where i = 1, . . . , n. Using this representation, we prove the following result:

Proposition 1.6. For any point θk ∈ J k (π), k ≥ 1, the Cartan plane

is of the form Cθk = (πk,k’1 )’1 (Lθk ), where Lθk is the R-plane at the

k k

Cθ k —

6

It is clear that for any regular point there exists a neighborhood of this point all

points of which are regular.

18 1. CLASSICAL SYMMETRIES

point πk,k’1 (θk ) ∈ J k’1 (π) determined by the point θk (see p. 5 for the

de¬nition of Lθk ).

k,•

Proof. Denote the vector (1.25) by vi . It is obvious that for any two

k,• k,•

sections •, • satisfying (1.23) the di¬erence vi ’ vi is a πk,k’1 -vertical

vector and any such a vector can be obtained in this way. On the other

k’1,•

hand, the vectors vi do not depend on • satisfying (1.23) and form a

basis in the space Lθk .

Remark 1.6. From the result proved it follows that the Cartan distri-

bution on J k (π) can be locally considered as generated by the vector ¬elds

m

[k]

uj i ‚uj , V„s = ‚us , |„ | = k, s = 1, . . . , m.

Di = ‚xi + σ „

σ+1

|σ|¤k’1 j=1

(1.26)

[k]

From here, by direct computations, it follows that [V„s , Di ] = V„s’1i , where

V(„1 ,...,„i ’1,...,„n ) , if „i > 0,

s

V(„1 ,...,„n )’1i =

0, otherwise.

j

But, as it follows from Proposition 1.6, vector ¬elds Vσ for |σ| ¤ k do not

lie in C k .

Let us consider the following 1-forms in special coordinates on J k+1 (π):

n

j def

uj i dxi ,

duj ’

ωσ = (1.27)

σ σ+1

i=1

where j = 1, . . . , m, |σ| < k. From the representation (1.26) we immediately

obtain the following important property of the forms introduced:

Proposition 1.7. The system of forms (1.27) annihilates the Cartan

j

distribution on J k (π), i.e., a vector ¬eld X lies in C k if and only if iX ωσ = 0

for all j = 1, . . . , m, |σ| < k.

Definition 1.12. The forms (1.27) are called the Cartan forms on

J k (π) associated to the special coordinate system xi , uj .

σ

Note that the Fk (π)-submodule generated in Λ1 (J k (π)) by the forms

(1.27) is independent of the choice of coordinates.

Definition 1.13. The Fk (π)-submodule generated in Λ1 (J k (π)) by the

Cartan forms is called the Cartan submodule. We denote this submodule by

CΛ1 (J k (π)).

Our last step is to describe maximal integral manifolds of the Cartan

distribution on J k (π). To do this, we start with the “in¬nitesimal estimate”.

Let N ‚ J k (π) be an integral manifold of the Cartan distribution. Then

from Proposition 1.7 it follows that the restriction of any Cartan form ω onto

2. NONLINEAR PDE 19

N vanishes. Similarly, the di¬erential dω vanishes on N . Therefore, if vector

¬elds X, Y are tangent to N , then dω |N (X, Y ) = 0.

Definition 1.14. Let Cθk be the Cartan plane at θ ∈ J k (π).

k

k

(i) We say that two vectors v, w ∈ Cθk are in involution, if the equality

dω |θk (v, w) = 0 holds for any ω ∈ CΛ1 (J k (π)).

k

(ii) A subspace W ‚ Cθk is said to be involutive, if any two vectors

v, w ∈ W are in involution.

(iii) An involutive subspace is called maximal , if it cannot be embedded

into other involutive subspace.

Consider a point θk = [•]k ∈ J k (π). Then from Proposition 1.7 it follows

x

that the direct sum decomposition

•

k v

Cθ k = T θ k • T θ k

v

is valid, where Tθk denotes the tangent plane to the ¬ber of the projection

•

πk,k’1 passing through the point θk , while Tθk is the tangent plane to the

graph of jk (•). Hence, the involutiveness is su¬cient to be checked for the

k

following pairs of vectors v, w ∈ Cθk :

v

(i) v, w ∈ Tθk ;

•

(ii) v, w ∈ Tθk ;

•

v

(iii) v ∈ Tθk , w ∈ Tθk .

v

Note now that the tangent space Tθk is identi¬ed with the tensor product

— —

S k (Tx ) — Ex , x = πk (θk ) ∈ M , where Tx is the ¬ber of the cotangent bundle

to M at the point x, Ex is the ¬ber of the bundle π at the same point while

S k denotes the k-th symmetric power. Then any tangent vector w ∈ Tx M

— —

determines the mapping δw : S k (Tx ) — Ex ’ S k’1 (Tx ) — Ex by

k

··· ρk ) — e = ··· ··· ρk — e,

δw (ρ1 ρ1 ρi , w

i=1

— —

where denotes multiplication in S k (Tx ), ρi ∈ Tx , e ∈ Ex , while ·, · is the

—

natural pairing between Tx and Tx .

k

Proposition 1.8. Let v, w ∈ Cθk . Then:

v

(i) All pairs v, w ∈ Tθk are in involution.

•

(ii) All pairs v, w ∈ Tθk are in involution too.

•

v

(iii) If v ∈ Tθk and w ∈ Tθk , then they are in involution if and only if

δπk,— (w) v = 0.

Proof. Note ¬rst that the involutiveness conditions are su¬cient to be

checked for the Cartan forms (1.27) only. All three results follow from the

representation (1.26) by straightforward computations.

20 1. CLASSICAL SYMMETRIES

Consider a point θk ∈ J k (π). Let Fθk be the ¬ber of the bundle πk,k’1

passing through the point θk and H ‚ Tx M be a subspace. De¬ne the

space7

Ann(H) = {v ∈ Fθk | δw v = 0, ∀w ∈ H}.

Then, as it follows from Proposition 1.8, the following description of maximal

involutive subspaces takes place:

Corollary 1.9. Let θk = [•]k , • ∈ “loc (π). Then any maximal invo-

x

k

lutive subspace V ‚ Cθk (π) is of the form

V = jk (•)— (H) • Ann(H)

for some H ‚ Tx M .

If V is a maximal involutive subspace, then the corresponding space

H is obviously πk,— (V ). We call dimension of H the type of the maximal

involutive subspace V and denote it by tp(V ).

Proposition 1.10. Let V be a maximal involutive subspace. Then

n’r+k’1

dim V = m + r,

k

where n = dim M , m = dim π, r = tp(V ).

Proof. Choose local coordinates in M in such a way that the vectors

‚x1 , . . . , ‚xr form a basis in H. Then, in the corresponding special system in

J k (π), coordinates along Ann(H) will consist of those functions uj , |σ| = k,

σ

for which σ1 = · · · = σr = 0.

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

bution on J k (π).

Let N ‚ J k (π) be such a manifold θk ∈ N . Then the tangent plane to

N at the point θk is a maximal involutive plane. Assume that its type is

equal to r(θk ).

Definition 1.15. The number

def

tp(N ) = max r(θk ).

θk ∈N

is called the type of the maximal integral manifold N of the Cartan distri-

bution.

Obviously, the set

def

g(N ) = {θk ∈ N | r(θk ) = tp(N )}

is everywhere dense in N . We call the points θk ∈ g(N ) generic. Let θk be

such a point and U be its neighborhood in N consisting of generic points.

Then:

7

Using the linear structure, we identify the ¬ber Fθk of the bundle πk,k’1 with its

tangent space.

2. NONLINEAR PDE 21

(i) N = πk,k’1 (N ) is an integral manifold of the Cartan distribution on

J k’1 (π);

(ii) dim(N ) = tp(N );

(iii) πk’1 |N : N ’ M is an immersion.

Theorem 1.11. Let N ‚ J k’1 (π) be an integral manifold of the Cartan

distribution on J k (π) and U ‚ N be an open domain consisting of generic

points. Then

U = {θk ∈ J k (π) | Lθk ⊃ Tθk’1 U },

where θk’1 = πk,k’1 (θk ), U = πk,k’1 (U).

