ments. Obviously, Hi (E) ⊃ Hi+1 (E).

2. SPECTRAL SEQUENCES AND GRADED EVOLUTIONARY DERIVATIONS 197

Proposition 5.6. For any equation E, the embedding

p p+1

‚C (Hi (E)) ‚ Hi+1 (E)

takes place.

To prove this we need some auxiliary facts.

Lemma 5.7. For any vector ¬elds X1 , . . . , Xp ∈ Dv (E) and an element

˜ ∈ Λ— (E) — D v (E) the equality

‚C (˜) = (’1)p ‚C (X1

X1 ... Xp ... Xp ˜)

p

(’1)p+i X1

+ ... Xi’1 ‚C (Xi ) Xi+1 ... Xp ˜ (5.23)

i=1

holds.

Proof. Recall that for any „¦ ∈ Λ— (E) — D v (E) one has

„¦ UE = „¦ (5.24)

and, by (5.17)

˜ ’ (’1)„¦ ‚C („¦

„¦ ‚C (˜) = ‚C („¦) ˜). (5.25)

In particular, taking „¦ = X ∈ D v (E), we get

˜ ’ ‚C (X

X ‚C ˜ = ‚C (X) ˜). (5.26)

This proves (5.23) for p = 1. The proof is ¬nished by induction on p starting

with (5.26).

Lemma 5.8. Consider vertical vector ¬elds X1 , . . . , Xp+1 ∈ Dv (E) and

p

an element ˜ ∈ H0 (E) = Λp (E) — D v (E). Then

X1 ... Xp+1 (‚C ˜) = 0,

p p+1

i.e., ‚C (H0 (E)) ‚ H1 (E).

This result is a direct consequence of (5.23).

Recall that a form θ ∈ Λp (E) is said to be horizontal if the identity

X θ = 0 holds for any X ∈ D v (E); the set of such forms is denoted by

Λp (E). It is easy to see that Hi (E) = Λi (E) § H0 (E), i.e., any element

p p’i

h

h

p

˜ ∈ Hi (E) can be represented as

ρs § ˜ s ,

˜= (5.27)

s

where ρs ∈ Λp (E), ˜s ∈ Λp’i (E) — D v (E). Applying (5.19) and (5.20) to

h

is the Cartan connection C, we get

(5.27) in the case when

’dh (ρs ) § ˜s + (’1)i ρs § ‚C (˜s ) .

‚C (˜) = (5.28)

C

s

198 5. DEFORMATIONS AND RECURSION OPERATORS

Lemma 5.9. 1 Let ξ : P ’ M be a ¬ber bundle with a ¬‚at connection

: D(M ) ’ D(P ) and

Λ— (P ) = {ρ ∈ Λ— (P ) | Y ρ = 0, Y ∈ D v (P )}

h

be the module of horizontal forms on P . Then for any form ρ ∈ Λi (P ) one

0

has

dh (ρ) ∈ Λi+1 (P ).

h

Proof. Let „¦ ∈ Λ— (P ) — D(P ), ρ ∈ Λ— (P ), and Y ∈ D(P ). Then

standard computations show that

ρ) ’ (’1)„¦ [[„¦, Y ]]fn

+ (’1)„¦ L„¦ (Y

Y (L„¦ ρ) = L(Y „¦) ρ ρ. (5.29)

and Y ∈ D v (P ), using (5.16) one has

In particular, if „¦ = U

(LU ρ) = Y (ρ) ’ LU (Y

Y ρ) + ‚ (Y ) ρ,

from where it follows that

dh (ρ) = ’dh (Y ρ) ’ ‚ (Y )

Y ρ,

since, by de¬nition, dh = d ’ LU .

Hence, if Y ∈ D v (P ) and ρ ∈ Λ— (P ), then one has ‚ (Y ) ∈ Λ1 (P ) —

h

v (P ) and Y h (ρ) = 0.

D d

Proposition 5.6 now follows from Lemmas 5.8, 5.9 and identity (5.28).

