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5.2. Case 2: = 0, »’1 = 0. The complete Lie algebra of symmetries
for the Dirac equations with nonvanishing rest mass is spanned by four-
teen generators, including ten in¬nitesimal generators of the Poincar´ group
e
X1 , . . . , X10 and the generators X19 , X20 , X23 , X16 . There is also a contin-
1 4
uous part generated by the functions F u , . . . , F v dependent on x1 , . . . , x4 ,
which satisfy Dirac equations (1.83) with nonvanishing rest mass.

5.3. Case 3: = 0, »’1 = 0. The complete Lie algebra in this situation
is spanned by fourteen generators. These generators are X1 , . . . , X10 , X19 ,
X20 , X23 , and X11 ’ X16 /2.

5.4. Case 4: = 0, »’1 = 0. The complete Lie algebra of symmetries
for the nonlinear Dirac equations with nonvanishing rest mass is spanned
by thirteen generators. The generators in this case are the ten in¬nitesimal
generators of the Poincar´ group, X1 , . . . , X10 , and X19 , X20 , X23 . This
e
result generalizes the result by Steeb [94] where X20 was found as additional
symmetry to the generators of the Poincar´ group.
e

6. Symmetries of the self-dual SU (2) Yang“Mills equations
We study here classical symmetries of the self-dual SU (2) Yang“Mills
equations. Two cases are considered: the general one and of the so-called
static gauge ¬elds. In the ¬rst case we obtain two instanton solutions (the
Belavin“Polyakov“Schwartz“Tyupkin [6] and ™t Hooft instantons [84]) as
invariant solutions for a special choice of symmetry subalgebras. In a similar
way, for the second case we derive a monopole solution [83].
We start with a concise description of the SU (2)-gauge theory referring
the reader to the survey paper by M. K. Prasad [83] for a more extensive
exposition.

6.1. Self-dual SU (2) Yang“Mills equations. Let M be a 4-dimen-
sional Euclidean space with the coordinates x1 , . . . , x4 . Due to nondegen-
erate metric in M , we make no distinction between contravariant and co-
variant indices, xµ = xµ . The basic object in the gauge theory is the Yang“
Mills gauge potential. The gauge potential is a set of ¬elds Aa ∈ C ∞ (M ),
µ
a = 1, . . . , 3, µ = 1, . . . , 4. It is convenient to introduce a matrix-valued
vector ¬eld Aµ (x), by setting
σa
a
Aa , a
Aµ = gT T= , a = 1, . . . , 3, µ = 1, . . . , 4, (1.93)
µ
2i
where σ a are the Pauli matrices
0 ’i
01 10
σ1 = σ2 = σ3 =
, , , (1.94)
0 ’1
10 i0
g being a constant, called the gauge coupling constant. Throughout this
section we shall use the Einstein summation convention when an index oc-
curs twice. From the matrix gauge potential Aµ dxµ one constructs the
44 1. CLASSICAL SYMMETRIES

matrix-valued ¬eld strength Fµν (x) by
‚ ‚
Aν ’
Fµν = Aµ + [Aµ , Aν ], µ, ν = 1, . . . , 4, (1.95)
‚xµ ‚xν
where [Aµ , Aν ] = Aµ Aν ’ Aν Aµ . If one de¬nes the covariant derivative

Dµ = + Aµ , (1.96)
‚xµ
then (1.95) is rewritten as
Fµν = [Dµ , Dν ]. (1.97)
In explicit component form, one has
Fµν = gT a Fµν ,
a
(1.98)
where
‚a ‚a
a bc
Aν ’
Fµν = A +g abc Aµ Aν (1.99)
‚xν µ
‚xµ
and
±
+1 if abc is an even permutation of (1,2,3),

= ’1 if abc is an odd permutation of (1,2,3), (1.100)
abc


0 otherwise.
We shall use the expression static gauge ¬eld to refer to gauge potentials
that are independent of x4 (x4 to be considered as time), i.e.,

Aµ (x) = 0, µ = 1, . . . , 4. (1.101)
‚x4
For gauge potentials that depend on all four coordinates x1 , . . . , x4 , the
action functional is de¬ned by
1
Fµν Fµν d4 x,
a a
S= (1.102)
4
the integral taken over R4 , while for static gauge ¬elds we de¬ne the energy
functional by
1
Fµν Fµν d3 x,
a a
E= (1.103)
4
whereas in (1.103) the integral is taken over R3 .
The extremals of the action S (or of the energy E for static gauge ¬elds)
are found by standard calculus of variations techniques leading to the Euler“
Lagrange equations

