uit = Dx (ut ), vit = Dx (vt ). (2.130)

94 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

From (2.127) we derive the universal linearization operator as a 2 — 2 matrix

operator by of the form

3

vDx + v1 ±Dx + uDx + u1

= . (2.131)

∆

Dx vDx + v1

To construct higher symmetries for equations (2.127), we start from a

vertical vector ¬eld of evolutionary type, i.e.,

∞ ∞

‚ ‚

Dx (Y u )

i

Dx (Y v )

i

Y’ = + . (2.132)

Y

‚ui ‚vi

i=0 i=0

From this and the presentation of the universal linearization operator we

derive the condition for Y = (Y u , Y v ) to be a higher symmetry of (2.127),

i.e.,

vDx Y u + v1 Y u + (±Dx + uDx + u1 )Y v = 0,

3

Dx Y u + (vDx + v1 )Y v = 0. (2.133)

It is quite of interest to make some remarks here on the construction

of solutions of this overdetermined system of partial di¬erential equations

for Y u , Y v . Recall that we require Y u and Y v to be dependent of a ¬nite

number variables. Equations (2.127) are graded, i.e., they admit a scaling

symmetry,

‚ ‚ ‚ ‚

’x ’ 2t + 2u +v ,

‚x ‚t ‚u ‚v

from where we have

‚

deg(x) = ’1, = ’2,

deg(u) = 2, deg

‚u

‚

deg(t) = ’2, = ’1.

deg(v) = 1, deg

‚v

Due to the grading of (2.127), equations (2.132) and (2.133) are graded too

and we require

Y u to dependent on x, t, v, u, v1 , . . . , u4 , v5 , u5 , v6 ,

Y v to dependent on x, t, v, u, v1 , . . . , u4 , v5 .

The general solution of (2.133) is then given by the following eight vector

¬elds

= (Yiu ,Yiv ) , i = 1, . . . , 8,

Yi

where

Y1u = ±v3 + u1 v + v1 u,

Y1v = u1 + v1 v;

Y2u = u1 ,

Y2v = v1 ;

5. THE CLASSICAL BOUSSINESQ EQUATION 95

Y3u = tu1 ,

Y3v = tv1 + 1;

1

Y4u = xu1 + t(±v3 + u1 v + v1 u) + u,

2

1 1

Y4v = xv1 + t(u1 + v1 v) + v;

2 2

1 3 3

Y5u = u1 (v 2 + 2u)

x(±v3 + u1 v + v1 u) + t u3 + v3 v + 3v2 v1 +

2± 2 4±

3 3 1

v1 vu + v 2 + vu,

+

2± 2 ±

1 3 3 12 1

Y5v = v1 (v 2 + 2u) +

x(u1 + v1 v) + t v3 + u1 v + v + u;

2± 2± 4± 4± ±

Y6u = 2±v5 + 4u3 v + v3 (3v 2 + 5u) + 9u2 v1 + 10v2 u1 + 12v2 v1 v

1 3

+ u1 v(v 2 + 6u) + 3v1 + v1 u(v 2 + u),

3

± ±

3 1

Y6v = 2u3 + 4v3 v + 7v2 v1 + u1 (v 2 + u) + v1 v(v 2 + 6u);

± ±

5 15 5 25

Y7u = ±u5 + v5 v + ±v4 v1 + u3 (v 2 + u) + ±v3 v2

2 2 2 2

5 45 25 5

+ v3 v(v 2 + 5u) + 5u2 u1 + u2 v1 v + v2 u1 v + v2 v1 (3v 2 + 5u)

4 4 2 2

75 5 15 3 5

2

u1 (v 4 + 12v 2 u + 6u2 ) + v1 v + v1 vu(v 2 + 3u),

+ u 1 v1 +

8 16± 4 4

5 5 35

Y7v = ±v5 + u3 v + v3 (v 2 + u) + 5u2 v1 + 5v2 u1 + v2 v1 v

2 2 4

5 15 3 5

+ u1 v(v 2 + 3u) + v1 + v1 (v 4 + 12v 2 u + 6u2 );

4 8 16

3 3 3

Y8u = u3 + v3 v + 3v2 v1 + u1 (v 2 + 2u) + v1 vu,

2 4± 2±

3 3

Y8v = v3 + v1 (v 2 + 2u).

u1 v + (2.134)

2± 4±

The Lie algebra structure of these symmetries is constructed by comput-

ing the Jacobi brackets of the respective generating functions Yi = (Yiu , Yiv ).

The commutators of the associated vector ¬elds are given then in Fig. 2.2.

The generating function Y9 is de¬ned here by

5 15 5 105

Y9u = ±v7 + u5 v + v5 (15v 2 + 14u) + 25u4 v1 + v4 u 1

2 2 8 4

225 175 25 175 375

u3 v2 + u3 v(v 2 + 3u) +

+ v4 v1 v + v3 u 2 + v3 v2 v

4 4 4 4 4

1125 25 75

2

v3 (3v 4 + 30v 2 u + 14u2 ) +

+ v3 v1 + u2 u1 v

16 32± 2±

96 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

[Yi , Yj ] Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8

0 ’Y2 ’Y1 ’Y8

Y1 0 0 0

1 1

’ 2 Y2 ’ 2± Y4

Y2 0 0 0 0

1 1 5 3

Y3 2 Y3 ± Y4 4Y8 4 Y6 2± Y1

1 5 3

Y4 2 Y5 2Y6 2 Y7 2 Y8

4 3

Y5 ± Y7 Y9 4± Y6

Y6 0 0

Y7 0

Y8

Figure 2.2. Commutator table for symmetries of the

Boussinesq equation

25 375 2 25

u2 v1 (27v 2 + 26u) + v2 u1 (15v 2 + 14u)

+ v2 v1 +

16± 4 8±

75 125 2 1125

v2 v1 v(v 2 + 5u) + 2

+ u 1 v1 + u 1 v1 v

4± 4± 16±

15 75 3 2

u1 v(v 4 + 20v 2 u + 30u2 ) +

+ v (3v + 5u)

16± 1

32±2

75

v u(v 4 + 6v 2 u + 2u2 ),

+ 21

32±

5 15 25 125

Y9v = u5 + v5 v + 20v4 v1 + u3 (3v 2 + 2u) + v3 v2

2 2 8± 4

25 25 75 75

v3 v(v 2 + 3u) +

+ u2 u1 + u 2 v1 v + v2 u 1 v

4± 2± 2± 2±

25 425 75

v2 v1 (21v 2 + 22u) + 2

u (v 4 + 6v 2 u + 2u2 )

+ u 1 v1 + 21

16± 16± 32±

225 3 15

v v(v 4 + 20v 2 u + 30u2 ).

+ v1 v + (2.135)

21

16± 32±

In order to transform the Lie algebra we introduce

Z1 = ±Y5 , Z0 = Y 4 , Z’1 = Y3 ,

1 1

W1 = Y 2 , W2 = Y1 , W3 = ±Y8 ,

2 2

3 3 3

W6 = ± 2 Y9 ,

W4 = ±Y6 , W5 = ±Y7 , (2.136)

8 2 2

which results in the Lie algebra structure presented in Fig. 2.3.

It is very interesting to note that the classical Boussinesq equation ad-

mits a higher symmetry Z1 (see (2.134)) which is local and which has the

property of acting as a recursion operator for the (x, t)-independent symme-

tries of the classical Boussinesq equation, thus giving rise to in¬nite series

of higher symmetries. In Chapter 5 we shall construct the associated

recursion operator by deformations of the equation structure of the classical

Boussinesq equation.