Proof. Let V = πk’1 (U ) ‚ M . Denote its dimension (which equals

the number tp(N )) by r and choose local coordinates in M in such a way

that the submanifold V is determined by the equations xr+1 = · · · = xn = 0

in these coordinates. Then, since U ‚ J k’1 (π) is an integral manifold

and πk’1 |U : U ’ V is a di¬eomorphism, in the corresponding special

coordinates the manifold U is given by the equations

± |σ| j

‚ •

, if σ = (σ1 , . . . , σr , 0, . . . , 0),

uj = ‚xσ

σ

0, otherwise,

for all j = 1, . . . , m, |σ| ¤ k’1 and some smooth function • = •(x1 , . . . , xr ).

Hence, the tangent plane H to U at θk’1 is spanned by the vectors of the

form (1.25) with i = 1, . . . , r. Consequently, a point θk , such that Lθk ⊃ H,

is determined by the coordinates

± |σ| j

‚ •

, if σ = (σ1 , . . . , σr , 0, . . . , 0),

j

uσ = ‚xσ

arbitrary real numbers, otherwise,

where j = 1, . . . , m, |σ| ¤ k. Hence, if θk , θk are two such points, then the

vector θk ’ θk lies in Ann(H), as it follows from the proof of Proposition

1.10. As it is easily seen, any integral manifold of the Cartan distribution

projecting onto U is contained in U, which ¬nishes the proof.

Remark 1.7. Note that maximal integral manifolds N of type dim M

are exactly graphs of jets jk (•), • ∈ “loc (π). On the other hand, if tp(N ) =

0, then N coincides with a ¬ber of the projection πk,k’1 : J k (π) ’ J k’1 (π).

2.3. Symmetries. The last remark shows that the Cartan distribution

on J k (π) is in a sense su¬cient to restore the structures speci¬c to the jet

manifolds. This motivates the following de¬nition:

Definition 1.16. Let U, U ‚ J k (π) be open domains.

(i) A di¬eomorphism F : U ’ U is called a Lie transformation, if it

preserves the Cartan distribution, i.e.,

k k

F— (Cθk ) = CF (θk )

for any point θk ∈ U.

22 1. CLASSICAL SYMMETRIES

Let E, E ‚ J k (π) be di¬erential equations.

(ii) A Lie transformation F : U ’ U is called a (local ) equivalence, if

F (U © E) = U © E .

(iii) A (local) equivalence is called a (local ) symmetry, if E = E and

U = U . Such symmetries are also called classical 8 .

Below we shall not distinguish between local and global versions of the

concepts introduced.

Remark 1.8. There is an alternative approach to the concept of a sym-

metry. Namely, we can introduce the Cartan distribution on E by setting

def

Cθ (E) = Cθ © Tθ E, θ ∈ E,

and de¬ne interior symmetries of E as a di¬eomorphism F : E ’ E preserv-

ing C(E). In general, the group of these symmetries does not coincide with

the above introduced. A detailed discussion of this matter can be found in

[60].

Example 1.11. Consider the case J 0 (π) = E. Then, since any n-di-

mensional horizontal plane in Tθ E is tangent to some section of the bundle

0

π, the Cartan plane Cθ coincides with the whole space Tθ E. Thus the Car-

tan distribution is trivial in this case and any di¬eomorphism of E is a Lie

transformation.

Example 1.12. Since the Cartan distribution on J k (π) is locally deter-

mined by the Cartan forms (1.27), the condition of F to be a Lie transfor-

mation cam be reformulated as

m

F — ωσ =

j

»j,± ω„ ,

±

|σ| < k,

j = 1, . . . , m, (1.28)

σ,„

±=1 |„ |<k

where »j,± are smooth functions on J k (π). Equations (1.28) are the base

σ,„

for computations in local coordinates.

In particular, if dim π = 1 and k = 1, equations (1.28) reduce to the only

condition F — ω = »ω, where ω = du ’ n u1i dxi . Hence, Lie transforma-

i=1

tions in this case are just contact transformations of the natural contact

structure in J 1 (π).

Example 1.13. Let F : J 0 (π) ’ J 0 (π) be a di¬eomorphism (which can

be considered as a general change of dependent and independent coordi-

nates). Let us construct a Lie transformation F (1) of J 1 (π) such that the

8

Contrary to higher, or generalized, symmetries which will be introduced in the next

chapter.

2. NONLINEAR PDE 23

diagram

F (1)

1

’ J 1 (π)

J (π)

π1,0 π1,0

“ “

F

J 0 (π) ’ J 0 (π)

is commutative, i.e., π1,0 —¦ F (1) = F —¦ π1,0 . To do this, introduce local

coordinates x1 , . . . , xn , u1 , . . . , um in J 0 (π) and consider the corresponding

special coordinates in J 1 (π) denoting the functions uj i by pj . Express the

1 i

transformation F in the form

xi ’ Xi (x1 , . . . , xn , u1 , . . . , um ), uj ’ U j (x1 , . . . , xn , u1 , . . . , um ),

i = 1, . . . , n, j = 1, . . . , m, in these coordinates. Then, due to (1.28), to ¬nd

F (1) : pj ’ Pij (x1 , . . . , xn , u1 , . . . , um , p1 , . . . , pm ),

1 n

i

one needs to solve the system

n m n

Pij

j j,± ±

p± dxi ),

dU ’ (du ’

dXi = » i

±=1

i=1 i=1

j = 1, . . . , m, with respect to the functions Pij for arbitrary smooth coe¬-

cients »j,± . Using matrix notation p = pj , P = Pij and » = »±β , we

i

see that

‚U ‚X

’P —¦

»=

‚u ‚u

and

’1

‚U ‚U ‚X ‚X

—¦p —¦ —¦p

P= + + , (1.29)

‚x ‚u ‚x ‚u

where

‚U ± ‚U ±

‚X ‚X± ‚X ‚X± ‚U ‚U

= , = , = , =

‚uβ ‚uβ

‚x ‚xβ ‚u ‚x ‚xβ ‚u

denote Jacobi matrices. Note that the transformation F (1) , as it follows

from (1.29), is unde¬ned at some points of J 1 (π), i.e., at the points where

the matrix ‚X/‚x + ‚X/‚u —¦ p is not invertible.

Example 1.14. Let π : Rn —Rn ’ Rn , i.e., dim π = dim M and consider

the transformation ui ’ xi , xi ’ ui , i = 1, . . . , n. This transformation is

called the hodograph transformation. From (1.29) it follows that the corre-

sponding transformation of the functions pj is de¬ned by P = p’1 .

i

24 1. CLASSICAL SYMMETRIES

Example 1.15. Let Ed be the equation determined by the de Rham

di¬erential (see Example 1.6), i.e., Ed = {dω = 0}, ω ∈ Λi (M ). Then for

any di¬eomorphism F : M ’ M one has F — (dω) = d(F — ω) which means

that F determines a symmetry of Ed . Symmetries of this type are called

gauge symmetries.

The construction of Example 1.13 can be naturally generalized. Let

F : J k (π) ’ J k (π) be a Lie transformation. Note that from the de¬-

nition it follows that for any maximal integral manifold N of the Cartan

distribution on J k (π), the manifold F (N ) possesses the same property. In

particular, graph of k-jets are taken to n-dimensional maximal integral man-

ifolds. Let now θk+1 be a point of J k+1 (π) and let us represent θk+1 as a pair

(θk , Lθk+1 ), or, which is the same, as a class of graphs of k-jets tangent to

each other at θk . Then, since di¬eomorphisms preserve tangency, the image

F— (Lθk+1 ) will almost always (cf. Example 1.13) be an R-plane at F (θk ).

Denote the corresponding point in J k+1 (π) by F (1) (θk+1 ).

Definition 1.17. Let F : J k (π) ’ J k (π) be a Lie transformation. The

above de¬ned mapping F (1) : J k+1 (π) ’ J k+1 (π) is called the 1-lifting of F .