Fµν + [Aµ , Fµν ] ≡ [Dµ , Fµν ] = 0, (1.104)
‚xµ
or in components
‚a
Fµν + g abc Ab Fµν = 0.
c
(1.105)
µ
‚xµ
6. SYMMETRIES OF THE SELF-DUAL SU (2) YANG“MILLS EQUATIONS 45

Equations (1.105) is a system of second order nonlinear partial dif-
ferential equations for the twelve unknown functions Aa , a = 1, . . . , 3,
µ
µ = 1, . . . , 4, that seems hard to solve.
Then one introduces the dual gauge ¬eld strength — Fµν as
1

Fµν = µν»ρ F»ρ , (1.106)
2
where µν»ρ is the completely antisymmetric tensor on M de¬ned by
±
+1 if µν»ρ is an even permutation of (1,2,3,4),

µν»ρ = (1.107)
’1 if µν»ρ is an odd permutation of (1,2,3,4),


0 otherwise.
Since the ¬elds Dµ (1.96) satisfy the Jacobi identity
[D» , [Dµ , Dν ]] + [Dµ , [Dν , D» ]] + [Dν , [D» , Dµ ]] = 0, (1.108)
multiplication of (1.108) by and summation result in
µν»ρ
[Dµ , — Fµν ] = 0. (1.109)
If we compare (1.104) with (1.109), we see that any gauge ¬eld which is
self-dual , i.e., for which

Fµν = Fµν , (1.110)
automatically satis¬es (1.101). Consequently, the only equations to solve are
(1.110) with — Fµν given by (1.106). This is a system of ¬rst order nonlinear
partial di¬erential equations.
Instanton solutions for general Yang“Mills equations and monopole so-
lutions for static gauge ¬elds satisfy (1.110) under the condition that S
(1.102) or E (1.103) are ¬nite.
Written in components, (1.110) takes the form
F13 = ’F24 ,
F12 = F34 , F14 = F23 . (1.111)
So in components, the self-dual Yang“Mills equations are described as a
system of nine nonlinear partial di¬erential equations,
’A1 + A1 ’ A1 + A1 ’ g(A2 A3 ’ A2 A3 + A2 A3 ’ A2 A3 ) = 0,
4,1 3,2 2,3 1,4 14 23 32 41
’A2 + A2 ’ A2 + A2 + g(A1 A3 ’ A1 A3 + A1 A3 ’ A1 A3 ) = 0,
4,1 3,2 2,3 1,4 14 23 32 41
’A3 + A3 ’ A3 + A3 ’ g(A1 A2 ’ A1 A2 + A1 A2 ’ A1 A2 ) = 0,
4,1 3,2 2,3 1,4 14 23 33 41
A1 + A1 ’ A1 ’ A1 + g(A2 A3 + A2 A3 ’ A2 A3 ’ A2 A3 ) = 0,
3,1 4,2 1,3 2,4 13 24 31 42
A2 + A2 ’ A2 ’ A2 ’ g(A1 A3 ’ A1 A3 ’ A1 A3 ’ A1 A3 ) = 0,
3,1 4,2 1,3 2,4 13 24 31 42
A3 + A3 ’ A3 ’ A3 + g(A1 A2 + A1 A2 ’ A1 A2 ’ A1 A2 ) = 0,
3,1 4,2 1,3 2,4 13 24 31 42
A1 ’ A1 ’ A1 + A1 + g(A2 A3 ’ A2 A3 ’ A2 A3 + A2 A3 ) = 0,
2,1 1,2 4,3 3,4 12 21 34 43
A2 ’ A2 ’ A2 + A2 ’ g(A1 A3 ’ A1 A3 ’ A1 A3 + A1 A3 ) = 0,
2,1 1,2 4,3 3,4 12 21 34 43
A3 ’ A3 ’ A3 + A3 + g(A1 A2 ’ A1 A2 ’ A1 A2 + A1 A2 ) = 0,
2,1 1,2 4,3 3,4 12 21 34 43
(1.112)
46 1. CLASSICAL SYMMETRIES