Remark 5.4. From the de¬nition of the di¬erential dh it immediately

C

follows that its restriction on Λ— (E), denoted by dh , coincides with the hor-

h

izontal de Rham complex of the equation E (see Chapter 2). As it follows

from (5.22), in local coordinates this restriction is completely determined by

the equalities

dh (f ) = Di (f ) dxi , dh (dxi ) = 0, (5.30)

i

where i = 1, . . . , n, f ∈ F(E) and D1 , . . . , Dn are total derivatives. One can

— —

show that the action L of HC (E) can be restricted onto the module Hh (E)

of horizontal cohomologies. In fact, if ρ ∈ Λ— (E) and X, Y ∈ D v (E), then

0

X Y (ρ) = Y (X ρ) + [X, Y ] ρ = 0.

On the other hand, if „¦ ∈ Λ— (E) — D v (E), then from (5.29) it follows that

Y (L„¦ ρ) = L(Y „¦) ρ.

Hence, by induction,

(Λ— (E)) ‚ Λ— (E).

L h h

Λ— (E)—D v (E)

On the other hand, the operator LUE is exactly the Cartan di¬erential of

the equation E (see also Chapter 2).

2. SPECTRAL SEQUENCES AND GRADED EVOLUTIONARY DERIVATIONS 199

Let us now de¬ne a ¬ltration in Λ— (E) — D v (E) by setting

p

F l (Λp (E) — D v (E)) = Hp+l (E). (5.31)

Obviously,

F l (Λp (E) — D v (E)) ⊃ F l+1 (Λp (E) — D v (E))

and Proposition 5.6 is equivalent to the fact that

‚C F l (Λp (E) — D v (E)) ‚ F l (Λp+1 (E) — D v (E)).

Thus (5.31) de¬nes a spectral sequence for the complex (5.9) which we call

H-spectral. Its term E0 is of the form

p,q p+q p+q

E0 = H2p+q (E)/H2p+q+1 (E), (5.32)

where p = 0, ’1, . . . , q = ’2p, . . . , ’2p + n.

p,q

To express E0 in more suitable terms, let us recall the splitting

Λ1 (E) = Λ1 (E) • CΛ1 (E),

h

where CΛ1 (E) is the set of all 1-forms vanishing on the Cartan distribution

on E. Let

C i Λ(E) = CΛ1 (E) § · · · § CΛ1 (E) .

i times

Then for any p the module Λp (E) can be represented as

p

p

C p’i Λ(E) § Λi (E).

Λ (E) = h

i=0

Thus

p

p

C p’i Λ(E) § Λi (E) — Dv (E)

Hi (E) = h

i=0

from where it follows that

E0 = C ’p Λ(E) § Λ2p+q (E) — D v (E).

p,q

h

The con¬guration of the term E0 for the H-spectral sequence is presented

on Fig. 5.1, where D v = Dv (E), Λi = Λi (E), etc.

h h

The second spectral sequence to be de¬ned is in a sense complementary

to the ¬rst one. Namely, we say that an element ˜ ∈ Λp (E) — D v (E) is

(p ’ i + 1)-Cartan, if X1 . . . Xi ˜ = 0 for any X1 , . . . , Xi ∈ CD(E), and

p

denote the set of all such elements by Ci (E) ‚ Λp (E) — D v (E). Obviously,

p p

Ci (E) ‚ Ci+1 (E).

Proposition 5.10. For any equation E ‚ J k (π) one has

p p+1

‚C (Ci (E)) ‚ Ci+1 (E).

To prove this proposition, we need some preliminary facts.

200 5. DEFORMATIONS AND RECURSION OPERATORS

C 2 Λ § Λn — D v

h

C 2 Λ § Λn’1 — Dv

h

C 2 Λ § Λn’2 — Dv C 1 Λ § Λn — D v

h

h

C 1 Λ § Λn’1 — Dv

... h

C 1 Λ § Λn’2 — Dv Λn — D v

... q=n

h

h

Λn’1 — Dv q = n’1

... ... h

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

C 2 Λ § Λ1 — D v ... ... ...

h

C 2 Λ — Dv C 1 Λ § Λ2 — D v ... ...

h

C 1 Λ § Λ1 — D v ... ...

h

C 1 Λ — Dv Λ2 — D v q=2

h

Λ1 — D v q=1

h

Dv q=0

p = ’2 p = ’1 p=0

Figure 5.1. The H-spectral sequence con¬guration (term E0 ).

