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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

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