The mapping F (1) is a Lie transformation at the domain of its de¬nition,

since almost everywhere it takes graphs of (k + 1)-jets to graphs of the same

def

kind. Hence, for any l ≥ 1 we can de¬ne F (l) = (F (l’1) )(1) and call this

map the l-lifting of F .

Theorem 1.12. Let π : E ’ M be an m-dimensional vector bundle over

an n-dimensional manifold M and F : J k (π) ’ J k (π) be a Lie transforma-

tion. Then:

(i) If m > 1 and k > 0, the mapping F is of the form F = G(k) for some

di¬eomorphism G : J 0 (π) ’ J 0 (π);

(ii) If m = 1 and k > 1, the mapping F is of the form F = G(k’1) for

some contact transformation G : J 1 (π) ’ J 1 (π).

Proof. Recall that ¬bers of the projection πk,k’1 : J k (π) ’ J k’1 (π)

for k ≥ 1 are the only maximal integral manifolds of the Cartan distribution

of type 0 (see Remark 1.7). Further, from Proposition 1.10 it follows that

in the cases m > 1, k > 0 and m = 1, k > 1 they are integral manifolds

of maximal dimension, provided n > 1. Therefore, the mapping F is πk,µ -

¬berwise, where µ = 0 for m > 1 and µ = 1 for m = 1.

Thus there exists a mapping G : J µ (π) ’ J µ (π) such that πk,µ —¦ F =

G —¦ πk,µ and G is a Lie transformation in an obvious way. Let us show

that F = G(k’µ) . To do this, note ¬rst that in fact, by the same reasons,

the transformation F generates a series of Lie transformations G l : J l (π) ’

J l (π), l = µ, . . . , k, satisfying πl,l’1 —¦Gl = Gl’1 —¦πl,l’1 and Gk = F , Gµ = G.

(1)

Let us compare the mappings F and Gk’1 .

From Proposition 1.6 and the de¬nition of Lie transformations we obtain

F— ((πk,k’1 )’1 (Lθk )) = F— (Cθk ) = CF (θk ) = (πk,k’1 )’1 (LF (θk ) )

k

— —

2. NONLINEAR PDE 25

for any θk ∈ J k (π). But F— ((πk,k’1 )’1 (Lθk )) = (πk,k’1 )’1 (Gk’1,— (Lθk )) and

— —

consequently Gk’1,— (Lθk ) = LF (θk ) . Hence, by the de¬nition of 1-lifting we

(1)

have F = Gk’1 . Using this fact as a base of elementary induction, we obtain

the result of the theorem for dim M > 1.

Consider the case n = 1, m = 1 now. Since all maximal integral man-

ifolds are one-dimensional in this case, it should treated in a special way.

Denote by V the distribution consisting of vector ¬elds tangent to the ¬bers

of the projection πk,k’1 . Then

F— V = V (1.30)

for any Lie transformation F , which is equivalent to F being πk,k’1 -¬berwise.

Let us prove (1.30). To do it, consider an arbitrary distribution P on a

manifold N and introduce the notation

PD = {X ∈ D(N ) | X lies in P} (1.31)

and

DP = {X ∈ D(N ) | [X, Y ] ∈ P, ∀Y ∈ PD}. (1.32)

Then one can show (using coordinate representation, for example) that

DV = DC k © D[DC k ,DC k ]

for k ≥ 2. But Lie transformations preserve the distributions at the right-

hand side of the last equality and consequently preserve DV.

We pass now to in¬nitesimal analogues of Lie transformations:

Definition 1.18. Let π : E ’ M be a vector bundle and E ‚ J k (π) be

a k-th order di¬erential equation.

(i) A vector ¬eld X on J k (π) is called a Lie ¬eld, if the corresponding

one-parameter group consists of Lie transformations.

(ii) A Lie ¬eld is called an in¬nitesimal classical symmetry of the equa-

tion E, if it is tangent to E.

It should be stressed that in¬nitesimal classical symmetries play an im-

portant role in applications of di¬erential geometry to particular equations.

Since in the sequel we shall deal with in¬nitesimal symmetries only, we

shall skip the adjective in¬nitesimal and call them just symmetries. By

de¬nition, one-parameter groups of transformations corresponding to sym-

metries preserve generalized solutions.

Remark 1.9. Similarly to the above considered situation, we may in-

troduce the concepts both of exterior and interior in¬nitesimal symmetries

(see Remark 1.8), but we do not treat the second ones below.

Let X be a Lie ¬eld on J k (π) and Ft : J k (π) ’ J k (π) be its one-param-

(l)

eter group. The we can construct l-liftings Ft : J k+l (π) ’ J k+l (π) and

the corresponding Lie ¬eld X (l) on J k+l (π). This ¬eld is called the l-lifting

of the ¬eld X. As we shall see a bit later, liftings of Lie ¬elds, as opposed

26 1. CLASSICAL SYMMETRIES

to those of Lie transformations, are de¬ned globally and can be described

explicitly.

An immediate consequence of the de¬nition and of Theorem 1.12 is the

following result:

Theorem 1.13. Let π : E ’ M be an m-dimensional vector bundle over

an n-dimensional manifold M and X be a Lie ¬eld on J k (π). Then:

(i) If m > 1 and k > 0, the ¬eld X is of the form X = Y (k) for some

vector ¬eld Y on J 0 (π);

(ii) If m = 1 and k > 1, the ¬eld X is of the form X = Y (k’1) for some

contact vector ¬eld Y on J 1 (π).

Coordinate expressions for Lie ¬elds can be obtained as follows. Let

x1 , . . . , xn , . . . , uj , . . . be a special coordinate system in J k (π) and ωσ be

j

σ

the corresponding Cartan forms. Then X is a Lie ¬eld if and only if the

following equations hold

m

j

»j,± ω„ ,

±

|σ| < k,

L X ωσ = j = 1, . . . , m, (1.33)

σ,„

±=1 |„ |<k

where »j,± are arbitrary smooth functions. Let the vector ¬eld X be repre-

σ,„

sented in the form

n m

‚ ‚

j

X= Xi + Xσ .

‚uj

‚xi σ

i=1 j=1 |σ|¤k

Then from (1.33) it follows that the coe¬cients of the ¬eld X are related by

the following recursion equalities

n

j

uj ± Di (X± ),

j

’

Xσ+1i = Di (Xσ ) (1.34)

σ+1

±=1

where

m

‚ ‚

uj i

Di = + (1.35)

σ+1

‚uj

‚xi σ

j=1 |σ|≥0

are the so-called total derivatives.

Recall now that a contact ¬eld X on J 1 (π), dim π = 1, is completely

def

determined by its generating function which is de¬ned as f = iX ω, where

ω = du’ i u1i dxi is the Cartan (contact) form on J 1 (π). The contact ¬eld

corresponding to a function f ∈ F1 (π) is denoted by Xf and is expressed as

n n

‚f ‚ ‚f ‚

Xf = ’ + f’ u1i

‚u1i ‚xi ‚u1i ‚u

i=1 i=1

n

‚f ‚f ‚

+ + u 1i (1.36)

‚xi ‚u ‚u1i

i=1

2. NONLINEAR PDE 27

in local coordinates.

Thus, starting with a ¬eld (1.36) in the case dim π = 1 or with an

arbitrary ¬eld on J 0 (π) for dim π > 1 and using (1.34), we can obtain

e¬cient expressions for Lie ¬elds.

Remark 1.10. Note that in the case dim π > 1 we can introduce

def

vector-valued generating functions by setting f j = iX ω j , where ω j =

duj ’ i uj i dxi are the Cartan forms on J 1 (π). Such a function may be

1

understood as an element of the module F1 (π, π). The local conditions that

a section f ∈ F1 (π, π) corresponds to a Lie ¬eld is as follows:

‚f ± ‚f β ‚f ±

= , = 0, ± = β.

‚u±i β

‚uβi

‚u1i

1 1

In Chapter 2 we shall generalize the theory and get rid of these conditions.

We call f the generating section (or generating function, depending on

the dimension of π) of the Lie ¬eld X, if X is a lifting of the ¬eld Xf .