Lemma 5.11. For any vector ¬elds X1 , . . . , Xp ∈ CD(E) and an element

˜ ∈ Λ— (E) — D v (E) the equality

‚C (˜) = (’1)p ‚C (X1

X1 ... Xp ... Xp ˜)

p

˜]]fn

(’1)p+i+1 X1

+ ... Xi’1 [[Xi , Xi+1 ... Xp UE . (5.33)

i=1

holds.

Proof. We proceed by induction on p. Let X ∈ CD(E). Then, since

X UE = 0 and [[X, UE ]]fn = 0, from equality (4.45) on p. 175 it follows that

˜) ’ [[X, ˜]]fn

‚C (˜) = ’‚C (X

X UE , (5.34)

which gives us the starting point of induction.

Suppose now that (5.33) is proved for all s ¤ r. Then by (5.34) we have

X1 X2 ... Xr+1 ‚C (˜) = X1 (X2 ... Xr+1 ‚C (˜))

= (’1)r X1 ‚C (X2 ... Xr+1 ˜)

r+1

˜]]fn

(’1)r+i X2

+ X1 ... Xi’1 [[Xi , Xi+1 ... Xr+1 UE

i=2

2. SPECTRAL SEQUENCES AND GRADED EVOLUTIONARY DERIVATIONS 201

˜]]fn

= (’1)r ’‚C (X1 ˜) ’ [[X1 , X2

... Xr+1 ... Xr+1 UE

r+1

˜]]fn

(’1)r+i X1

+ X2 ... Xi’1 [[Xi , Xi+1 ... Xr+1 UE

i=2

= (’1)r+1 ‚C (X1 ... Xr+1 ˜)

r+1

˜]]fn

(’1)r+i+1 X1

+ ... Xi’1 [[Xi , Xi+1 ... Xr+1 UE ,

i=1

which ¬nishes the proof of lemma.

p

Lemma 5.12. For any X ∈ CD(E) and ˜ ∈ Ci (E) we have

(i)

p’1

˜ ∈ Ci’1 (E)

X

and

(ii)

p

[[X, ˜]]fn ∈ Ci (E).

Proof. The ¬rst statement is obvious. To prove the second one, note

that from equality (4.45) on p. 175 it follows that for any X, X1 ∈ D(E) and

˜ ∈ Λ— (E) — D v (E) one has

[[X, ˜]]fn = [[X, X1 ˜]]fn + [[X1 , X]]fn

X1 ˜.

Now, by an elementary induction one can conclude that

[[X, ˜]]fn = [[X, X1 ˜]]fn

X1 ... Xi ... Xi

i

[[Xs , X]]fn

+ X1 ... Xs’1 Xs ... Xi ˜ (5.35)

s=1

for any X1 , . . . , Xi ∈ D(E).

p

Consider vector ¬elds X, X1 , . . . , Xi ∈ CD(E) and an element ˜ ∈ Ci (E).

Then, since [[Xs , X]]fn = [Xs , X] ∈ CD(E), all the summands on the right-

hand side of (5.35) vanish.

p

Proof of Proposition 5.10. Consider an element ˜ ∈ Ci (E) and

¬elds X1 , . . . , Xi+1 ∈ CD(E). Then, by (5.33), one has

‚C (˜) = (’1)i+1 ‚C (X1

X1 ... Xi+1 ... Xi+1 ˜)

i+1

˜]]fn

(’1)i+s X1

+ ... Xs’1 [[Xs , Xs+1 ... Xi+1 UE . (5.36)

s=1

The ¬rst summand on the right-hand side vanishes by de¬nition while the

rest of them, due to equality (4.31) on p. 173 and since UE ∈ Λ1 (E) — D v (E),

can be represented in the form

˜]]fn

(’1)i+s X1 ... Xs’1 [[Xs , Xs+1 ... Xi+1 UE .

202 5. DEFORMATIONS AND RECURSION OPERATORS

p

Since ˜ ∈ Ci (E) and X1 , . . . , Xi+1 ∈ CD(E), we have

p’i+s’1

˜ ∈ Cs’1

Xs+1 ... Xi+1 (E)

and by Lemma 5.12 (ii) the element [[Xs , Xs+1 . . . Xi+1 ˜]]fn belongs

p’i+s’1

to Cs’1 (E) as well. Hence, all the summands in (5.36) vanish.