5. THE CLASSICAL BOUSSINESQ EQUATION 97

[—, —] Z1 Z0 Z’1 W1 W2 W3 W4 W5

1

0 ’ 2 Z1 ’Z0

Z1 W2 W3 W4 W5 W6

1 1 3 5

Z0 0 0 2 Z1 2 W1 W2 2 W3 2W4 2 W5

1 3

Z’1 0 0 0 0 2 W1 2 W2 3W3 5W4

W1 0 0 0 0 0 0 0 0

W2 0 0 0 0 0 0 0 0

W3 0 0 0 0 0 0 0 0

W4 0 0 0 0 0 0 0 0

W5 0 0 0 0 0 0 0 0

Figure 2.3. Commutator table for symmetries of the

Boussinesq equation (2)

98 2. HIGHER SYMMETRIES AND CONSERVATION LAWS

CHAPTER 3

Nonlocal theory

The facts exposed in this chapter constitute a formal base to introduce

nonlocal variables to the di¬erential setting, i.e., variables of the type • dx,

• being a function on an in¬nitely prolonged equation. These variables are

essential for introducing nonlocal symmetries of PDE as well as for existence

of recursion operators. A detailed exposition of this material can be found

in [62, 61] and [12].

1. Coverings

We start with ¬xing up the setting. To do this, extend the universum

of in¬nitely prolonged equations in the following way. Let N be a chain of

„i+1,i

smooth maps · · · ’ N i+1 ’ ’ N i ’ · · · , i.e., an object of the category

’’

∞ (see Chapters 1 and 2), where N i are smooth ¬nite-dimensional mani-

M

folds. As before, let us de¬ne the algebra F(N ) of smooth functions on N as

—

„i+1,i

C ∞ (N i ) ’ ’ C ∞ (N i+1 ) ’

the direct limit of the homomorphisms · · · ’ ’’

— : C ∞ (N i ) ’ F(N ) and

· · · . Then there exist natural homomorphisms „∞,i

the algebra F(N ) may be considered to be ¬ltered by the images of these

maps. Let us consider calculus (cf. Subsection 1.3 of Chapter 1) over F(N )

agreed with this ¬ltration. We de¬ne the category DM∞ as follows:

1. The objects of the category DM∞ are the above introduced chains

N endowed with integrable distributions DN ‚ D(F(N )), where the

word “integrable” means that [DN , DN ] ‚ DN .

2. If N1 = {N1 , „i+1,i }, N2 = {N2 , „i+1,i } are two objects of DM∞ , then

i1 i2

i+±

a morphism • : N1 ’ N2 is a system of smooth mappings •i : N1 ’

i 2

N2 , where ± ∈ Z is independent of i, satisfying „i+1,i —¦ •i+1 = •i —¦

1

„i+±+1,i+± and such that •—,θ (DN1 ,θ ) ‚ DN2 ,•(θ) for any point θ ∈ N1 .

Definition 3.1. A morphism • : N1 ’ N2 is called a covering in the

category DM∞ , if •—,θ |DN ,θ : DN1 ,θ ’ DN2 ,•(θ) is an isomorphism for any

1

point θ ∈ N1 .

In particular, manifolds J ∞ (π) and E ∞ endowed with the corresponding

Cartan distributions are objects of DM∞ and we can consider coverings

over these objects.

Example 3.1. Let ∆ : “(π) ’ “(π ) be a di¬erential operator of order

(l)

¤ k. Then the system of mappings ¦∆ : J k+l (π) ’ J l (π ) (see De¬nition 1.6

99

100 3. NONLOCAL THEORY

on p. 6) is a morphism of J ∞ (π) to J ∞ (π ). Under unrestrictive conditions

of regularity, its image is of the form E ∞ for some equation E in the bundle

π while the map J ∞ (π) ’ E ∞ is a covering.

Definition 3.2. Let • : N ’ N and • : N ’ N be two coverings.

1. A morphism ψ : N ’ N is said to be a morphism of coverings, if

• = • —¦ ψ.

2. The coverings • , • are called equivalent, if there exists a morphism

ψ : N ’ N which is a di¬eomorphism.

Assume now that • : N ’ N is a linear (i.e., vector) bundle and denote

by L(N ) ‚ F(N ) the subset of functions linear along the ¬bers of the

mapping •.

Definition 3.3. A covering • : N ’ N is called linear, if

1. The mapping • is a linear bundle.

2. Any element X ∈ D(N ) preserves L(N ).

Example 3.2. Let E ‚ J k (π) be a formally integrable equation and E ∞

be its in¬nite prolongation and T E ∞ ’ E ∞ be its tangent bundle. Denote

by „ v : V E ∞ ’ E ∞ the subbundle whose sections are π∞ -vertical vector

¬elds. Obviously, any Cartan form ωf = dC (f ), f ∈ F(E ∞ ) (see (2.13) on

p. 66) can be understood as a ¬ber-wise linear function on V E ∞ :

def

Y ∈ “(„ v ),

ωf (Y ) = Y ωf , (3.1)

and any function • ∈ L(V E ∞ ) is a linear combination of the above ones

(with coe¬cients in F(E)).

Take the Cartan distribution C for the distribution DE ∞ and let us de¬ne

the action of any vector ¬eld Z lying in this distribution on the functions of

the form (3.1) by

def

Z(ωf ) = LZ ωf .

Since any Z under consideration is (at least locally) of the form Z =

i fi CXi , X ∈ D(M ), fi ∈ F(E), one has

f LCXi dC f + dfi § iCXi (dC f )

Z(ωf ) = LPi fi CXi ωf =

i

= dC (CXi f ) = fi ωCXi (f ) .

i i

But de¬ned on linear functions, you obtain a vector ¬eld Z on the entire

manifold V E ∞ . Obviously, the distribution spanned by all Z is integrable

and projects to the Cartan distribution on E ∞ isomorphically. Thus we

obtain a linear covering structure in „ v : V E ∞ ’ E ∞ which is called the

(even) Cartan covering.

1. COVERINGS 101

Remark 3.1. In Chapter 6 we shall introduce a similar construction

where the functions ωf will play the role of odd variables. This explains the

adjective even in the above de¬nition.

If the equation E ‚ J k (π) is locally presented in the form E = {F =

0}, then the object V E ∞ is isomorphic to the in¬nite prolongation of the

equation

±

F = 0,

(3.2)

j,σ ‚F wσ = 0,

j

‚ujσ

def

j

where wσ = ωuj . Thus, V E ∞ corresponds to the initial equation together

σ

with its linearization.

Let N be an object of DM∞ and W be a smooth manifold. Consider

the projection „W : N — W ’ N to the ¬rst factor. Then we can make a

covering of „W by lifting the distribution DN to N — W in a trivial way.

Definition 3.4. A covering „ : N ’ N is called trivial, if it is equiva-

lent to the covering „W for some W .

Let again • : N ’ N , • : N ’ N be two coverings. Consider the

commutative diagram

• — (• )

N —N N ’N

• — (• ) •

“ “

•

N ’N

where

N —N N = { (θ , θ ) ∈ N — N | • (θ ) = • (θ ) }

while • — (• ), • — (• ) are the natural projections. The manifold N —N N

is supplied with a natural structure of an object of DM∞ and the mappings

(• )— (• ), (• )— (• ) become coverings.

Definition 3.5. The composition

— —

• —N • = • —¦ • (• ) = • —¦ • (• ) : N —N N ’ N

is called the Whitney product of the coverings • and • .

Definition 3.6. A covering is said to be reducible, if it is equivalent to

a covering of the form • —N „ , where „ is a trivial covering. Otherwise it is

called irreducible.

From now on, all coverings under consideration will be assumed to be

smooth ¬ber bundles. The ¬ber dimension is called the dimension of the

covering • under consideration and is denoted by dim •.

