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