Let us now de¬ne a ¬ltration in Λ— (E) — D v (E) by setting

p

F l (Λp (E) — D v (E)) = Cp’l+1 (E). (5.37)

Obviously,

F l (Λp (E) — D v (E)) ⊃ F l+1 (Λp (E) — D v (E))

and, by Proposition 5.10,

‚C F l (Λp (E) — D v (E)) ‚ F l (Λp+1 (E) — D v (E)).

Thus, ¬ltration (5.37) de¬nes a spectral sequence for the complex (5.9) which

we call the C-spectral sequence for the equation E.

Remark 5.5. As it was already mentioned before, C-spectral sequences

were introduced by A.M. Vinogradov (see [102]). As A.M. Vinogradov

noted (a private communication), the H-spectral sequence can also be viewed

as a C-spectral sequence constructed with respect to ¬bers of the bundle

π∞ : E ∞ ’ M . It is similar to the classical Leray“Serre sequence.

The term E0 of the C-spectral sequence is of the form

p,q p+q p+q

E0 = Cq+1 (E)/Cq (E), p = 0, 1, . . . , q = 0, 1, . . . , n.

To describe these modules explicitly, note that

p+q

C i Λ(E) § Λp+q’i (E) — Dv (E)

p+q

Cq (E) = h

i=p

while

p+q

p+q

C i Λ(E) § Λp+q’i (E) — Dv (E).

Cq+1 (E) = h

i=p+1

Thus

E0 = C p Λ(E) § Λq (E) — D v (E).

p,q

h

The con¬guration of the term E0 for the C-spectral sequence is given on

Fig. 5.2.

Remark 5.6. The 0-th column of the term E0 coincides with the hori-

zontal de Rham complex for the equation E with coe¬cients in the bundle of

vertical vector ¬elds. Complexes of such a type were introduced by T. Tsu-

jishita in [97].

2. SPECTRAL SEQUENCES AND GRADED EVOLUTIONARY DERIVATIONS 203

Λn — D v Λn § C 1 Λ — D v Λn § C p Λ — D v

q=n ... ...

h h h

q = n’1 Λn’1 — Dv Λn’1 § C 1 Λ — Dv Λn’1 § C p Λ — Dv

... ...

h h h

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

Λq — D v Λq § C 1 Λ — D v Λq § C p Λ — D v

... ... ...

h h h

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

Λ1 — D v Λ1 § C 1 Λ — D v Λ1 § C p Λ — D v

q=1 ... ...

h h h

Dv C 1 Λ — Dv C p Λ — Dv

q=0 ... ...

p=0 p=1 ... ... ...

Figure 5.2. The C-spectral sequence con¬guration (term E0 ).

Consider, as before, a formally integrable equation E ‚ J k (π) and the

corresponding algebra F(E) ¬ltered by its subalgebras Fi (E).

We say that an element ˜ ∈ Λp (E) — D v (E) is i-vertical if

L˜ |Fi’k’1 (E) = 0 (5.38)

and denote by Vip (E) the set of all such elements. Obviously, Vip (E) ⊃

p p

Vi+1 (E) and V0 (E) = Λp (E) — D v (E).

Proposition 5.13. For any equation E the embedding

‚C (Vip (E)) ‚ Vi’1 (E)

p+1

takes place.

Proof. Obviously, LUE (Fj (E)) ‚ Fj+1 (E) for any j ≥ ’k ’1. Consider

elements ˜ ∈ Vip (E) and φ ∈ Fi’k’2 (E). Then, by de¬nition,

L‚C (˜) (φ) = L[[UE ,˜]]fn (φ) = LUE (L˜ (φ)) ’ (’1)˜ L˜ (LUE (φ)) = 0,

which ¬nishes the proof.

Let us de¬ne a ¬ltration in Λ— (E) — D v (E) by setting

p

F l (Λp (E) — D v (E)) = Vl’p (E). (5.39)

Obviously,

F l (Λp (E) — D v (E)) ‚ F l+1 (Λp (E) — D v (E))

and

‚C F l (Λp (E) — D v (E)) ‚ F l (Λp+1 (E) — D v (E)).