102 3. NONLOCAL THEORY

Proposition 3.1. Let E ‚ J k (π) be an equation in the bundle π : E ’

M and • : N ’ E ∞ be a smooth ¬ber bundle. Then the following statements

are equivalent:

1. The bundle • is equipped with a structure of a covering.

2. There exists a connection C • in the bundle π∞ —¦• : N ’ M , C • : X ’

X • , X ∈ D(M ), X • ∈ D(N ), such that

(a) [X • , Y • ] = [X, Y ]• , i.e., C • is ¬‚at, and

(b) any vector ¬eld X • is projectible to E ∞ under •— and •— (X • ) =

CX, where C is the Cartan connection on E ∞ .

The proof reduces to the check of de¬nitions.

Using this result, we shall now obtain coordinate description of coverings.

Namely, let x1 , . . . , xn , u1 , . . . , um be local coordinates in J 0 (π) and assume

that internal coordinates in E ∞ are chosen. Suppose also that over the

neighborhood under consideration the bundle • : N ’ E ∞ is trivial with the

¬ber W and w 1 , w2 , . . . , ws , . . . are local coordinates in W . The functions

wj are called nonlocal coordinates in the covering •. The connection C •

puts into correspondence to any partial derivative ‚/‚xi the vector ¬eld

˜

C • (‚/‚xi ) = Di . By Proposition 3.1, these vector ¬elds are to be of the

form

‚

˜

Di = Di + Xiv = Di + Xi± ± , i = 1, . . . , n, (3.3)

‚w

±

where Di are restrictions of total derivatives to E ∞ , and satisfy the condi-

tions

˜˜

[Di , Di ] = [Di , Dj ] + [Di , Xj ] + [Xiv , Dj ] + [Xiv , Xj ]

v v

= [Di , Xj ] + [Xiv , Dj ] + [Xiv , Xj ] = 0 (3.4)

v v

for all i, j = 1, . . . , n.

We shall now prove a number of facts that simplify checking of triviality

and equivalence of coverings.

Proposition 3.2. Let •1 : N1 ’ E ∞ and •2 : N2 ’ E ∞ be two cover-

ings of the same dimensions r < ∞. They are equivalent if and only if there

exists a submanifold X ‚ N1 —E ∞ N2 such that

1. The equality codim X = r holds.

2. The restrictions •— (•2 ) |X and •— (•1 ) |X are surjections.

1 2

3. One has (DN1 —E ∞ N2 )θ ‚ Tθ X for any point θ ∈ X.

Proof. In fact, if ψ : N1 ’ N2 is an equivalence, then its graph

Gψ = { (y, ψ(y)) | y ∈ N1 }

is the needed manifold X. Conversely, if X is a manifold satisfying the

assupmtions of the proposition, then the correspondence

y ’ •— (•2 ) (•— (•2 ))’1 (y) © X

1 1

is an equivalence.

2. NONLOCAL SYMMETRIES AND SHADOWS 103

Submanifolds X satisfying assumption (3) of the previous proposition are

called invariant.

Proposition 3.3. Let •1 : N1 ’ E ∞ and •2 : N2 ’ E ∞ be two irre-

ducible coverings of the same dimension r < ∞. Assume that the Whitney

product of •1 and •2 is reducible and there exists an invariant submanifold

X in N1 —E ∞ N2 of codimension r. Then •1 and •2 are equivalent almost

everywhere.

Proof. Since •1 and •2 are irreducible, X is to be mapped surjectively

almost everywhere by •— (•2 ) and •— (•1 ) to N1 and N2 respectively (other-

1 2

wise, their images would be invariant submanifolds). Hence, the coverings

are equivalent by Proposition 3.2.

Corollary 3.4. If •1 and •2 are one-dimensional coverings over E ∞

and their Whitney product is reducible, then they are equivalent.

Proposition 3.5. Let • : N ’ E ∞ be a covering and U ‚ E ∞ be a

˜

domain such that the the manifold U = •’1 (U) is represented in the form

U — Rr , r ¤ ∞, while •|U is the projection to the ¬rst factor. Then the

˜

covering • is locally irreducible if the system

• •

D1 (f ) = 0, . . . , Dn (f ) = 0 (3.5)

has constant solutions only.

Proof. Suppose that there exists a solution f = const of (3.5). Then,

since the only solutions of the system

D1 (f ) = 0, . . . , Dn (f ) = 0,

where Di is the restriction of the i-th total derivative to E ∞ , are constants, f

depends on one nonlocal variable w ± at least. Without loss of generality, we

may assume that ‚f /‚w 1 = 0 in a neighborhood U — V , U ‚ U, V ‚ Rr .

De¬ne the di¬eomorphism ψ : U ‚ U ’ ψ(U ‚ U) by setting

ψ(. . . , xi , . . . , pj , . . . , w± , . . . ) = (. . . , xi , . . . , pj , . . . , f, w2 , . . . , w± , . . . ).

σ σ

•

Then ψ— (Di ) = Di + ±>1 Xi± ‚/‚w± and consequently • is reducible.

Let now • be a reducible covering, i.e., • = • —E ∞ „ , where „ is trivial.

Then, if f is a smooth function on the total space of the covering „ , the

—

function f — = „ — (• ) (f ) is a solution of (3.5). Obviously, there exists an

f such that f — = const.

2. Nonlocal symmetries and shadows

Let N be an object of DM∞ with the integrable distribution P = PN .

De¬ne

DP (N ) = { X ∈ D(N ) | [X, P] ‚ P }

and set sym N = DP (N )/PN . Obviously, DP (N ) is a Lie R-algebra and D

is its ideal. Elements of the Lie algebra sym N are called symmetries of the

object N .

104 3. NONLOCAL THEORY

Definition 3.7. Let • : N ’ E ∞ be a covering. A nonlocal •-

symmetry of E is an element of sym N . The Lie algebra of such symmetries

is denoted by sym• E.

Example 3.3. Consider the even Cartan covering „ v : V E ∞ ’ E ∞ (see

Example 3.2) and a symmetry X ∈ sym E of the equation E. Then we can

de¬ne a vector ¬eld X e on V E ∞ by setting X e (f ) = X(f ) for any function

f ∈ F(E) and

X e (ωf ) = LX (dC f ) = dC (Xf ) = ωXf .

Then, by obvious reasons, X e ∈ sym„ v E and „— X e = X. In other, words

v

X e is a nonlocal symmetry which is obtained by lifting the corresponding

higher symmetry of E to V E ∞ .

On the other hand, we can de¬ne a ¬eld X o by X o (f ) = 0 and

X o (ωf ) = iX (dC f ) = X(f ).

Again, X o is a nonlocal symmetry in „ v , but as a vector ¬eld it is „ v -vertical.

So, in a sense, this symmetry is “purely nonlocal”.

Due to identities [LX , LY ] = L[X,Y ] , [LX , iY ] = i[X,Y ] , and [iX , iY ] = 0,

we have

[X e , Y e ] = [X, Y ]e , [X e , Y o ] = [X, Y ]e , [X o , Y o ] = 0.

A base for computation of nonlocal symmetries is the given by following

two results.

Theorem 3.6. Let • : N ’ E ∞ be a covering. The algebra sym• E is

isomorphic to the Lie algebra of vector ¬elds X on N such that

1. The ¬eld X is vertical, i.e., X(•— (f )) = 0 for any function f ∈

C ∞ (M ) ‚ F(E).

