

integrable equation E ‚ J k (π) one has:

(i) The Cartan plane Cθ (π) is n-dimensional at any point θ ∈ J ∞ (π).

(ii) Any point θ ∈ E ∞ is generic, i.e., Cθ (π) ‚ Tθ E ∞ and thus Cθ (E ∞ ) =

Cθ (J ∞ ).

(iii) Both distributions, C(J ∞ ) and C(E ∞ ), are integrable.

Proof. Let θ ∈ J ∞ (π) and π∞ (θ) = x ∈ M . Then the point θ com-

pletely determines all partial derivatives of any section • ∈ “loc (π) such that

its graph passes through θ. Consequently, all such graphs have a common

tangent plane at this point which coincides with Cθ (π). This proves the ¬rst

statement.

To prove the second one, recall Example 2.1: locally, any vector ¬eld Di

j

is tangent to E ∞ . But as it follows from (1.27) on p. 18, one has iDi ωσ = 0

j

for any Di and any Cartan form ωσ . Hence, linear independent vector ¬elds

D1 , . . . , Dn locally lie both in C(π) and in C(E ∞ ) which gives the result.

Finally, as it follows from the above said, the module

def

CD(π) = {X ∈ D(π) | X lies in C(π)} (2.3)

is locally generated by the ¬elds D1 , . . . , Dn . But it is easily seen that

[D± , Dβ ] = 0 for all ±, β = 1, . . . , n and consequently [CD(π), CD(π)] ‚

CD(π). The same reasoning is valid for

def

CD(E) = {X ∈ D(E ∞ ) | X lies in C(E ∞ )}. (2.4)

This ¬nishes the proof of the proposition.

We shall describe now maximal integral manifolds of the Cartan distri-

butions on J ∞ (π) and E ∞ .

Proposition 2.2. Maximal integral manifolds of the Cartan distribu-

tion C(π) are graph of j∞ (•), • ∈ “loc (π).

Proof. Note ¬rst that graphs of in¬nite jets are integral manifolds of

the Cartan distribution of maximal dimension (equaling to n) and that any

integral manifold projects onto J k (π) and M without singularities.

def

Let now N ‚ J ∞ (π) be an integral manifold and N k = π∞,k N ‚ J k (π),

def

N = π∞ N ‚ M . Hence, there exists a di¬eomorphism • : N ’ N 0 such

that π —¦ • = idN . Then by the Whitney theorem on extension for smooth

functions [71], there exists a local section • : M ’ E satisfying • |N = •

and jk (•)(M ) ⊃ N k for all k > 0. Consequently, j∞ (•)(M ) ⊃ N .

Corollary 2.3. Maximal integral manifolds of the Cartan distribution

on E ∞ coincide locally with graphs of in¬nite jets of solutions.

We use the results obtained here in the next subsection.

1. BASIC STRUCTURES 61

1.3. Cartan connection. Consider a point θ ∈ J ∞ (π) and let x =

π∞ (θ) ∈ M . Let v be a tangent vector to M at the point x. Then, since

the Cartan plane Cθ isomorphically projects onto Tx M , there exists a unique

tangent vector Cv ∈ Tθ J ∞ (π) such that π∞,— (Cv) = v. Hence, for any vector

def

¬eld X ∈ D(M ) we can de¬ne a vector ¬eld CX ∈ D(π) by setting (CX)θ =