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