Thus, (5.39) de¬nes a spectral sequence for the complex (5.9) which we call

V-spectral. The term E0 for this spectral sequence is of the form

p,q p+q p+q

E0 = V’q (E)/V1’q (E), p = 0, 1, . . . , q = 0, ’1, . . . , ’p.

204 5. DEFORMATIONS AND RECURSION OPERATORS

p=0 p=1 ... ... ...

FV F V — Λ1 (π) F V — Λp (π)

q=0 ... ...

F V — S 1 D(M ) F V — Λp’1 (π) — S 1 D(M )

q = ’1 ... ...

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

F V — S p D(M )

q = ’p ...

... ...

Figure 5.3. The V-spectral sequence con¬guration for

J ∞ (π) (term E0 ).

— —

Now we shall compute the algebra HC (E) = HC (π) for the “empty equa-

tion” J ∞ (π) using the V-spectral sequence.

p,q

First, we shall represent elements of the modules E0 in a more conve-

nient way. Denote ’q by r and consider the bundle πr,r’1 : J r (π) ’ J r’1 (π)

and the subbundle πr,r’1,V : T v (J r (π)) ’ J r (π) of the tangent bundle

T (J r (π)) ’ J r (π) consisting of πr,r’1 -vertical vectors. Then we have the

induced bundle:

—

π∞,r (T v (J r (π)) ’ T v (J r (π))

—

π∞,r (πr,r’1,V ) πr,r’1,V

“ “

J ∞ (π) ’ J r (π)

and obviously,

p,’r —

= Λp’r (π) —F (π) “(π∞,r (πr,r’1,V )).

E0

—

On the other hand, the bundle π∞,r (πr,r’1,V ) can be described in the fol-

lowing way. Consider the tangent bundle „ : T (M ) ’ M , its rth symmetric

power S r („ ) : S r T (M ) ’ M and the bundle

πV — π — (S r („ )) : T v (J 0 (π)) — π — (S r T (M )) ’ J 0 (π),

where πV : T v (J 0 (π)) ’ J 0 (π) is the bundle of π-vertical vectors. Then, at

least locally,

πr,r’1,V ≈ πp,0 (πV — π — (S r („ ))).

—

It means that locally we have an isomorphism

p,’r

≈ F(π, πV ) —F (π) Λp’r (π) —C ∞ (M ) S r (D(M )).

µ : E0

Thus the term E0 of the V-spectral sequence is of the form which is presented

def

on Fig. 5.3, where F V = F(π, πV ).

2. SPECTRAL SEQUENCES AND GRADED EVOLUTIONARY DERIVATIONS 205

Let (x1 , . . . , xn ) be local coordinates in M , pj be the coordinates arising

σ

∞ (π), and ξ = ‚/‚x , . . . , ξ = ‚/‚x be the local basis in

naturally in J 1 1 n n

T (M ) corresponding to (x1 , . . . , xn ). Denote also by v j , j = 1, . . . , m, local

vector ¬elds ‚/‚uj , where uj = pj (0,...,0) are coordinates along the ¬ber of

p,’r

the bundle π. Then any element ˜ ∈ E0 is of the form

m

‚

j

θσ ∈ Λp’r (π),

j

θσ —

˜= ,

‚pj

σ

j=1 |σ|=r

while the identi¬cation µ can be represented as

m

1σ

v j — θσ —

j

„¦ = µ(˜) = ξ,

σ!

j=1 |σ|=r

σ

where σ = (σ1 , . . . , σn ), σ! = σ1 ! · · · · · σn !, ξ σ = ξ1 1 · · · · · ξnn .

σ

Let us now represent „¦ in the form

p’r

ρi § ω i — Q i ,

„¦=

i=0

where ρi ∈ F(π, πV ) — C p’r’i Λ(π), ωi ∈ Λi (π), and Qi = Qi (ξ) are homo-

h

geneous polynomials in ξ1 , . . . , ξn of the power q.