whereas in (1.112)
‚a
Aa = A, a = 1, . . . , 3, µ, ν = 1, . . . , 4. (1.113)
µ,ν
‚xν µ
Thus, we obtain a system E ‚ J 1 (π) for π : R12 — R4 ’ R4 .
6.2. Classical symmetries of self-dual Yang“Mills equations. In
order to construct the Lie algebra of classical symmetries of (1.112), we start
at a vector ¬eld V given by
‚ ‚ 1‚ 3‚
V = V x1 + · · · + V x4 + V A1 + · · · + V A4 . (1.114)
‚A1 ‚A3
‚x1 ‚x4 1 4
The condition for V to be a symmetry of equations (1.112) now leads to an
overdetermined system of partial di¬erential equations for the components
1 3
V x1 , . . . , V x4 , V A1 , . . . , V A4 , which are functions dependent of the variables
x1 , . . . , x4 , A 1 , . . . , A 3 .
1 4
The general solution of this overdetermined system of partial di¬erential
equations constitutes a Lie algebra of symmetries, generated by the vector
¬elds
1‚ 1‚ 1‚ 1‚
1
V1f = fx1 + f x2 + f x3 + f x4
‚A1 ‚A1 ‚A1 ‚A1
1 2 3 4
‚ ‚ ‚ ‚
+ f 1 gA3 2 + f 1 gA3 2 + f 1 gA3 2 + f 1 gA3 2
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚ ‚
’ f 1 gA2 3 ’ f 1 gA2 3 ’ f 1 gA2 3 ’ f 1 gA2 3 ,
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚ ‚
2
V2f = ’f 2 gA2 1 ’ f 2 gA2 1 ’ f 2 gA2 1 ’ f 2 gA2 1
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚ ‚
+ f 2 gA1 2 + f 2 gA1 2 + f 2 gA1 2 + f 2 gA1 2
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
2‚ 2‚ 2‚ 2‚
’ f x1 ’ f x2 ’ f x3 ’ f x4 ,
‚A3 ‚A3 ‚A3 ‚A3
1 2 3 4
‚ ‚ ‚ ‚
3
V3f = f 3 gA3 1 + f 3 gA3 1 + f 3 gA3 1 + f 3 gA3 1
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
3‚ 3‚ 3‚ 3‚
’ f x1 ’ f x2 ’ f x3 ’ f x4
‚A2 ‚A2 ‚A2 ‚A2
1 2 3 4
‚ ‚ ‚ ‚
’ f 3 gA1 3 ’ f 3 gA1 3 ’ f 3 gA1 3 ’ f 3 gA1 3 ,
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚ ‚
V4 = , V5 = , V6 = , V7 = ,
‚x1 ‚x2 ‚x3 ‚x4
‚ ‚ ‚ ‚
+ A1 1 ’ A1 1
’ x1
V8 = x 2 2 1
‚x1 ‚x2 ‚A1 ‚A2
‚ ‚ ‚ ‚
+ A2 2 ’ A2 2 + A3 3 ’ A3 3 ,
2 1 2 1
‚A1 ‚A2 ‚A1 ‚A2
6. SYMMETRIES OF THE SELF-DUAL SU (2) YANG“MILLS EQUATIONS 47