Let us ¬nally write down the conditions of a Lie ¬eld to be a symmetry.

Assume that an equation E is given by the relations F 1 = 0, . . . , F r = 0,

where F j ∈ Fk (π). Then X is a symmetry of E if and only if

r

j

»j F ± ,

X(F ) = j = 1, . . . , r,

±

±=1

where »j are smooth functions, or

±

X(F j ) |E = 0, j = 1, . . . , r. (1.37)

These conditions can be rewritten in terms of generating sections and we

shall do it in Chapter 2 in a more general situation.

Let E ‚ J k (π) be a di¬erential equation and X be its symmetry. Then

for any solution • of this equation, the one-parameter group {At } corre-

sponding to X transforms • to some new solution •t almost everywhere. In

special local coordinates, evolution of • is governed by the following evolu-

tionary equation:

‚• ‚• ‚•

= f (x1 , . . . , xn , •, ,..., ), (1.38)

‚t ‚x1 ‚xn

if π is one-dimensional and f is the generating function of X, or by a system

of evolutionary equations of the form

‚•j 1 ‚•m

m ‚•

j 1

= f (x1 , . . . , xn , • , . . . , • , ,..., ), (1.39)

‚t ‚x1 ‚xn

where j = 1, . . . , m = dim π and f j are the components of the generating

section.

In particular, we say that a solution is invariant with respect to X, if

it is transformed by {At } to itself, which means that it has to satisfy the

28 1. CLASSICAL SYMMETRIES

equation

‚• ‚•

f (x1 , . . . , xn , •, ,..., )=0 (1.40)

‚x1 ‚xn

or a similar system of equations when dim π > 1. If g is a subalgebra in the

symmetry algebra of E, we can also de¬ne g-invariant solutions as solutions

invariant with respect to all elements of g.

2.4. Prolongations. The idea of prolongation originates from a simple

observation that, a di¬erential equation given, not all relations between

dependent variables are explicitly encoded in this equation. To reconstruct

these relations, it needs to analyze “di¬erential consequences” of the initial

equations.

Example 1.16. Consider the system

2

uxxy = vy , uxyy = vx + uy .

Then, di¬erentiating the ¬rst equation with respect to y and the second one

with respect to x, we obtain

uxxyy = 2vy vyy , uxxyy = vxx + uxy

and consequently

2vy vyy = vxx + uxy .

Example 1.17. Let

1

v t = u2 + u x .

vx = u,

2

Then

ut = uux + uxx

by a similar procedure.

Example 1.18. Consider equations (1.19) from Example 1.8 on p. 15.

Then as consequences of these equations we obtain equations (1.20) which

may be viewed at as compatibility conditions for equations (1.19). One can

see that if the functions j satisfy (1.20), i.e., if the connection is ¬‚at,

i

then these conditions are void; otherwise we obtain functional relations on

the variables uj .

i

Geometrically, the process of computation of di¬erential consequences is

expressed by the following de¬nition:

Definition 1.19. Let E ‚ J k (π) be a di¬erential equation of order k.

De¬ne the set

E 1 = {θk+1 ∈ J k+1 (π) | πk+1,k (θk+1 ) ∈ E, Lθk+1 ‚ Tπk+1,k (θk+1 ) E}

and call it the ¬rst prolongation of the equation E.

2. NONLINEAR PDE 29

If the ¬rst prolongation E 1 is a submanifold in J k+1 (π), we de¬ne the

second prolongation of E as (E 1 )1 ‚ J k+2 (π), etc. Thus the l-th prolongation

is a subset E l ‚ J k+l (π).

Let us rede¬ne the notion of l-th prolongation directly. Namely, take a

point θk ∈ E and consider a section • ∈ “loc (π) such that the graph of jk (•)

is tangent to E with order l. Let πk (θk ) = x ∈ M . Then [•]k+l is a point of

x

k+l (π) and the set of all points obtained in such a way obviously coincides

J

with E l , provided all intermediate prolongations E 1 , . . . , E l’1 be well de¬ned

in the sense of De¬nition 1.19.

Assume now that locally E is given by the equations

F 1 = 0, . . . , F r = 0, F j ∈ Fk (π)

and θk ∈ E is the origin of the chosen special coordinate system. Let u1 =

•1 (x1 , . . . , xn ), . . . , um = •m (x1 , . . . , xn ) be a local section of the bundle π.

Then

‚ |σ| •±

— j j

jk (•) F = F (x1 , . . . , xn , . . . , ,...)

‚xσ

n

‚F j ‚ |σ|+1 •±

‚F j

= + xi + o(x),

‚u± ‚xσ+1i

‚xi σ

±,σ

i=1 θk

where the sums are taken over all admissible indices. From here it follows,

that the graph of jk (•) is tangent to E at the point under consideration if

and only if

n

‚F j ‚ |σ|+1 •±

‚F j

+ = 0.

‚u± ‚xσ+1i

‚xi σ

±,σ

i=1 θk

Hence, the equations of the ¬rst prolongation are

n

‚F j ‚F j ±

+ u = 0, i = 1, . . . , n.

‚u± σ+1i

‚xi σ

±,σ

i=1

From here and by comparison with the coordinate representation of prolon-

gations for nonlinear di¬erential operators (see Subsection 1.2), we obtain

the following result:

Proposition 1.14. Let E ‚ J k (π) be a di¬erential equation. Then

(i) If the equation E is determined by a di¬erential operator ∆ : “(π) ’

“(π ), then its l-th prolongation is given by the l-th prolongation

∆(l) : “(π) ’ “(πl ) of the operator ∆.

(ii) If E is locally described by the system of equations

F 1 = 0, . . . , F r = 0, F j ∈ Fk (π),

then the system

Dσ F j = 0, |σ| ¤ l, j = 1, . . . , r, (1.41)

30 1. CLASSICAL SYMMETRIES

def σ

where Dσ = D1 1 —¦ · · · —¦ Dnn , corresponds to E l . Here Di stands for

σ

the i-th total derivative (see (1.35)).

From the de¬nition it follows that for any l ≥ l ≥ 0 one has the

embeddings πk+l,k+l (E l ) ‚ E l and consequently one has the mappings

πk+l,k+l : E l ’ E l .

Definition 1.20. An equation E ‚ J k (π) is called formally integrable,

if

(i) all prolongations E l are smooth manifolds

and

(ii) all the mappings πk+l+1,k+l : E l+1 ’ E l are smooth ¬ber bundles.

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

The rest of this chapter is devoted to classical symmetries of some par-

ticular equations of mathematical physics.

3. Symmetries of the Burgers equation

As a ¬rst example, we shall discuss the computation of classical symme-

tries for the Burgers equation, which is described by

ut = uux + uxx . (1.42)

The equation holds on J 2 (x, t; u) = J 2 (π) for the trivial bundle π : R —

R2 ’ R2 with x, t being coordinates in R2 (independent variables) and u a

coordinate in the ¬ber (dependent variable). The total derivative operators

are given by

‚ ‚ ‚

Dx = + ux + uxx

‚x ‚u ‚ux

‚ ‚ ‚ ‚

+ ··· ,

+ uxt + uxxx + uxxt + uxtt

‚ut ‚uxx ‚uxt ‚utt

‚ ‚ ‚

Dt = + ut + uxt

‚t ‚u ‚ux

‚ ‚ ‚ ‚

+ ···

+ utt + uxxt + uxtt + uttt (1.43)

‚ut ‚uxx ‚uxt ‚utt

We now introduce the vector ¬eld V of the form

‚ ‚ ‚ ‚

V =Vx +Vt +Vu + · · · + V utt , (1.44)

‚x ‚t ‚u ‚utt

where in (1.44) V x , V t , V u are functions depending on x, t, u, while the

components with respect to ‚/‚ux , ‚/‚ut , ‚/‚uxx , ‚/‚uxt , ‚/‚utt , which

are denoted by V ux , V ut , V uxx , V uxt , V utt , are given by formula (1.34) and

are of the form

V ux = Dx (V u ’ ux V x ’ ut V t ) + uxx V x + uxt V t ,

V ut = Dt (V u ’ ux V x ’ ut V t ) + uxt V x + utt V t ,

3. SYMMETRIES OF THE BURGERS EQUATION 31

V uxx = Dx (V u ’ ux V x ’ ut V t ) + uxxx V x + uxxt V t ,

2

V uxt = Dx Dt (V u ’ ux V x ’ ut V t ) + uxxt V x + uxtt V t ,

V utt = Dt (V u ’ ux V x ’ ut V t ) + uxtt V x + uttt V t .