From equality (5.15) it follows that in this representation the di¬erential

p,’r p,’r+1

’ E0

‚0 : E 0 in the following way

p’r n

‚Q

(’1)p’r’i ρi § dxs § ωi —

‚0 („¦) = . (5.40)

‚ξs

s=1

i=0

Thus, the di¬erential ‚0 reduces to δ-Spencer operators (see [93]) from which

p,0

it follows that all its cohomologies are trivial except for the terms E0 . But

as it is easily seen from (5.40) and from the previous constructions,

p,0 p,0 p,1

E1 = E0 /‚0 (E0 ) = F(π, πV ) —F (π) C p Λ(π).

Hence, only the 0-th row survives in the term E1 and it is of the form

0,0

‚1

0 ’ F(π, πV ) ’ ’ F(π, πV ) — C 1 Λ(π) ’ · · ·

’

p,0

‚1

’ F(π, πV ) — C Λ(π) ’ ’ F(π, πV ) — C p+1 Λ(π) ’ · · ·

p

’

Recall now that ‚1 is induced by the di¬erential ‚π and that the latter

increases the degree of horizontality for the elements from Λ— (π) — D v (π)

(Proposition 5.6). Again, we see that ‚1 is trivial. Thus, we have proved

the following

Theorem 5.14. The V-spectral sequence for the “empty” equation

E∞ = J ∞ (π) stabilizes at the term E1 , i.e., E1 = E2 = · · · = E∞ , and

C-cohomologies for this equation are of the form

p

HC (π) ≈ F(π, πV ) —F (π) C p Λ(π).

206 5. DEFORMATIONS AND RECURSION OPERATORS

Remark 5.7. When π is a vector bundle, then F(π, πV ) ≈ F(π, π) and

we have the isomorphism

p

HC (π) ≈ F(π, π) —F (π) C p Λ(π).

This result allows to generalize the notion of evolutionary derivations

and to introduce graded (or super ) evolutionary derivations. Namely, we

choose a canonical coordinate system (x, pj ) in J ∞ (π) and for any element

σ

1 , . . . , ω m ) ∈ F(π, π ) — C p Λ(π), ω j ∈ C p Λ(π), set

ω = (ω V

‚

Dσ (ω j ) — ∈ Λp (π) — D v (π).

= (5.41)

ω

‚pj

σ

j,σ

We call ω a graded evolutionary derivation with the generating form ω ∈

F(π, πV ) — C p Λ(π). Denote the set of such derivations by κp (π).

The following local facts are obvious:

(i)

(F(π)) ‚ C p Λ(π),

L ω

(ii)

p

∈ ker(‚π ),

ω

(iii) the correspondence ω ’ splits the natural projection

ω

p

p

ker(‚π ) ’ HC (π)

and thus

ker(‚π ) = im(‚π ) • κp (π).

p p’1

We shall show now that De¬nition (5.41) is independent of local coor-

dinates. The proposition below, as well as its proof, is quite similar to that

one which has been proved in [60] for “ordinary” evolutionary derivations

(see also Chapter 2).

Proposition 5.15. Any element „¦ ∈ Λ— (π) — D v (π) which satis¬es

the conditions (i) and (ii) above, i.e., for which L„¦ (F(π)) ‚ C p Λ(π) and

‚π („¦) = 0, is uniquely determined by the restriction of L„¦ onto F0 (π) =

C ∞ (J 0 (π)).

Proof. First recall that „¦ is uniquely determined by the derivation

L„¦ ∈ Dgr (Λ— ) (see Proposition 4.20). Further, since L„¦ is a graded deriva-

tion and due to the fact that

L„¦ (dθ) = (’1)„¦ d(L„¦ (θ)) (5.42)

for any θ ∈ Λ— (π) (Proposition 4.20), L„¦ is uniquely determined by its

restriction onto F(π) = Λ0 (π).

Now, from the equality ‚π („¦) = 0 it follows that

0 = [[Uπ , „¦]]fn (φ) = LUπ (L„¦ (φ)) ’ (’1)„¦ L„¦ (LUπ (φ)). (5.43)

2. SPECTRAL SEQUENCES AND GRADED EVOLUTIONARY DERIVATIONS 207

Let „¦ be such that L„¦ |F0 (π) = 0 and suppose that we have proved that

L„¦ |Fr (π) = 0. Then taking φ = pj , |σ| = r, and using equality (5.43), we

σ

obtain

n

pj i dxi = LUπ (L„¦ (pj )) = 0.

