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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

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