2

(1.45)

The symmetry condition (1.37) on V , which is just the invariance condition

of the hypersurface E ‚ J 2 (x, t; u) given by (1.42) under the vector ¬eld V ,

results in the equation

V ut ’ ux V u ’ uV ux ’ V uxx = 0. (1.46)

Calculation of the quantities V ut , V ux , V uxx required in (1.46) yields

‚V u ‚V u ‚V x ‚V x ‚V t ‚V t

ux

’ ux ’ ut

V = + ux + ux + ux ,

‚x ‚u ‚x ‚u ‚x ‚u

‚V u ‚V u ‚V x ‚V x ‚V t ‚V t

V ut ’ ux ’ ut

= + ut + ut + ut ,

‚t ‚u ‚t ‚u ‚t ‚u

‚2V u ‚2V u 2u ‚V u

2‚ V

V uxx = + 2ux + ux + uxx

‚x2 ‚u2

‚x‚u ‚u

x x t ‚V t

‚V ‚V ‚V

’ 2uxx ’ 2uxt

+ ux + ux

‚x ‚u ‚x ‚u

‚2V x ‚2V x 2x ‚V x

2‚ V

’ ux + 2ux + ux + uxx

‚x2 ‚u2

‚x‚u ‚u

‚2V t ‚2V t 2t ‚V t

2‚ V

’ ut + 2ux + ux + uxx . (1.47)

‚x2 ‚u2

‚x‚u ‚u

Substitution of these expressions (1.47) together with

ut = uux + uxx ,

uxt = u2 + uuxx + uxxx , (1.48)

x

into (1.46) leads to a polynomial expression with respect to the variables

uxxx , uxx , ux , the coe¬cients of which should vanish.

The coe¬cient at uxxx , which arises solely from the term uxt in V uxx ,

leads to the ¬rst condition

‚V t ‚V t

+ ux = 0, (1.49)

‚x ‚u

from which we immediately obtain that ‚V t /‚x = 0, ‚V t /‚u = 0, or

V t (x, t, u) = F0 (t), (1.50)

i.e., the function V t is dependent just on the variable t.

Remark 1.11. Although V t is a function dependent just on one variable

t, we prefer to write in the sequel partial derivatives instead of ordinary

derivatives.

32 1. CLASSICAL SYMMETRIES

Now, using the obtained result for the function V t (x, t, u) we obtain

from (1.46), (1.47), (1.48), (1.49) that the coe¬cients at the corresponding

terms vanish:

‚V x

uxx ux : 2 = 0,

‚u

‚V t ‚V x

uxx : ’ +2 = 0,

‚t ‚x

‚2V x

3

ux : = 0,

‚u2

‚2V u ‚2V x

2

ux : ’ +2 = 0,

‚u2 ‚x‚u

‚V x ‚V t ‚V x ‚2V u ‚2V x

u

ux : ’ ’u ’V +u ’2 + = 0,

‚x2

‚t ‚t ‚x ‚x‚u

‚V u ‚V u ‚ 2 V u

’u ’

1: = 0. (1.51)

‚x2

‚t ‚x

From the ¬rst and the fourth equation in (1.51) we have

V x = F1 (x, t), V u = F2 (x, t) + F3 (x, t)u. (1.52)

Substitution of this result into the second, ¬fth and sixth equation of (1.51)

leads to

‚F0 (t) ‚F1 (x, t)

’2 = 0,

‚t ‚x

‚F1 (x, t) ‚F0 (t)

+u + F2 (x, t) + uF3 (x, t)

‚t ‚t

‚F3 (x, t) ‚ 2 F1 (x, t)

‚F1 (x, t)

’u ’

+2 = 0,

‚x2

‚x ‚x

‚F2 (x, t) ‚ 2 F2 (x, t)

‚F2 (x, t)

’u ’

‚x2

‚t ‚x

‚F3 (x, t) ‚ 2 F3 (x, t)

‚F3 (x, t)

’u ’

+u = 0. (1.53)

‚x2

‚t ‚x

We now ¬rst solve the ¬rst equation in (1.53):

x ‚F0 (t)

F1 (x, t) = + F4 (t), (1.54)

2 ‚t

The second equation in (1.53) is an equation polynomial with respect to u,

so we obtain from this the following relations:

‚F0 (t)

+ 2F3 (x, t) = 0,

‚t

‚ 2 F0 (t)

‚F3 (x, t) ‚F4 (t)

4 +2 +x + 2F2 (x, t) = 0, (1.55)

‚t2

‚x ‚t

while from the third equation in (1.53) we obtain

‚F3 (x, t)

= 0,

‚x

3. SYMMETRIES OF THE BURGERS EQUATION 33

‚F3 (x, t) ‚ 2 F3 (x, t) ‚F2 (x, t)

’ ’ = 0,

‚x2

‚t ‚x

‚F2 (x, t) ‚ 2 F2 (x, t)

’ = 0. (1.56)

‚t ‚x

From (1.55) we can obtain the form of F3 (x, t) and F2 (x, t), i.e.,

1 ‚F0 (t)

F3 (x, t) = ’ ,

2 ‚t

‚F4 (t) x ‚ 2 F0 (t)

F2 (x, t) = ’ ’ . (1.57)

2 ‚t2

‚t

The ¬rst and second equation in (1.56) now ful¬ll automatically, while the

third equation is a polynomial with respect to x; hence we have

‚ 2 F4 (t) x ‚ 3 F0 (t)

+ = 0, (1.58)

‚t2 2 ‚t3

from which we ¬nally arrive at

F0 (t) = c1 + c2 t + c3 t2 , F4 (t) = c4 + c5 t. (1.59)

Combining the obtained results we ¬nally have:

1

V x (x, t, u) = c4 + c2 x + c5 t + c3 xt,

2

V (x, t, u) = c1 + c2 t + c3 t2 ,

t

1

V u (x, t, u) = ’c5 ’ c3 x ’ c2 u ’ c3 tu,

2

which are the components of the vector ¬eld V , whereas c1 , . . . , c5 are arbi-

trary constants.

From (1.59) we have that the Lie algebra of classical symmetries of the

Burgers equation is generated by ¬ve vector ¬elds

‚

V1 = ,

‚t

1‚ ‚ 1‚

+t ’ u ,

V2 = x

2 ‚x ‚t 2 ‚u

‚ ‚ ‚

+ t2 ’ (x + tu) ,

V3 = xt

‚x ‚t ‚u

‚

V4 = ,

‚x

‚ ‚

’

V5 = t . (1.60)

‚x ‚u

The commutator table for the generators (1.60) is presented on Fig. 1.1.

Note that the generating functions •i = Vi (du ’ ux dx ’ ut dt) corre-

sponding to symmetries (1.60) are

•1 = ’ut ,

1

•2 = ’ (u + xux + 2tut ),

2

34 1. CLASSICAL SYMMETRIES

[Vi , Vj ] V1 V2 V3 V4 V5

V1 0 V1 2V2 0 V4

1 1

0 V 3 ’ 2 V4

V2 2 V5

’V5

V3 0 0

V4 0 0

V5 0

Figure 1.1. Commutator table for classical symmetries of

the Burgers equation

•3 = ’(x + tu + xtux + t2 ut ),

•4 = ’ux ,

•5 = ’(tux + 1). (1.61)

The computations carried through in this application indicate the way

one has to take to solve overdetermined systems of partial di¬erential equa-

tions for the components of a vector ¬eld arising from the symmetry con-

dition (1.37). We also refer to Chapter 8 for description of computer-based

computations of symmetries.