•

2. The identities [X, Di ] = 0 hold for all i = 1, . . . , n.

Proof. Note that the ¬rst condition means that in coordinate repre-

sentation the coe¬cients of the ¬eld X at all ‚/‚xi vanish. Hence the

intersection of the set of vertical ¬elds with D vanish. On the other hand, in

any coset [X] ∈ sym• E there exists one and only one vertical element X v .

•

In fact, let X be an arbitrary element of [X]. Then X v = X ’ i ai Di ,

where ai is the coe¬cient of X at ‚/‚xi .

Theorem 3.7. Let • : N = E ∞ — Rr ’ E ∞ be the covering locally de-

termined by the ¬elds

r

‚

•

Xi± Xi± ∈ F(N ),

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

‚w±

±=1

where w1 , w2 , . . . are coordinates in Rr (nonlocal variables). Then any non-

local •-symmetry of the equation E = {F = 0} is of the form

r

‚

˜ ψ,a = ˜ ψ + a± , (3.6)

‚w±

±=1

3. RECONSTRUCTION THEOREMS 105

where ψ = (ψ 1 , . . . , ψ m ), a = (a1 , . . . , ar ), ψ i , a± ∈ F(N ) are functions

satisfying the conditions

˜F (ψ) = 0, (3.7)

D• (a± ) = ˜ ψ,a (Xi± ) (3.8)

i

while

‚

˜ψ = •

Dσ (ψ) (3.9)

‚uj

σ

j,σ

•

and ˜F is obtained from by changing total derivatives Di for Di .

F

Proof. Let X ∈ sym• E. Using Theorem 3.6, let us write down the

¬eld X in the form

r

‚ ‚

bj a±

X= + , (3.10)

σ

‚uj ‚w±

σ ±=1

σ,j

where “prime” over the ¬rst sum means that the summation extends on

internal coordinates in E ∞ only. Then, equating to zero the coe¬cient at

‚/‚uj in the commutator [X, Di ], we obtain the following equations

•

σ

bj , if uj is an internal coordinate,

σi σi

•

Di (bj ) =

σ

X(uj ) otherwise.

σi

Solving these equations, we obtain that the ¬rst summand in (3.10) is of the

form ˜ ψ , where ψ satis¬es (3.7).

Comparing the result obtained with the description of local symmetries

(see Theorem 2.15 on p. 72), we see that in the nonlocal setting an additional

obstruction arises represented by equation (3.8). Thus, in general, not every

solution of (3.7) corresponds to a nonlocal •-symmetry. We call vector ¬elds

˜ ψ of the form (3.9), where ψ satis¬es equation (3.7), •-shadows. In the next

subsection it will be shown that for any •-shadow ˜ ψ there exists a covering

• : N ’ N and a nonlocal • —¦ • -symmetry S such that •— (S) = ˜ ψ .

3. Reconstruction theorems

Let E ‚ J k (π) be a di¬erential equation. Let us ¬rst establish relations

between horizontal cohomology of E (see De¬nition 2.7 on p. 65) and cover-

ings over E ∞ . All constructions below are realized in a local chart U ‚ E ∞ .

Let us consider a horizontal 1-form ω = n Xi dxi ∈ Λ1 (E) and de¬ne

i=1 h

on the space E ∞ — R the vector ¬elds

‚

ω

, Xi ∈ F(E),

Di = D i + X i (3.11)

‚w

where w is a coordinate along R. By direct computations, one can easily see

ω ω

that the conditions [Di , Dj ] = 0 are ful¬lled if and only if dh ω = 0. Thus,

(3.11) determines a covering structure in the bundle • : E ∞ — R ’ E ∞ and

106 3. NONLOCAL THEORY

this covering is denoted by •ω . It is also obvious that the coverings •ω and

•ω are equivalent if and only if the forms ω and ω are cohomologous, i.e.,

if ω ’ ω = dh f for some f ∈ F(E).

Definition 3.8. A covering over E ∞ constructed by means of elements

1

of Hh (E) is called Abelian.

Let [ω1 ], . . . , [ω ± ], . . . be an R-basis of the vector space Hh (E). Let us

1

de¬ne the covering a1,0 : A1 (E) ’ E ∞ as the Whitney product of all •ω± .

It can be shown that the equivalence class of a1,0 does not depend on the

basis choice. Now, literary in the same manner as it was done in De¬nition

2.7 for E ∞ , we can de¬ne horizontal cohomology for A1 (E) and construct

the covering a2,1 : A2 (E) ’ A1 (E), etc.

Definition 3.9. The inverse limit of the chain

ak,k’1 a1,0

· · · ’ Ak (E) ’ ’ ’ Ak’1 (E) ’ · · · ’ A1 (E) ’ ’ E ∞

’’ ’ (3.12)

is called the universal Abelian covering of the equation E and is denoted by

a : A(E) ’ E ∞ .

1

Obviously, Hh (A(E)) = 0.

Theorem 3.8 (see [43]). Let a : A(E) ’ E ∞ be the universal Abelian

covering over the equation E = {F = 0}. Then any a-shadow reconstructs

up to a nonlocal a-symmetry, i.e., for any solution ψ = (ψ 1 , . . . , ψ m ), ψ j ∈

F(A(E)), of the equation ˜F (ψ) = 0 there exists a set of functions a = (a±,i ),

where a±,i ∈ F(A(E)), such that ˜ ψ,a is a nonlocal a-symmetry.

Proof. Let w j,± , j ¤ k, be nonlocal variables in Ak (E) and assume

that the covering structure in a is determined by the vector ¬elds Di =

a

Di + j,± Xij,± ‚/‚wj,± , where, by construction, Xij,± ∈ F(Aj’1 (E)), i.e.,

the functions Xij,± do not depend on w k,± for all k ≥ j.

Our aim is to prove that the system

Di (aj,± ) = ˜ ψ,a (X j,± )

a

(3.13)

i

is solvable with respect to a = (aj,± ) for any ψ ∈ ker ˜F . We do this by

induction on j. Note that

‚

Di (aj,± ) ’ ˜ ψ,a (Xij,± )

[Di , ˜ ψ,a ] =

a a

‚wj,±

j,±

1,±

for any set of functions (aj,± ). Then for j = 1 one has [Di , ˜ ψ,a ](Xk ) = 0,

a

or

Di ˜ ψ,a (X 1,± ) = ˜ ψ,a Di (X 1,± ) ,

a a

k k

1,±

since Xk are functions on E ∞ .

But from the construction of the covering a one has the following equal-

ity:

Di (Xk ) = Dk (Xi1,± ),

1,±

a a

3. RECONSTRUCTION THEOREMS 107

and we ¬nally obtain

1,± 1,±

a a

Di ψ (Xk ) = Dk ψ (Xi ) .

1

Note now that the equality Hh (A(E)) = 0 implies existence of functions a1,±

satisfying

1,±

a

Di (a1,± ) = ψ (Xi ),

i.e., equation (3.13) is solvable for j = 1.

Assume now that solvability of (3.13) was proved for j < s and the func-

tions (a1,± , . . . , aj’1,± ) are some solutions. Then, since [Di , ˜ ψ,a ] Aj’1 (E) =

a

0, we obtain the needed aj,± literally repeating the proof for the case

j = 1.

Let now • : N ’ E ∞ be an arbitrary covering. The next result shows

that any •-shadow is reconstructable.

Theorem 3.9 (see also [44]). For any •-shadow, i.e., for any solution

ψ = (ψ 1 , . . . , ψ m ), ψ j ∈ F(N ), of the equation ˜F (ψ) = 0, there exists a

¯

ψ •

covering •ψ : Nψ ’ N ’ E ∞ and a •ψ -symmetry Sψ , such that Sψ |E ∞ =

’ ’

˜ ψ |E ∞ .