‚ ‚ ‚ ‚
’ A1 1 + A1 1
V9 = ’x3 + x1 3 1
‚x1 ‚x3 ‚A1 ‚A3
‚ ‚ ‚ ‚
’ A2 2 + A2 2 ’ A3 3 + A3 3 ,
3 1 3 1
‚A1 ‚A3 ‚A1 ‚A3
‚ ‚ ‚ ‚
’ A1 1 + A1 1
V10 = ’x4 + x1 4 1
‚x1 ‚x4 ‚A1 ‚A4
‚ ‚ ‚ ‚
’ A2 2 + A2 2 ’ A3 3 + A3 3 ,
4 1 4 1
‚A1 ‚A4 ‚A1 ‚A4
‚ ‚ ‚ ‚
’ A1 1 + A1 1
V11 = ’x3 + x2 3 2
‚x2 ‚x3 ‚A2 ‚A3
‚ ‚ ‚ ‚
’ A2 2 + A2 2 ’ A3 3 + A3 3 ,
3 2 3 2
‚A2 ‚A3 ‚A2 ‚A3
‚ ‚ ‚ ‚
+ A1 1 ’ A1 1
’ x2
V12 = x4 4 2
‚x2 ‚x4 ‚A2 ‚A4
‚ ‚ ‚ ‚
+ A2 2 ’ A2 2 + A3 3 ’ A3 3 ,
4 2 4 2
‚A2 ‚A4 ‚A2 ‚A4
‚ ‚ ‚ ‚
’ A1 1 + A1 1
V13 = ’x4 + x3 4 3
‚x3 ‚x4 ‚A3 ‚A4
‚ ‚ ‚ ‚
’ A2 2 + A2 2 ’ A3 3 + A3 3 ,
4 3 4 3
‚A3 ‚A4 ‚A3 ‚A4
‚ ‚ ‚ ‚
V14 = x1 + x2 + x3 + x4
‚x1 ‚x2 ‚x3 ‚x4
‚ ‚ ‚ ‚
’ A1 1 ’ A1 1 ’ A1 1 ’ A1 1
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚ ‚
’ A2 2 ’ A2 2 ’ A2 2 ’ A2 2
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚ ‚
’ A3 3 ’ A3 3 ’ A3 3 ’ A3 3 ,
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚ ‚
V15 = (’x2 + x2 + x2 + x2 ) ’ 2x1 x2 ’ 2x1 x3 ’ 2x1 x4
1 2 3 4
‚x1 ‚x2 ‚x3 ‚x4
‚ ‚
+ 2(x1 A1 + x2 A1 + x3 A1 + x4 A1 ) 1 + 2(x1 A1 ’ x2 A1 ) 1
1 2 3 4 2 1
‚A1 ‚A2
‚ ‚
+ 2(x1 A1 ’ x3 A1 ) 1 + 2(x1 A1 ’ x4 A1 ) 1
3 1 4 1
‚A3 ‚A4
‚ ‚
+ 2(x1 A2 + x2 A2 + x3 A2 + x4 A2 ) 2 + 2(x1 A2 ’ x2 A2 ) 2
1 2 3 4 2 1
‚A1 ‚A2
‚ ‚
+ 2(x1 A2 ’ x3 A2 ) 2 + 2(x1 A2 ’ x4 A2 ) 2
3 1 4 1
‚A3 ‚A4
48 1. CLASSICAL SYMMETRIES

‚ ‚
+ 2(x1 A3 + x2 A3 + x3 A3 + x4 A3 ) + 2(x1 A3 ’ x2 A3 ) 3
1 2 3 4 2 1
‚A3 ‚A2
1
‚ ‚
+ 2(x1 A3 ’ x3 A3 ) 3 + 2(x1 A3 ’ x4 A3 ) 3 ,
3 1 4 1
‚A3 ‚A4
‚ ‚ ‚ ‚
+ (x2 ’ x2 + x2 + x2 )
V16 = ’2x2 x1 ’ 2x2 x3 ’ 2x2 x4
1 2 3 4
‚x1 ‚x2 ‚x3 ‚x4
‚ ‚
+ 2(’x1 A1 + x2 A1 ) 1 + 2(x1 A1 + x2 A1 + x3 A1 + x4 A1 ) 1
2 1 1 2 3 4
‚A1 ‚A2
‚ ‚
+ 2(x2 A1 ’ x3 A1 ) 1 + 2(x2 A1 ’ x4 A1 ) 1
3 2 4 2
‚A3 ‚A4
‚ ‚
+ 2(’x1 A2 + x2 A2 ) 2 + 2(x1 A2 + x2 A2 + x3 A2 + x4 A2 ) 2
2 1 1 2 3 4
‚A1 ‚A2
‚ ‚
+ 2(x2 A2 ’ x3 A2 ) 2 + 2(x2 A2 ’ x4 A2 ) 2
3 2 4 2
‚A3 ‚A4
‚ ‚
+ 2(’x1 A3 + x2 A3 ) 3 + 2(x1 A3 + x2 A3 + x3 A3 + x4 A3 ) 3
2 1 1 2 3 4
‚A1 ‚A2
‚ ‚
+ 2(x2 A3 ’ x3 A3 ) 3 + 2(x2 A3 ’ x4 A3 ) 3 ,
3 2 4 2
‚A3 ‚A4
‚ ‚ ‚ ‚
+ (x2 + x2 ’ x2 + x2 )
V17 = ’2x3 x1 ’ 2x3 x2 ’ 2x3 x4
1 2 3 4
‚x1 ‚x2 ‚x3 ‚x4
‚ ‚
+ 2(’x1 A1 + x3 A1 ) 1 + 2(’x2 A1 + x3 A1 ) 1
3 1 3 2
‚A1 ‚A2
‚ ‚
+ 2(x1 A1 + x2 A1 + x3 A1 + x4 A1 ) 1 + 2(x3 A1 ’ x4 A1 ) 1
1 2 3 4 4 3
‚A3 ‚A4
‚ ‚
+ 2(’x1 A2 + x3 A2 ) 2 + 2(’x2 A2 + x3 A2 ) 2
3 1 3 2
‚A1 ‚A2
‚ ‚
+ 2(x1 A2 + x2 A2 + x3 A2 + x4 A2 ) 2 + 2(x3 A2 ’ x4 A2 ) 2
1 2 3 4 4 3
‚A3 ‚A4
‚ ‚
+ 2(’x1 A3 + x3 A3 ) 3 + 2(’x2 A3 + x3 A3 ) 3
3 1 3 2
‚A1 ‚A2
‚ ‚
+ 2(x1 A3 + x2 A3 + x3 A3 + x4 A3 ) 3 + 2(x3 A3 ’ x4 A3 ) 3 ,
1 2 3 4 4 3
‚A3 ‚A4
‚ ‚ ‚ ‚
+ (x2 + x2 + x2 ’ x2 )
V18 = ’2x4 x1 ’ 2x4 x2 ’ 2x4 x3 1 2 3 4
‚x1 ‚x2 ‚x3 ‚x4
‚ ‚
+ 2(’x1 A1 + x4 A1 ) 1 + 2(’x2 A1 + x4 A1 ) 1
4 1 4 2
‚A1 ‚A2
‚ ‚
+ 2(’x3 A1 + x4 A1 ) 1 + 2(x1 A1 + x2 A1 + x3 A1 + x4 A1 ) 1
4 3 1 2 3 4
‚A3 ‚A4
6. SYMMETRIES OF THE SELF-DUAL SU (2) YANG“MILLS EQUATIONS 49