4. Symmetries of the nonlinear di¬usion equation

The (3 + 1)-nonlinear di¬usion equation is given by

∆(up+1 ) + kuq = ut , (1.62)

where u = u(x, y, z, t), ∆ = ‚ 2 /‚x2 + ‚ 2 /‚y 2 + ‚ 2 /‚z 2 , p, k, q ∈ Q, and

p = ’1.

We shall state the results for the Lie algebras of symmetries for all

distinct values of p, k, q.

First of all we derived that there are no contact symmetries, i.e., the

coe¬cients of any symmetry V ,

‚ ‚ ‚ ‚ ‚

V =Vx +Vy +Vz +Vt +Vu ,

‚x ‚y ‚z ‚t ‚u

V x , V y , V z , V t , V u depend on x, y, z, t, u only.

Remark 1.12. Such symmetries are called point symmetries contrary

to general contact symmetries whose coe¬cients at ‚/‚xi and ‚/‚u may

depend on coordinates in J 1 (π) (see Theorem 1.12 (ii)).

Secondly, for any value of p, k, q, equation (1.62) admits the following

seven symmetries:

‚ ‚ ‚ ‚

V1 = , V2 = , V3 = , V4 = ,

‚x ‚y ‚z ‚t

‚ ‚ ‚ ‚ ‚ ‚

’ x , V6 = z ’ x , V7 = z ’y .

V5 = y (1.63)

‚x ‚y ‚x ‚z ‚y ‚z

4. SYMMETRIES OF THE NONLINEAR DIFFUSION EQUATION 35

We now summarize the ¬nal results, while the complete Lie algebras are

given for all the cases that should be distinguished.

4.1. Case 1: p = 0, k = 0. The complete Lie algebra of symmetries of

the equation

∆(u) = ut (1.64)

is spanned by the vector ¬elds V1 , . . . , V7 given in (1.63) and

‚

V8 = u ,

‚u

‚ ‚

’ xu ,

V9 = 2t

‚x ‚u

‚ ‚

’ yu ,

V10 = 2t

‚y ‚u

‚ ‚

’ zu ,

V11 = 2t

‚z ‚u

‚ ‚ ‚ ‚

V12 = x +y +z + 2t ,

‚x ‚y ‚z ‚t

‚ ‚ ‚ ‚ 1 ‚

+ t2 + u(’x2 ’ y 2 ’ z 2 ’ 6t)

V13 = xt + yt + zt (1.65)

‚x ‚y ‚z ‚t 4 ‚u

together with the continuous part F (x, y, z, t)‚/‚u, where F (x, y, z, t) is an

arbitrary function which has to satisfy (1.64). In fact, all linear equations

possess symmetries of this type.

4.2. Case 2: p = 0, k = 0, q = 1. The complete Lie algebra of

symmetries of the equation

∆(u) + ku = ut (1.66)

is spanned by the ¬elds V1 , . . . , V7 given in (1.63) and

‚

V8 = u ,

‚u

‚ ‚

’ xu ,

V9 = 2t

‚x ‚u

‚ ‚

’ yu ,

V10 = 2t

‚y ‚u

‚ ‚

’ zu ,

V11 = 2t

‚z ‚u

‚ ‚ ‚ ‚ ‚

V12 = x +y +z + 2t + 2kut ,

‚x ‚y ‚z ‚t ‚u

‚ ‚ ‚ ‚ 1 ‚

+ t2 + u(4kt2 ’ x2 ’ y 2 ’ z 2 ’ 6t) .

V13 = xt + yt + zt

‚x ‚y ‚z ‚t 4 ‚u

(1.67)

36 1. CLASSICAL SYMMETRIES

Since (1.66) is a linear equation, it also possesses symmetries of the form

F (x, y, z, t)‚/‚u, where

∆(F ) + kF = Ft . (1.68)

4.3. Case 3: p = 0, k = 0, q = 1. The complete Lie algebra of

symmetries of the equation

∆(u) + kuq = ut (1.69)

is spanned by V1 , . . . , V7 given in (1.63) and the ¬eld

‚ ‚ ‚ ‚ 2 ‚

+ 2t ’

V8 = x +y +z u. (1.70)

‚t q ’ 1 ‚u

‚x ‚y ‚z

4.4. Case 4: p = ’4/5, k = 0. The complete Lie algebra of symmetries

of

∆(u1/5 ) = ut (1.71)

is spanned by V1 , . . . , V7 given in (1.63) together with the ¬elds

‚ ‚

V8 = 4t + 5u ,

‚t ‚u

‚ ‚ ‚ ‚

’ 5u ,

V9 = 2x + 2y + 2z

‚x ‚y ‚z ‚u

‚ ‚ ‚ ‚

V10 = (x2 ’ y 2 ’ z 2 ) ’ 5xu ,

+ 2xy + 2xz

‚x ‚y ‚z ‚u

‚ ‚ ‚ ‚

+ (’x2 + y 2 ’ z 2 ) ’ 5yu ,

V11 = 2xy + 2yz

‚x ‚y ‚z ‚u

‚ ‚ ‚ ‚

+ (’x2 ’ y 2 + z 2 ) ’ 5zu .

V12 = 2xz + 2yz (1.72)

‚x ‚y ‚z ‚u

4.5. Case 5: p = ’4/5, p = 0, k = 0. The complete Lie algebra of

symmetries of the equation

∆(up+1 ) = ut (1.73)

is spanned by V1 , . . . , V7 given in (1.63) and two additional vector ¬elds

‚ ‚

V8 = ’pt +u ,

‚t ‚u

‚ ‚ ‚ ‚

V9 = px + py + pz + 2u . (1.74)

‚x ‚y ‚z ‚u

4.6. Case 6: p = ’4/5, k = 0, q = 1. The complete Lie algebra of

symmetries of the equation

∆(u1/5 ) + ku = ut (1.75)

is spanned by V1 , . . . , V7 given in (1.63) and

‚ 4kt ‚

4kt

V8 = e + kue 5 ,

5

‚t ‚u

5. THE NONLINEAR DIRAC EQUATIONS 37

‚ ‚ ‚ ‚

’ 5u ,

V9 = 2x + 2y + 2z

‚x ‚y ‚z ‚u

‚ ‚ ‚ ‚

= (x2 ’ y 2 ’ z 2 ) ’ 5xu ,

V10 + 2xy + 2xz

‚x ‚y ‚z ‚u

‚ ‚ ‚ ‚

+ (’x2 + y 2 ’ z 2 ) ’ 5yu ,

V11 = 2xy + 2yz

‚x ‚y ‚z ‚u

‚ ‚ ‚ ‚

+ (’x2 ’ y 2 + z 2 ) ’ 5zu .

V12 = 2xz + 2yz (1.76)

‚x ‚y ‚z ‚u

4.7. Case 7: p = 0, p = ’4/5, k = 0, q = 1. The complete Lie algebra

of symmetries of the equation

∆(up+1 ) + ku = ut (1.77)

is spanned by V1 , . . . , V7 given in (1.63) and by

‚ ‚

V8 = e’pkt + 4ku ,

‚t ‚u

‚ ‚ ‚ ‚

V9 = px + py + pz + 2u . (1.78)

‚x ‚y ‚z ‚u

4.8. Case 8: p = 0, p = ’4/5, q = p + 1. The complete Lie algebra of

symmetries of the equation

∆(up+1 ) + kup+1 = ut (1.79)

is spanned by V1 , . . . , V7 given in (1.63) and by the ¬eld

‚ ‚

’u .

V8 = pt (1.80)

‚t ‚u

4.9. Case 9: p = 0, p = ’4/5, q = 1, q = p + 1. The complete Lie

algebra of symmetries of the equation

∆(up+1 ) + kuq = ut (1.81)

is spanned by V1 , . . . , V7 given in (1.63) and by the ¬eld

‚ ‚ ‚ ‚ ‚

V8 = (’p + q ’ 1) x + 2(q ’ 1)t ’ 2u .