Proof. Let locally the covering • be represented by the vector ¬elds

r

‚

•

Xi±

Di = Di + ,

‚w±

±=1

r ¤ ∞ being the dimension of •. Consider the space R∞ with the coordi-

nates wl , ± = 1, . . . , r, l = 0, 1, 2, . . . , w0 = w± , and set Nψ = N — R∞

± ±

with

˜ ψ + Sw (X ± ) ‚ ,

l

•

Di ψ = D i + (3.14)

i ±

‚wl

l,±

where

‚ ‚

˜ψ = Dσ (ψ k )

• ±

, Sw = wl+1 (3.15)

±

‚uk ‚wl

σ

σ,k ±,l

and “prime”, as before, denotes summation over internal coordinates.

Set Sψ = ˜ ψ + Sw . Then

˜ ψ (¯k ) ‚ + ‚

l+1

• ˜ ψ + Sw (Xi± )

[Sψ , Di ψ ] = uσi ±

‚uk ‚wl

σ

σ,k l,±

‚ ‚

l+1

• ˜ ψ + Sw

Di ψ (Dσ (ψ k ))

•

(Xi± )

’ ’ ±

‚uk ‚wl

σ

σ,k l,±

‚

˜ ψ (¯k ) ’ D• (ψ k )

= uσi = 0.

σi

‚ukσ

σ,k

108 3. NONLOCAL THEORY

•

Here, by de¬nition, uk = Di (uk ) |N .

¯σi σ

Now, using the above proved equality, one has

‚

• • • •

l l

Dj ψ ˜ ψ + Sw (Xj ) ’ Dj ψ ˜ ψ + Sw (Xi± )

±

[Di ψ , Dj ψ ] = ±

‚wl

l,±

‚

• •

l

˜ ψ + Sw Di ψ (Xj ) ’ Dj ψ (Xi± )

±

= ± = 0,

‚wl

l,±

• •• •

since Di ψ (Xj ) ’ Dj ψ (Xi± ) = Di (Xj ) ’ Dj (Xi± ) = 0.

± ±

•

¯ •

Let now • : N ’ E ∞ be a covering and • : N ’ N ’ E ∞ be another

’ ’

one. Then, by obvious reasons, any •-shadow ψ is a • -shadow as well.

Applying the construction of Theorem 3.9 to both • and • , we obtain two

coverings, •ψ and •ψ respectively.

Lemma 3.10. The following commutative diagram of coverings

Nψ ’ Nψ

¯ ¯

ψ ψ

“ “

•¯ •

’ E∞

N ’N

takes place. Moreover, if Sψ and Sψ are nonlocal symmetries corresponding

in Nψ and Nψ constructed by Theorem 3.9, then Sψ = Sψ .

F (Nψ )

Proof. It su¬ces to compare expressions (3.14) and (3.15) for the cov-

erings Nψ and Nψ .

As a corollary of Theorem 3.9 and of the previous lemma, we obtain the

following result.

Theorem 3.11. Let • : N ’ E ∞ , where E = { F = 0 }, be an arbitrary

covering and ψ1 , . . . , ψs ∈ F(N ) be solutions of the equation ˜F (ψ) = 0.

•

Then there exists a covering •Ψ : NΨ ’ N ’ E ∞ and •Ψ -symmetries

’

Sψ1 , . . . , Sψs , such that Sψs |E ∞ = ˜ ψi |E ∞ , i = 1, . . . , s.

•ψ

¯ •

’1

Proof. Consider the section ψ1 and the covering •ψ1 : Nψ1 ’ ’ N ’ ’

∞ together with the symmetry S

E ψ1 constructed in Theorem 3.9. Then ψ2

is a •ψ1 -shadow and we can construct the covering

•ψ

¯ •ψ

,ψ

•ψ1 ,ψ2 : Nψ1 ,ψ2 ’ ’ ’ Nψ1 ’ ’ E ∞

’1’2 ’1

with the symmetry Sψ2 . Applying this procedure step by step, we obtain

the series of coverings

•ψ1 ,...,ψs’1

¯

•ψ

¯ •ψ

¯ •ψ

¯ •

,...,ψs ,ψ

’ ’’ Nψ1 ,...,ψs’1 ’ ’ ’ ’ . . . ’ ’ ’ Nψ1 ’ ’ N ’ E ∞

Nψ1 ,...,ψs ’’1 ’ ’1’2 ’1

’’’’ ’

4. NONLOCAL SYMMETRIES OF THE BURGERS EQUATION 109

with the symmetries Sψ1 , . . . , Sψs . But ψ1 is a •ψ1 ,...,ψs -shadow and we can

(1) (1)

construct the covering •ψ1 : Nψ1 ’ Nψ1 ,...,ψs ’ E ∞ with the symmetry Sψ1

(1)

satisfying Sψ1 F (Nψ1 ) = Sψ1 (see Lemma 3.10), etc. Passing to the inverse

limit, we obtain the covering NΨ we need.

4. Nonlocal symmetries of the Burgers equation

Consider the Burgers equation E given by

ut = uxx + uux (3.16)

and choose internal coordinates on E ∞ by setting u = u0 = u(0,0) , uk =

u(k,0) . Below we use the method described in [60]. The Lie algebra of

higher symmetries of the Burgers equation is well known and is described

in Section 3 of Chapter 2.

The total derivative operators Dx , Dt are given by

∞

‚ ‚

Dx = + ui+1 ,

‚x ‚ui

k=0

∞

‚ ‚

i

Dt = + Dx (u2 + uu1 ) . (3.17)

‚t ‚ui

k=0

We now start from the only one existing conservation law for Burgers

equation, i.e.,

Dt (2u) = Dx (u2 + 2u1 ). (3.18)

From (3.18) we introduce the new formal variable p by de¬ning its partial

derivatives as follows:

pt = u2 + 2u1 ,

px = 2u, (3.19)

which is in a formal sense equivalent to

p= (2u) dx, (3.20)

from which we have p is a nonlocal variable. Note at this moment that (3.18)

is just the compatibility condition on px , pt . We can now put the question:

What are symmetries of equation E which is de¬ned by

ut = uxx + uux ,

px = 2u,

pt = (u2 + 2ux ). (3.21)

In e¬ect (3.21) is a system of partial di¬erential equations for two depen-

dent variables, u and p, as functions of x and t. The in¬nite prolongation of

110 3. NONLOCAL THEORY

E, denoted by E ∞ , admits internal coordinates x, t, u, p, u1 , u2 , . . . , while

the total derivative operators Dx and Dt are given by

∞

‚ ‚ ‚

Dx = + 2u + ui+1 ,

‚x ‚p ‚ui

k=0

∞

‚ ‚ ‚

+ (u2 + 2u1 ) i

Dt = + Dx (u2 + uu1 ) . (3.22)

‚t ‚p ‚ui

k=0

In order to search for higher symmetries, we search for vertical vector

¬elds with generating function • = (•u , •p ), where •u , •p are functions

dependent on the internal coordinates x, t, u, p, u1 , u2 . . . .

The remarkable result is a symmetry • whose generating function • =

u , •v ) is

(•

‚g(x, t)

+ g(x, t)u e’p/4

•u = ’2

‚x

•p = ’4g(x, t)e’p/4 , (3.23)

where g(x, t) is an arbitrary solution to the heat equation

‚g(x, t) ‚ 2 g(x, t)

’ = 0. (3.24)

‚x2

‚t

If we now contract the vector ¬eld • , • given by (3.23), with the Cartan

one-form associated to the nonlocal variable p, i.e.,

dC (u) = du ’ ux dx ’ (ux x + uux )dt, (3.25)

we obtain an additional condition to E, (3.21), i.e.,

‚g(x, t)

’2 + g(x, t)u = 0, (3.26)

‚x

or equivalently,

‚g(x, t)

u = 2(g(x, t))’1 . (3.27)

‚x

Substitution of (3.27) into (3.16) yields the fact that any function u(x, t)

of the form (3.27), where g(x, t) is a solution of the heat equation (3.24),

is a solution of Burgers equation (3.16). Note that (3.27) is the well-known

Cole“Hopf transformation.