‚ ‚
+ 2(’x1 A2 + x4 A2 ) + 2(’x2 A2 + x4 A2 )
4 1 4 2
‚A2 ‚A2
1 2
‚ ‚
+ 2(’x3 A2 + x4 A2 ) 2 + 2(x1 A2 + x2 A2 + x3 A2 + x4 A2 )
4 3 1 2 3 4
‚A2
‚A3 4
‚ ‚
+ 2(’x1 A3 + x4 A3 ) 3 + 2(’x2 A3 + x4 A3 )
4 1 4 2
‚A3
‚A1 2
‚ ‚
+ 2(’x3 A3 + x4 A3 ) 3 + 2(x1 A3 + x2 A3 + x3 A3 + x4 A3 ) .
4 3 1 2 3 4
‚A3
‚A3 4
(1.115)

The functions F 1 , F 2 , F 3 in the symmetries V1 , V2 , V3 are arbitrary,
depending on the variables x1 , x2 , x3 , x4 . The vector ¬elds V1 , V2 , V3 are
just the generators of the gauge transformations.
The vector ¬elds V4 , V5 , V6 , V7 are generators of translations while the
¬elds V8 , . . . , V13 refer to in¬nitesimal rotations in R4 , X4 , . . . , X18 being the
in¬nitesimal generators of the conformal group.

6.3. Instanton solutions. In order to construct invariant solutions
associated to symmetries of the self-dual Yang“Mills equations (1.112), we
start from the vector ¬elds X1 , X2 , X3 de¬ned by
1 2 3
f1 f1 f1
X1 = V 8 + V1 + V2 + V3 ,
f1 f2 f3
X2 = V 9 + V 1 2 + V 2 2 + V 3 2 ,
f1 f2 f3
X3 = V10 + V1 3 + V2 3 + V3 3 , (1.116)
i.e., we take a combination of a rotation and a special choice for the
gauge transformations choosing particular values fij of arbitrary func-
tions f j . We also construct commutators of the vector ¬elds X1 , X2 , X3 ,
[X1 , X2 ], [X1 , X3 ], [X2 , X3 ] (1.117)
and make the following choice for the gauge transformations
1 2 3
f1 = ’1,
f1 = 0, f1 = 0,
1 2 3
f2 = ’1,
f2 = 0, f2 = 0,
1 2 3
f3 = ’1, f3 = 0, f3 = 0. (1.118)
In order to derive invariant solutions (see equations (1.40) on p. 28), we
impose the additional conditions. Namely, we compute generating functions
(•i )j = Yi ωA j , j = 1, . . . , 3, µ = 1, . . . , 4, (1.119)
µ µ