+y +z (1.82)

‚x ‚y ‚z ‚t ‚u

The results in these nine cases are a generalization of the results of

other authors [13]. We leave to the reader to describe the corresponding Lie

algebra structures in the cases above.

5. The nonlinear Dirac equations

In this section, we consider the nonlinear Dirac equations and compute

their classical symmetries [33]. Symmetry classi¬cation of these equations

leads to four di¬erent cases: linear Dirac equations with vanishing and non-

vanishing rest mass, nonlinear Dirac equation with vanishing rest mass, and

38 1. CLASSICAL SYMMETRIES

general nonlinear Dirac equation (with nonvanishing rest mass). We con-

tinue to study the last case in the next chapter (Subsection 2.2) and compute

there conservation laws associated to some symmetries.

We shall only give here a short idea of the solution procedure, since all

computations follow to standard lines. The Dirac equations are of the form

[11]:

3

‚(γk ψ) ‚(γ4 ψ)

’i + m0 cψ + n0 ψ(ψψ) = 0, (1.83)

‚xk ‚x4

k=1

where

x4 = ct,

ψ = (ψ1 , ψ2 , ψ3 , ψ4 )T ,

— — — —

ψ = (ψ1 , ψ2 , ’ψ3 , ’ψ4 ), (1.84)

T stands for transposition, — is complex conjugate and γ1 , γ2 , γ3 , γ4 are

4 — 4-matrices de¬ned by

« «

0 0 0 ’i 0 0 0 ’1

¬0 0 ’i 0 · ¬0 0 1 0·

¬ ·, γ2 = ¬ ·

γ1 = 0 1 0 0 ,

0

0i 0

’1 0 0 0

i00 0

« «

0 0 ’i 0 10 0 0

¬0 0 0 i· ¬0 1 0 0·

γ3 = ¬ ·, γ4 = ¬ ·

0 0 ’1 0 . (1.85)

i 0 0 0

0 ’i 0 0 0 0 0 ’1

After introduction of the parameter

»= , (1.86)

m0 c

we obtain

3

‚ ‚

(γ4 ψ) + ψ + »3 ψ(ψψ) = 0.

(γk ψ) ’ »i

» (1.87)

‚xk ‚x4

k=1

In computation of the symmetry algebra of (1.87) we have to distinguish

the following cases:

1. = 0, »’1 = 0: Dirac equations with vanishing rest mass,

2. = 0, »’1 = 0: Dirac equations with nonvanishing rest mass,

3. = 0, »’1 = 0: nonlinear Dirac equations with vanishing rest mass,

4. = 0, »’1 = 0: nonlinear Dirac equations.

These cases are equivalent to the respective choices of m0 and n0 in (1.83):

e.g., = 0, »’1 = 0 is the same as m0 = n0 = 0, etc.

We put ψj = uj + iv j , j = 1, . . . , 4, and obtain a system of eight coupled

partial di¬erential equations

»v1 ’ »u4 + »v3 + »v4 + (1 + »3 K)u1 = 0,

4 3 1

2

5. THE NONLINEAR DIRAC EQUATIONS 39

»v1 + »u3 ’ »v3 + »v4 + (1 + »3 K)u2 = 0,

3 4 2

2

’»v1 + »u2 ’ »v3 ’ »v4 + (1 + »3 K)u3 = 0,

2 1 3

2

’»v1 ’ »u1 + »v3 ’ »v4 + (1 + »3 K)u4 = 0,

1 2 4

2

’»u4 ’ »v2 ’ »u3 ’ »u1 + (1 + »3 K)v 1 = 0,

4

1 3 4

’»u3 + »v2 + »u4 ’ »u2 + (1 + »3 K)v 2 = 0,

3

1 3 4

»u2 + »v2 + »u1 + »u3 + (1 + »3 K)v 3 = 0,

2

1 3 4

»u1 ’ »v2 ’ »u2 + »u4 + (1 + »3 K)v 4 = 0,

1

(1.88)

1 3 4

where

‚uj ‚v j

uj = j

, vk = , j, k = 1, . . . , 4,

k ‚xk ‚xk

and

K = (u1 )2 + (u2 )2 ’ (u3 )2 ’ (u4 )2 + (v 1 )2 + (v 2 )2 ’ (v 3 )2 ’ (v 4 )2 . (1.89)

Thus (1.87) is a determined system E ‚ J 1 (π) in the trivial bundle π : R8 —

R4 ’ R 4 .

Using relations (1.34) and symmetry conditions (1.37), we construct the

overdetermined system of partial di¬erential equations for the coe¬cients of

the vector ¬eld V

‚ ‚ 1‚ 4‚

V = F x1 + · · · + F x4 + Fu + ··· + Fv . (1.90)

‚u1 ‚v 4

‚x1 ‚x4

From the resulting overdetermined system of partial di¬erential equations

we derive in a straightforward way the following intermediate result:

F x1 , . . . , F x4 are independent of u1 , . . . , v 4 ,

1:

F x1 , . . . , F x4 are polynomials of degree 3 in x1 , . . . , x4 ,

2:

1 4

Fu ,...,Fv are linear with respect to u1 , . . . , v 4 .

3: (1.91)

Combination of this intermediate result (1.91) with the remaining system of

partial di¬erential equations leads to the following description of symmetry

algebras in the four speci¬c cases.

= 0, »’1 = 0. The complete Lie algebra of classical

5.1. Case 1:

symmetries for the Dirac equations with vanishing rest mass is spanned by 23

generators. In addition, there is a continuous part generated by functions

1 4

F u , . . . , F v dependent on x1 , . . . , x4 and satisfying the Dirac equations

(1.88) due to the linearity of these equations. The Lie algebra contains the

¬fteen in¬nitesimal generators of the conformal group X1 , . . . , X15 and eight

vertical vector ¬elds X16 , . . . , X23 :

‚ ‚ ‚ ‚

X1 = , X2 = , X3 = , X4 = ,

‚x1 ‚x2 ‚x3 ‚x4

‚ ‚ ‚ ‚ ‚ ‚

’ v1 1 + v2 2 ’ v3 3 + v4 4

’ 2x1

X5 = 2x2

‚x1 ‚x2 ‚u ‚u ‚u ‚u

40 1. CLASSICAL SYMMETRIES

‚ ‚ ‚ ‚

+ u1 ’ u2 2 + u3 3 ’ u4 4 ,

‚v 1 ‚v ‚v ‚v

‚ ‚ ‚ ‚ ‚ ‚

’ u2 1 + u1 2 ’ u4 3 + u3 4

’ 2x1

X6 = 2x3

‚x1 ‚x3 ‚u ‚u ‚u ‚u

‚ ‚ ‚ ‚

’ v2 1 + v1 2 ’ v4 3 + v3 4 ,

‚v ‚v ‚v ‚v

‚ ‚ ‚ ‚ ‚ ‚

+ v2 1 + v1 2 + v4 3 + v3 4

= ’2x3

X7 + 2x2

‚x2 ‚x3 ‚u ‚u ‚u ‚u

‚ ‚ ‚ ‚

’ u2 1 ’ u1 2 ’ u4 3 ’ u3 4 ,

‚v ‚v ‚v ‚v

‚ ‚ ‚ ‚ ‚ ‚

+ u4 1 + u3 2 + u2 3 + u1 4

X8 = 2x4 + 2x1

‚x1 ‚x4 ‚u ‚u ‚u ‚u

‚ ‚ ‚ ‚

+ v4 1 + v3 2 + v2 3 + v1 4 ,

‚v ‚v ‚v ‚v

‚ ‚ ‚ ‚ ‚ ‚

+ v4 1 ’ v3 2 + v2 3 ’ v1 4

X9 = 2x4 + 2x2

‚x2 ‚x4 ‚u ‚u ‚u ‚u

‚ ‚ ‚ ‚

’ u4 1 + u3 2 ’ u2 3 + u1 4 ,

‚v ‚v ‚v ‚v

‚ ‚ ‚ ‚ ‚ ‚

+ u3 1 ’ u4 2 + u1 3 ’ u2 4

X10 = 2x4 + 2x3

‚x3 ‚x4 ‚u ‚u ‚u ‚u

‚ ‚ ‚ ‚

+ v3 1 ’ v4 2 + v1 3 ’ v2 4 ,

‚v ‚v ‚v ‚v

‚ ‚ ‚ ‚

X11 = x1 + x2 + x3 + x4 ,

‚x1 ‚x2 ‚x3 ‚x4

‚ ‚ ‚ ‚

= (x2 ’ x2 ’ x2 + x2 )