This rather simple example of the notion of nonlocal symmetry indicates

its signi¬cance in the study of geometrical structures of partial di¬erential

equations. Further applications of the nonlocal theory, which are more in-

tricate, will be treated in the next sections.

5. NONLOCAL SYMMETRIES OF THE KDV EQUATION 111

5. Nonlocal symmetries of the KDV equation

In order to demonstrate how to handle calulations concerning the con-

struction of nonlocal symmetries and the calculation of Lie brackets of the

corresponding vertical vector ¬elds, or equivalently, the associated Jacobi

bracket of the generating functions, we discuss these features for the KdV

equation

ut = uux + uxxx . (3.28)

The in¬nite prolongation of (3.28), denoted by E ∞ , is given as

ut = uux + uxxx ,

uxt = Dx (uux + uxxx ) = u2 + uuxx + uxxxx ,

x

ux...xt = Dx . . . Dx (uux + uxxx ),

where total partial derivative operators Dx and Dt are given with respect

to the internal coordinates x, t, u, ux , uxx , uxxx , . . . as

‚ ‚ ‚ ‚

Dx = + ux + uxx + uxxx + ...,

‚x ‚u ‚ux ‚uxx

‚ ‚ ‚ ‚

Dt = + ut + uxt + uxxt + ...

‚t ‚u ‚ux ‚uxx

Classical symmetries of KdV Equation are given by

‚

V1 =’ ,

‚x

‚

V2 =’ ,

‚t

‚ ‚

V3 =t + ,

‚x ‚u

‚ ‚ ‚

V 4 = ’x ’ 3t + 2u ,

‚x ‚t ‚u

or equivalently, the generating functions associated to them, given by

V1u = ux ,

V2u = uux + uxxx ,

V3u = 1 ’ tux ,

V4u = xux + 3t(uux + uxxx ) + 2u.

Associated to (3.28), we can construct conservation laws Ax , At such that

Dt (Ax ) = Dx (At ), (3.29)

which leads to

A1 = u,

x

1

A1 = u2 + uxx ,

t

2

Ax = u 2 ,

2

112 3. NONLOCAL THEORY

2

A2 = u3 ’ u2 + 2uuxx . (3.30)

t x

3

A few higher conservation laws are given by

A3 = u3 ’ 3u2 ,

x x

3

A3 = (u4 + 4u2 uxx ’ 8uu2 + 4u2 ’ 8ux uxxx ),

t x xx

4

36

A4 = u4 ’ 12uu2 + u2 ,

x x

5 xx

4

A4 = u5 + 4u3 uxx ’ 18u2 u2 ’ 24uux uxxx + 12u2 uxx ,

t x x

5

96 72 36

+ uu2 + uxx uxxxx ’ u2 . (3.31)

xx

5 xxx

5 5

We now introduce nonlocal variables associated to two of conservation

laws (3.30) in the form

p1 = u dx,

(u2 ) dx.

p3 = (3.32)

We also introduce the grading to the polynomial functions on the KdV

equation by setting

[x] = ’1, [t] = ’3, [u] = 2, [ux ] = 3, [ut ] = 5, . . . (3.33)

Then the nonlocal variables p1 and p3 are of degree

[p1 ] = 1, [p3 ] = 3.

In order to study nonlocal symmetries of the KdV equation, we consider the

augmented system

ut = uux + uxxx ,

(p1 )x = u,

1

(p1 )t = u2 + uxx ,

2

(p3 )x = u2 ,

2

(p3 )t = u3 ’ u2 + 2uuxx . (3.34)

x

3

We note here that system (3.34) is in e¬ect a system of partial di¬eren-

tial equations in three dependent variables u, p1 , p3 and two independent

variables x, t. We choose internal coordinates on E ∞ — R2 as

x, t, u, p1 , p3 , ux , uxx , uxxx , uxxxx , uxxxxx , . . . , (3.35)

while the total derivative operators D x , Dt are given as

‚ ‚

+ u2

D x = Dx + u ,

‚p1 ‚p3

5. NONLOCAL SYMMETRIES OF THE KDV EQUATION 113

12 ‚ 23 ‚

u ’ u2 + 2uuxx

D t = Dt + u + uxx + . (3.36)

x

2 ‚p1 3 ‚p3

A vertical vector ¬eld V on E ∞ — R2 has as its generating functions V u ,

V p1 , V p3 . The symmetry conditions resulting from (3.34) are

3

Dt V u = V u ux + uDx V u + Dx V u ,

D x V p1 = V u ,

Dx V p3 = 2uV u . (3.37)

For the vertical vector ¬elds V1 , . . . , V4 we derive from this after a short

computation

V1u = ux , V2u = uux + uxxx ,

1

V1p1 = u, V2p1 = u2 + uxx ,

2

2

V1p3 = u2 , V2p3 = u3 + 2uuxx ’ u2 ,

x

3

V3u = 1 ’ tux , V4u = xux + 3t(uux + uxxx ) + 2u,

12

V3p1 = x ’ tu, V4p1 = xu + 3t u + uxx + p1 ,

2

23

V3p3 = 2p1 ’ tu2 , V4p3 = xu2 + 3t u + 2uuxx ’ u2 + 3p3 . (3.38)

x

3

It is a well-known fact [80] that the KdV equation (3.28) admits the Lenard

recursion operator for higher symmetries, i.e.,

2 1 ’1

2

L = D x + u + u x Dx . (3.39)

3 3

From this we have

L(V1u ) = V2u ,

5 10 5

L(V2u ) = V5u = uxxxxx + uxxx u + uxx ux + ux u2 ,

3 3 6

2 1 1

L(V3u ) = u + xux + t(uux + uxxx ) = V4u . (3.40)

3 3 3

We now compute the action of the Lenard recursion operator L on the

generating function V4u of the symmetry V4 . The result is

V5u = L(V4u ) = x(uxxx + uux )

5 10 5 4 1

+ 3t uxxxxx + uxxx u + uxx ux + ux u2 + 4uxx + u2 + ux p1 .

3 3 6 3 3

(3.41)

It is a straightforward check that V5u satis¬es the ¬rst condition of (3.37),

i.e.,

3

Dt (V5u ) = V5u ux + uDx V5u + Dx V5u . (3.42)

114 3. NONLOCAL THEORY

The component V5p1 can be computed directly from the second condition

in (3.37), i.e.,

Dx (V5p1 ) = V5u , (3.43)

which readily leads to

1

V5p1 = x uxx + u2

2

5 5 5 1 1

+ 3t uxxxx + uxx u + u2 + u3 + 3ux + up1 + p3 . (3.44)

x

3 6 18 3 2

The construction of the component V5p3 , which should result from the third

condition in (3.37), i.e.,

Dx (V5p3 ) = 2uV5u , (3.45)

causes a problem:

It is impossible to derive a formula for V5p3 in this setting.