whereas in (1.119) ωAj is the contact form associated to Aj , i.e.,
µ
µ


ωAj = dAj ’ Aj dxν ,
µ µ,ν
µ
50 1. CLASSICAL SYMMETRIES

while Yi refers to the ¬elds X1 , X2 , X3 , [X1 , X2 ], [X1 , X3 ], [X2 , X3 ]. Then
we impose additional equations
j
(x1 , . . . , x4 , . . . , Aj , . . . , Aj , . . . )
•i =0 (1.120)
µ µν
µ

and solve them together with the initial system. From conditions (1.119)
we arrive at a system of 6 — 12 = 72 equations.
The resulting system can be solved in a straightforward way, leading to
the following intermediate presentation
A1 = x4 F (r), A1 = x3 F (r), A1 = ’x2 F (r), A1 = ’x1 F (r),
1 2 3 4
A2 = ’x3 F (r), A2 = x4 F (r), A2 = x1 F (r), A2 = ’x2 F (r),
1 2 3 4
A3 = x2 F (r), A3 = ’x1 F (r), A3 = x4 F (r), A3 = ’x3 F (r),
1 2 3 4
(1.121)
where
1
r = (x2 + x2 + x2 + x2 ) 2 . (1.122)
1 2 3 4

When obtaining the monopole solution (see below), we shall discuss in some
more detail how to solve a system of partial di¬erential equations like (1.120).
Substitution of (1.122) in (1.95) yields an ordinary di¬erential equation for
the function F (r), i.e.,
dF (r)
+ grF (r)2 = 0, (1.123)
dr
the solution of which is given by
2g ’1
F (r) = 2 , (1.124)
r +C
C being a constant. The result (1.124) is just the Belavin“Polyakov“
Schwartz“Tyupkin instanton solution!
More general, if we choose
2 3 1
f1 = ±1, f2 = ±1, f3 = ±1, (1.125)
and
231
f1 f2 f3 = ’1, (1.126)
or equivalently
1 23
f3 = ’f1 f2 , (1.127)
we arrive at
A1 = x4 F (r), A1 = x3 F (r), A1 = ’x2 F (r), A1 = ’x1 F (r),
1 2 3 4
A2 = x3 F (r)f1 ,
2
A2 = ’x4 F (r)f1 , A2 = ’x1 F (r)f1 , A2 = ’x2 F (r)f1 ,
2 2 2
1 2 3 4
A3 = ’x2 F (r)f3 , A3 = x1 F (r)f3 ,
2 2
A3 = ’x4 F (r)f3 , A3 = ’x3 F (r)f3 ,
2 2
1 2 3 4
(1.128)
6. SYMMETRIES OF THE SELF-DUAL SU (2) YANG“MILLS EQUATIONS 51

while (1.128) with F (r) has to satisfy (1.112), which results in
‚F (r)
+ grf1 f3 F (r)2 = 0.
22
(1.129)
‚r
2 3 1
Choosing f1 , f2 , f3 as in (1.125) but with
231
f1 f2 f3 = +1, (1.130)
then the result is
A1 = x4 F (r), A1 = ’ x3 F (r), A1 = x2 F (r), A1 = ’ x1 F (r),
1 2 3 4
A2 = ’x3 F (r)f1 , A2 = ’ x4 F (r)f1 , A2 = x1 F (r)f1 ,
2 2 2
A2 =x2 F (r)f1 ,
2
1 2 3 4
A3 = x2 F (r)f3 ,
2
A3 = ’ x1 F (r)f3 , A3 = ’x4 F (r)f3 , A3 =x3 F (r)f3 ,
2 2 2
1 2 3 4
(1.131)
while F (r) has to satisfy
dF (r)
+ 4F (r) + gr 2 f1 f2 F (r)2 = 0.
23
r (1.132)
dr
The solution of (1.132)
a2
2 2 3 ’1
F (r) = ’ (f1 f2 ) (1.133)
(r2 + a2 )r2
g
together with (1.131) is just the ™t Hooft instanton solution with instanton
number k = 1. This solution can be obtained from (1.124) by a gauge
transformation.