X12 + 2x1 x2 + 2x1 x3 + 2x1 x4

1 2 3 4

‚x1 ‚x2 ‚x3 ‚x4

‚

’ (3x1 u1 ’ x2 v 1 ’ x3 u2 ’ x4 u4 ) 1

‚u

‚

’ (3x1 u2 + x2 v 2 + x3 u1 ’ x4 u3 ) 2

‚u

‚

’ (3x1 u3 ’ x2 v 3 ’ x3 u4 ’ x4 u2 ) 3

‚u

‚

’ (3x1 u4 + x2 v 4 + x3 u3 ’ x4 u1 ) 4

‚u

‚

’ (3x1 v 1 + x2 u1 ’ x3 v 2 ’ x4 v 4 ) 1

‚v

‚

’ (3x1 v 2 ’ x2 u2 + x3 v 1 ’ x4 v 3 ) 2

‚v

‚

’ (3x1 v 3 + x2 u3 ’ x3 v 4 ’ x4 v 2 ) 3

‚v

‚

’ (3x1 v 4 ’ x2 u4 + x3 v 3 ’ x4 v 1 ) 4 ,

‚v

5. THE NONLINEAR DIRAC EQUATIONS 41

‚ ‚ ‚ ‚

’ (x2 ’ x2 + x2 ’ x2 )

X13 = 2x1 x2 + 2x2 x3 + 2x2 x4

1 2 3 4

‚x1 ‚x2 ‚x3 ‚x4

‚

’ (3x2 u1 + x1 v 1 ’ x3 v 2 ’ x4 v 4 ) 1

‚u

‚

’ (3x2 u2 ’ x1 v 2 ’ x3 v 1 + x4 v 3 ) 2

‚u

‚

’ (3x2 u3 + x1 v 3 ’ x3 v 4 ’ x4 v 2 ) 3

‚u

‚

’ (3x2 u4 ’ x1 v 4 ’ x3 v 3 + x4 v 1 ) 4

‚u

‚

’ (3x2 v 1 ’ x1 u1 + x3 u2 + x4 u4 ) 1

‚v

‚

’ (3x2 v 2 + x1 u2 + x3 u1 ’ x4 u3 ) 2

‚v

‚

’ (3x2 v 3 ’ x1 u3 + x3 u4 + x4 u2 ) 3

‚v

‚

’ (3x2 v 4 + x1 u4 + x3 u3 ’ x4 u1 ) 4 ,

‚v

‚ ‚ ‚ ‚

’ (x2 + x2 ’ x2 ’ x2 )

X14 = 2x1 x3 + 2x2 x3 + 2x3 x4

1 2 3 4

‚x1 ‚x2 ‚x3 ‚x4

‚

’ (3x3 u1 + x2 v 2 + x1 u2 ’ x4 u3 ) 1

‚u

‚

’ (3x3 u2 + x2 v 1 ’ x1 u1 + x4 u4 ) 2

‚u

‚

’ (3x3 u3 + x2 v 4 + x1 u4 ’ x4 u1 ) 3

‚u

‚

’ (3x3 u4 + x2 v 3 ’ x1 u3 + x4 u2 ) 4

‚u

‚

’ (3x3 v 1 ’ x2 u2 + x1 v 2 ’ x4 v 3 ) 1

‚v

‚

’ (3x3 v 2 ’ x2 u1 ’ x1 v 1 + x4 v 4 ) 2

‚v

‚

’ (3x3 v 3 ’ x2 u4 + x1 v 4 ’ x4 v 1 ) 3

‚v

‚

’ (3x3 v 4 ’ x2 u3 ’ x1 v 3 + x4 v 2 ) 4 ,

‚v

‚ ‚ ‚ ‚

+ (x2 + x2 + x2 + x2 )

X15 = 2x1 x4 + 2x2 x4 + 2x3 x4 1 2 3 4

‚x1 ‚x2 ‚x3 ‚x4

‚

’ (3x4 u1 ’ x2 v 4 ’ x3 u3 ’ x1 u4 ) 1

‚u

‚

’ (3x4 u2 + x2 v 3 + x3 u4 ’ x1 u3 ) 2

‚u

‚

’ (3x4 u3 ’ x2 v 2 ’ x3 u1 ’ x1 u2 ) 3

‚u

42 1. CLASSICAL SYMMETRIES

‚

’ (3x4 u4 + x2 v 1 + x3 u2 ’ x1 u1 )

‚u4

‚

’ (3x4 v 1 + x2 u4 ’ x3 v 3 ’ x1 v 4 ) 1

‚v

‚

’ (3x4 v 2 ’ x2 u3 + x3 v 4 ’ x1 v 3 ) 2

‚v

‚

’ (3x4 v 3 + x2 u2 ’ x3 v 1 ’ x1 v 2 ) 3

‚v

‚

’ (3x4 v 4 ’ x2 u1 + x3 v 2 ’ x1 v 1 ) 4 ,

‚v

‚ ‚ ‚ ‚ ‚

X16 = u1 1 + u2 2 + u3 3 + u4 4 + v 1 1

‚u ‚u ‚u ‚u ‚v

‚ ‚ ‚

+ v2 2 + v3 3 + v4 4 ,

‚v ‚v ‚v

‚ ‚ ‚ ‚ ‚

X17 = u2 1 ’ u1 2 ’ u4 3 + u3 4 ’ v 2 1

‚u ‚u ‚u ‚u ‚v

‚ ‚ ‚

+ v1 2 + v4 3 ’ v3 4 ,

‚v ‚v ‚v

‚ ‚ ‚ ‚ ‚

X18 = u3 1 + u4 2 + u1 3 + u2 4 + v 3 1

‚u ‚u ‚u ‚u ‚v

‚ ‚ ‚

+ v4 2 + v1 3 + v2 4 ,

‚v ‚v ‚v

‚ ‚ ‚ ‚ ‚

X19 = u4 1 ’ u3 2 ’ u2 3 + u1 4 ’ v 4 1

‚u ‚u ‚u ‚u ‚v

‚ ‚ ‚

+ v3 2 + v2 3 ’ v1 4 ,

‚v ‚v ‚v

‚ ‚ ‚ ‚ ‚

X20 = v 1 1 + v 2 2 + v 3 3 + v 4 4 ’ u1 1

‚u ‚u ‚u ‚u ‚v

‚ ‚ ‚

’ u2 2 ’ u3 3 ’ u4 4 ,

‚v ‚v ‚v

‚ ‚ ‚ ‚ ‚

X21 = v 2 1 ’ v 1 2 ’ v 4 3 + v 3 4 + u2 1

‚u ‚u ‚u ‚u ‚v

‚ ‚ ‚

’ u1 2 ’ u4 3 + u3 4 ,

‚v ‚v ‚v

‚ ‚ ‚ ‚ ‚

X22 = v 3 1 + v 4 2 + v 1 3 + v 2 4 ’ u3 1

‚u ‚u ‚u ‚u ‚v

‚ ‚ ‚

’ u4 2 ’ u1 3 ’ u2 4 ,

‚v ‚v ‚v

‚ ‚ ‚ ‚ ‚

X23 = v 4 1 ’ v 3 2 ’ v 2 3 + v 1 4 + u4 1

‚u ‚u ‚u ‚u ‚v

‚ ‚ ‚

’ u3 2 ’ u2 3 + u1 4 . (1.92)

‚v ‚v ‚v

The result is in full agreement with that of Ibragimov [5].

6. SYMMETRIES OF THE SELF-DUAL SU (2) YANG“MILLS EQUATIONS 43