The way out of this problem is to augment system (3.34) once more with

the nonlocal variable p5 resulting from

(p5 )x = u3 ’ 3u2 ,

x

3

(p5 )t = (u4 + 4u2 uxx ’ 8uu2 + 4u2 ’ 8ux uxxx ), (3.46)

x xx

4

or equivalently

(u3 ’ 3u2 ) dx,

p5 = (3.47)

x

and extending total derivative operators D x , Dt to

‚

Dx = Dx + (u3 ’ 3u2 ) ,

x

‚p5

3 ‚

Dt = Dt + (u4 + 4u2 uxx ’ 8uu2 + 4u2 ’ 8ux uxxx ) . (3.48)

x xx

4 ‚p5

Within this once more augmented setting, i.e., having a system of par-

tial di¬erential equations for u, p1 , p3 , and p5 , it is posssible to solve the

symmetry condition for p3 , (3.34):

Dx (V5p3 ) = 2uV5u , (3.49)

the result being the vertical vector ¬eld V5 whose generating functions are

given by (3.41), (3.44), and from (3.49) we obtain

5 10 5

V5u = x(uxxx + uux ) + 3t uxxxxx + uxxx u + uxx ux + ux u2

3 3 6

4 1

+ 4uxx + u2 + ux p1 ,

3 3

1 5 5 5

V5p1 = x uxx + u2 + 3t uxxxx + uxx u + u2 + u3

6 x 18

2 3

6. SYMMETRIES OF THE MASSIVE THIRRING MODEL 115

1 1

+ 3ux + up1 + p3 ,

3 2

1 1

V5p3 = 2x uuxx ’ u2 + u3

x

2 3

1 5 5

+ 6t uuxxxx ’ ux uxxx + u2 + u2 uxx + u5

2 xx 3 24

1 5

+ 6uux + u2 p1 + p5 . (3.50)

3 3

The outline above indicates that we are working in e¬ect in an aug-

mented system of partial di¬erential equations in which all nonlocal vari-

ables associated to all conservation laws for the KdV equation are incorpo-

rated (cf. Theorem 3.8).

The computation of Lie brackets of vertical vector ¬elds, or equivalently,

the computation of the Jacobi brackets for the associated generating func-

tions, is to be carried out in this augmented setting. To demonstrate this,

we want to compute the Lie bracket of the symmetry V1 and the nonlocal

symmetry V5 with the generating functions

V1u = ux ,

5 10 5

V5u = x(uxxx + uux ) + 3t uxxxxx + uxxx u + uxx ux + ux u2

3 3 6

4 1

+ 4uxx + u2 + ux p1 . (3.51)

3 3

The associated Jacobi bracket {V5u , V1u } is de¬ned as

V u = {V5u , V1u } = u u

’

V5 (V1 ) V1 (V5 ), (3.52)

which, using in this computation the equality V1p1 = u, results in

V u = uxxx + uux = V2u .

In a similar way the Jacobi bracket {V5u , V2u } equals

5 10 5

{V5u , V2u } = 3 uxxxxx + uxxx u + uxx ux + ux u2 ,

3 3 6

which is just the generating function of the classical ¬rst higher symmetry

of the KdV equation.

Remark 3.2. The functions Viu , i = 1, . . . , 5, are just the so-called shad-

ows (see the previous section) of the symmetries Vi , i = 1, . . . , 5, in the

augmented setting, including all nonlocal variables.

6. Symmetries of the massive Thirring model

We shall establish higher and nonlocal symmetries of the so-called mas-

sive Thirring model [32], which is de¬ned as the following system E 0 of

partial di¬erential equations de¬ned on J 1 (π), where π : R4 — R2 ’ R2 is

116 3. NONLOCAL THEORY

the trivial bundle with the coordinates u1 , v1 , u2 , v2 in the ¬ber (unknown

functions) and x, t in the base (independent variables):

‚u1 ‚u1

= mv2 ’ (u2 + v2 )v1 ,

2

’ + 2

‚x ‚t

‚u2 ‚u2

= mv1 ’ (u2 + v1 )v2 ,

2

+ 1

‚x ‚t

‚v1 ‚v1

= mu2 ’ (u2 + v2 )u1 ,

2

’ 2

‚x ‚t

‚v2 ‚v2

= mu1 ’ (u2 + v1 )u2 .

2

’ ’ (3.53)

1

‚x ‚t

For this system of equations we choose internal coordinates on E 1 as x, t, u1 ,

v1 , u2 , v2 , u1,1 , v1,1 , u2,1 , v2,1 , while internal coordinates on E 4 are chosen as

x, t, u1 , v1 , u2 , v2 , . . . , u1,4 , v1,4 , u2,4 , v2,4 , where ui,j , vi,j refer to ‚ j ui /‚xj ,

‚ j vi /‚xj , i = 1, 2, j = 1, . . . , 4. In a similar way coordinates can be choosen

on E ∞ .

6.1. Higher symmetries. According to Theorem 2.15 on p. 72, we

construct higher symmetries (symmetries of order 2) by constructing vertical

vector ¬elds • , where the generating functions •u1 , •v1 , •u2 , •v2 depend

on the local variables x, t, u1 , v1 , u2 , v2 , u1,1 , v1,1 , u2,1 , v2,1 , u1,2 , v1,2 , u2,2 ,

v2,2 [41]. The symmetry condition then is

’Dx •u1 + Dt •u1 = m•v2 ’ 2(u2 •u2 + v2 •v2 )v1 + (u2 + v2 )•v1 ,

2

2

Dx •u2 + Dt •u2 = m•v1 ’ 2(u1 •u2 + v1 •v2 )v2 + (u2 + v1 )•v2 ,

2

1

Dx •v1 ’ Dt •v1 = m•u2 ’ 2(u2 •u2 + v2 •v2 )u1 + (u2 + v2 )•u1 ,

2

2

’Dx •v2 ’ Dt •v2 = m•u1 ’ 2(u1 •u2 + v1 •v2 )u2 + (u2 + v1 )•u2 .

2

(3.54)

1

The result then is the existence of four symmetries X1 , . . . , X4 of order 1

the generating functions of which, •u1 , •v1 , •u2 , •v2 , i = 1, . . . , 4, are given

i i i i

as

1

•u1 = (’mv2 + v1 (u2 + v2 )),

2

2

1

2

1

•v1 = (mu2 ’ u1 (u2 + v2 )),

2

2

1

2

1

•u2 = (2u2,1 ’ mv1 + v2 (u2 + v1 )),

2

1

1

2

1

•v2 = (2v2,1 + mu1 ’ u2 (u2 + v1 )),

2

1

1

2

1

•u1 = (2u1,1 + mv2 ’ v1 (u2 + v2 )),

2

2

2

2

1

•v1 = (2v1,1 ’ mu2 + u1 (u2 + v2 )),

2

2

2

2

1

•u2 = (mv1 ’ v2 (u2 + v1 )),

2

1

2

2

6. SYMMETRIES OF THE MASSIVE THIRRING MODEL 117

1

•v2 = (’mu1 + u2 (u2 + v1 )),

2

1

2

2

1

•u1 = u1,1 (x + t) + mv2 x + u1 ’ v1 (u2 + v2 )x,

2

2

3

2

1

•v1 = v1,1 (x + t) ’ mu2 x + v1 + u1 (u2 + v2 )x,

2

2

3

2

1

•u2 = u2,1 (’x + t) + mv1 x ’ u2 ’ v2 (u2 + v1 )x,

2

1

3

2

1

•v2 = v2,1 (’x + t) + mu1 x ’ v2 + u2 (u2 + v1 )x,

2

1

3

2

•u 1 = v 1 ,

4

•v1 = ’u1 ,

4

•u 2 = v 2 ,

4

•v2 = ’u2 . (3.55)

4

Thus in e¬ect, the ¬elds X1 , X2 , X3 are of the ¬rst order, while X4 is of

order zero.

