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the static SU (2) gauge ¬eld, we proceed in a way analogously to the one for
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 =

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