6.4. Classical symmetries for static gauge ¬elds. The equations
for the static SU (2) gauge ¬eld are described by (1.109) and (1.101). The
symmetries for the static gauge ¬eld are obtained from those for the time-
dependent case or straightforwardly in the following way . The respective
computations then results in the following Lie algebra of symmetries for the
static self-dual SU (2) Yang“Mills equations
‚ ‚ ‚
1
V1C = Cx1
1 1 1
+ C x2 + C x3
‚A1 ‚A1 ‚A1
1 2 3
‚ ‚ ‚ ‚
+ C 1 gA3 2 + C 1 gA3 2 + C 1 gA3 2 + C 1 gA3 2
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚ ‚
’ C 1 gA2 3 ’ C 1 gA2 3 ’ C 1 gA2 3 ’ C 1 gA2 3 ,
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚ ‚
2
V2C = ’C 2 gA2 1 ’ C 2 gA2 1 ’ C 2 gA2 1 ’ C 2 gA2 1
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚ ‚
+ C 2 gA1 2 + C 2 gA1 2 + C 2 gA1 2 + C 2 gA1 2
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚
2 2 2
’ C x1 ’ C x2 ’ C x3 ,
‚A3 ‚A3 ‚A3
1 2 3
52 1. CLASSICAL SYMMETRIES

‚ ‚ ‚ ‚
3
V3C = C 3 gA3 + C 3 gA3 1 + C 3 gA3 1 + C 3 gA3 1
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
1
‚ ‚ ‚
3 3 3
’ C x1 ’ C x2 ’ C x3
‚A2 ‚A2 ‚A2
1 2 3
‚ ‚ ‚ ‚
’ C 3 gA1 3 ’ C 3 gA1 3 ’ C 3 gA1 3 ’ C 3 gA1 3 ,
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚
V4 = , V5 = , V6 = ,
‚x1 ‚x2 ‚x3
‚ ‚ ‚ ‚
+ A1 1 ’ A1 1
’ x1
V7 = x 2 2 1
‚x1 ‚x2 ‚A1 ‚A2
‚ ‚ ‚ ‚
+ A2 2 ’ A2 2 + A3 3 ’ A3 3 ,
2 1 2 1
‚A1 ‚A2 ‚A1 ‚A2
‚ ‚ ‚ ‚
+ A1 1 ’ A1 1
V8 = ’x3 + x1 1 3
‚x1 ‚x3 ‚A3 ‚A1
‚ ‚ ‚ ‚
’ A2 2 + A2 2 ’ A3 3 + A3 3 ,
3 1 3 1
‚A1 ‚A3 ‚A1 ‚A3
‚ ‚ ‚ ‚
’ A1 1 + A1 1
V9 = ’x3 + x2 3 2
‚x2 ‚x3 ‚A2 ‚A3
‚ ‚ ‚ ‚
’ A2 2 + A2 2 ’ A3 3 + A3 3 ,
3 2 3 2
‚A2 ‚A3 ‚A2 ‚A3
‚ ‚ ‚ ‚
V10 = x1 + x2 + x3 + x4
‚x1 ‚x2 ‚x3 ‚x4
‚ ‚ ‚ ‚
’ A1 1 ’ A1 1 ’ A1 1 ’ A1 1
1 2 3 4
‚A1 ‚A2 ‚A3 ‚A4
‚ ‚ ‚
’ A2 2 ’ A2 2 ’ A2 2
1 2 3
‚A1 ‚A2 ‚A3
‚ ‚ ‚ ‚ ‚
’ A2 2 ’ A3 3 ’ A3 3 ’ A3 3 ’ A3 3 . (1.134)
4 1 2 3 4
‚A4 ‚A1 ‚A2 ‚A3 ‚A4
In (1.134) C 1 , C 2 , C 3 are arbitrary functions of x1 , . . . , x3 , while V1 ,
V2 , V3 themselves are just the generators of the gauge transformations. The
¬elds V7 , V8 , V9 generate rotations, while V10 is the generator of the scale
change of variables.

6.5. Monopole solution. In order to construct invariant solutions to
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 =
C(Xπ∞ (θ) ). Then, by construction, the ¬eld CX is projected by π∞,— to X
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

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