In order to ¬nd symmetries of higher order, we take great advantage of

the fact that the massive Thirring model is a graded system, as is the case

with all equations possessing a scaling symmetry, i.e.,

deg(x) = deg(t) = ’2,

deg(u1 ) = deg(v1 ) = deg(u2 ) = deg(v2 ) = 1,

‚u1

deg(m) = 2, deg = 3, . . . (3.56)

‚x

Due to this grading, all equations in (3.53) are of degree three; the total

derivative operators Dx , Dt are graded too as is the symmetry condition

0

) = 0 mod E 3 .

• (E (3.57)

The solutions of (3.57) are graded too. Note that the ¬elds X1 , . . . , X4 are

of degrees 2, 2, 0, 0 respectively.

We now introduce the following notation:

[u] refers to u1 , v1 , u2 , v2 ,

[u]x refers to u1,1 , v1,1 , u2,1 , v2,1 ,

..............................

In our search for higher symmetries we are not constructing the general

solution of the overdetermined system of partial di¬erential equations for

the generating functions •u1 , •v1 , •u2 , •v2 , resulting from (3.57).

We are just looking for those (x, t)-independent functions which are of

degree ¬ve; so the presentation of these functions is as follows:

•— = [u]xx + ([u]2 + [m])[u]x + ([u]5 + [m][u]3 + [m]2 [u]). (3.58)

118 3. NONLOCAL THEORY

Using the presentation above, we derive two higher symmetries, X5 and X6

of degree 4 and order 2, whose generating functions are given as

1

•u1 = (2u2,1 (’m + 2v1 v2 ) ’ 4v2,1 u2 v1 ’ mv2 (R1 + R2 )

5

4

2

’ 2mv1 R + v1 (R2 + 2R1 R2 )),

1

•v1 = (2v2,1 (’m + 2u1 u2 ) ’ 4u2,1 v2 u1 + mu2 (R1 + R2 )

5

4

2

+ 2mu1 R ’ u1 (R2 + 2R1 R2 )),

1

•u2 = (’4v2,2 + 2u1,1 (’m + 2u1 u2 ) + 4u2,1 (R1 + R2 ) + 4v1,1 u2 v1

5

4

2

’ mv1 (R1 + R2 ) ’ 2mv2 R + v2 (R1 + 2R1 R2 )),

1

•v2 = (4u2,2 + 2v1,1 (’m + 2v1 v2 ) + 4v2,1 (R1 + R2 ) + 4u1,1 v2 u1

5

4

2

+ mu1 (R1 + R2 ) + 2mu2 R ’ u2 (R1 + 2R1 R2 )),

1

•u1 = (4v1,2 + 2u2,1 (’m + 2u1 u2 ) + 4u1,1 (R1 + R2 ) + 4v2,1 u1 v2

6

4

2

+ mv2 (R1 + R2 ) + 2mv1 R + v1 (R2 + 2R1 R2 )),

1

•v1 = (’4u1,2 + 2v2,1 (’m + 2v1 v2 ) + 4v1,1 (R1 + R2 ) + 4u2,1 u2 v1

6

4

2

’ mu2 (R1 + R2 ) ’ 2mu1 R + u1 (R2 + 2R1 R2 )),

1

•u2 = (2u1,1 (’m + 2v1 v2 ) ’ 4v1,1 u1 v2 + mv1 (R1 + R2 )

6

4

2

+ 2mv2 R ’ v2 (R1 + 2R1 R2 )),

1

•v2 = (2v1,1 (’m + 2u1 u2 ) ’ 4u1,1 u2 v1 ’ mu1 (R1 + R2 )

6

4

2

’ 2mu2 R + u2 (R1 + 2R1 R2 )), (3.59)

whereas in (3.59)

R1 = u2 + v1 ,

2

R2 = u 2 + v 2 ,

2

R = u 1 u 2 + v 1 v2 .

1 2

For third order higher symmetries the representation of the generating func-

tions, whose degree is seven, is

•— = [u]xxx + ([u]2 + [m])[u]xx + [u][u]2

x

+ ([u]4 + [m][u]2 + [m]2 )[u]x

+ ([u]7 + [m][u]5 + [m]2 [u]3 + [m]3 [u]).

After a massive computation, we arrive at the existence of higher symmetries

X7 and X8 of degree 6 and order 3, given by

1

•u1 = (8u2,2 u2 v1 + 4v2,2 (2v1 v2 ’ m) ’ 4u2 v1

2,1

7

8

2 2 2

+ 4u2,1 (m(R1 + R2 + v1 + v2 ) ’ 3v1 v2 (R1 + R2 )) ’ 4v2,1 v1

6. SYMMETRIES OF THE MASSIVE THIRRING MODEL 119

+ 4v2,1 (’m(u1 v1 + u2 v2 ) + 3u2 v1 (R1 + R2 )) + 4u1,1 mR

’ 2m2 v1 (R1 + R2 ) ’ 4v2 m2 R + 4v1 mR(R1 + 2R2 )

2 2 3 2 2

+ v2 m(R1 + 4R1 R2 + R2 ) ’ v1 (R2 + 6R2 R1 + 3R2 R1 )),

1

• v1 2

= (’8v2,2 v2 u1 ’ 4u2,2 (2u1 u2 ’ m) + 4v2,1 u1

7

8

+ 4v2,1 (m(R1 + R2 + u2 + u2 ) ’ 3u1 u2 (R1 + R2 ))

1 2

+ 4u2 u1 + 4u2,1 (’m(u1 v1 + u2 v2 ) + 3v2 u1 (R1 + R2 )) + 4v1,1 mR

2,1

+ 2m2 u1 (R1 + R2 ) + 4u2 m2 R ’ 4u1 mR(R1 + 2R2 )

2 2 3 2 2

’ u2 m(R1 + 4R1 R2 + R2 ) + u1 (R2 + 6R2 R1 + 3R2 R1 )),

1

•u 2 = (8u2,3 + 12v2,2 (R1 + R2 ) + 8u1,2 u1 v2 + 4v1,2 (2v1 v2 ’ m)

7

8

’ 12u2 v2 + 24u2,1 v2,1 u2 + 2u2,1 (10mR ’ 3R1 ’ 12R1 R2 ’ 3R2 )

2 2

2,1

+ 12v2,1 v2 + 24v2,1 u1,1 u1 + 24v2,1 v1,1 v1 + 8u2 v2

2

1,1

+ 4u1,1 (m(R1 + R2 + u2 + u2 ) ’ 3u1 u2 (R1 + R2 )) + 8v1,1 v2

2

1 2

+ 4v1,1 (m(u1 v1 + u2 v2 ) ’ 3u2 v1 (R1 + R2 )) ’ 4m2 v1 R

’ 2m2 v2 (R1 + R2 ) + mv1 (R2 + 4R1 R2 + R1 ) + 4mv2 R(R2 + 2R1 )

2 2

3 2 2

’ v2 (R1 + 6R1 R2 + 3R1 R2 )),

1

• v2 = (8v2,3 ’ 12u2,2 (R1 + R2 ) + 8v1,2 u2 v1 ’ 4u1,2 (2u1 u2 ’ m)

7

8

2 2 2

’ 12v2,1 u2 ’ 24u2,1 v2,1 v2 + 2v2,1 (10mR ’ 3R1 ’ 12R1 R2 ’ 3R2 )

+ 12u2 u2 + 24u2,1 v1,1 v1 + 24u2,1 u1,1 u1 ’ 8v1,1 u2

2

2,1

+ 4v1,1 (m(R1 + R2 + v1 + v2 ) ’ 3v1 v2 (R1 + R2 )) ’ 8u2 u2

2 2

1,1

+ 4u1,1 (m(u1 v1 + u2 v2 )) ’ 3v2 u1 (R1 + R2 ) + 4m2 u1 R