„¦

L„¦ dpj ’

(’1) σ σ

σ+1

i=1

In other words,

n n

pj i L„¦ (pj i ) dxi

L„¦ dxi =

σ+1 σ+1

i=1 i=1

= L„¦ (dpj ) = (’1)„¦ d(L„¦ (pj ) = 0.

σ σ

Since L„¦ (pj i ) ∈ C — Λ(π), we conclude that L„¦ (pj i ) = 0, i.e., we have

σ+1 σ+1

L„¦ |Fr+1 (π) = 0.

Remark 5.8. The element Uπ = j,σ dpj ’ i pj i dxi —‚/‚pj itself

σ σ

σ+1

1 , . . . , ω m ),

is an example of an evolutionary derivation: Uπ = ω , ω = (ω

where ω j = duj ’ i pj i dxi .

1

Since

F(π, πV ) — C — Λ(π) = F(π, πV ) — C i Λ(π)

i≥0

—

is identi¬ed with the module HC (π), it carries the structure of a graded Lie

algebra. The corresponding operation in F(π, πV ) — C — Λ(π) is denoted by

{·, ·} and is called the graded Jacobi bracket. Thus, for any elements ω ∈

F(π, πV ) — C p Λ(π) and θ ∈ F(π, πV ) — C q Λ(π) we have {ω, θ} ∈ F(π, πV ) —

C p+q Λ(π) and

{ω, θ} + (’1)pq {θ, ω} = 0,

(’1)(p+r)q {ω, {θ, ρ}} = 0,

where ρ ∈ F(π, πV ) — C r Λ(π) and , as before, denotes the sum of cyclic

permutations.

To express the graded Jacobi bracket in more e¬cient terms we prove

the following

Proposition 5.16. The space κ— (π) = i≥0 κi (π) of super evolutionary

derivations is a graded Lie subalgebra in Λ— (π) — D v (π), i.e., for any two

generating forms ω, θ ∈ F(π, πV ) — C — Λ(π) the bracket [[ ω , θ ]]fn is again

an evolutionary derivation and

fn

[[ ω, θ ]] = {ω,θ} . (5.44)

fn

Proof. First note that it is obvious that [[ ω, θ ]] lies in ker(‚π ).

208 5. DEFORMATIONS AND RECURSION OPERATORS

Consider a vector ¬eld X ∈ CD(π). Then, since X =X = 0,

ω θ

from equality (4.45) on p. 175 it follows that

fn fn fn

= (’1)ω [[X, ’ (’1)(ω+1)θ [[X,

X [[ ω, θ ]] ω ]] θ ]] ω.

θ

Let X = Di , where Di is the total derivative along xi in the chosen coordi-

nate system. Then we have

‚ ‚

[[Di , ω ]]fn = Dσ (ω j ) — Di , j + Dσ+1i (ω j ) — j = 0.

‚pσ ‚pσ

j,σ

Since any X ∈ CD(π) is a linear combination of the ¬elds Di , one has

fn

CD(π) [[ ω, θ ]] = 0,

fn

∈ C — Λ(π) — D v (π). Hence, Proposition 5.15 implies that the

i.e., [[ ω , θ ]]

fn

bracket [[ ω , θ ]] is an evolutionary derivation.

From (5.44) and from Proposition 5.16 it follows that if (ω 1 , . . . , ω m )

and (θ 1 , . . . , θm ) are local representations of ω and θ respectively then

m

j θi

{ω, θ}i = i

) ’ (’1)ω j

ω j (θ θ i (ω ), (5.45)

j=1

where i = 1, . . . , m.

For example, if ω = LUπ (f ) = df ’ i Di (f ) dxi and θ = LUπ (g), where

f, g ∈ “(π), then

‚g i ‚f i

i j j

{ω, θ} = LUπ (Dσ (f )) § LUπ + LUπ (Dσ (g )) § LUπ ,

‚pj ‚pj

σ σ

j,σ

where i = 1, . . . , m. In particular,

i j

{ωσ , ω„ } = 0, (5.46)

j

i

where ωσ , ω„ are the Cartan forms (see (1.27) on p. 18).

3. C-cohomologies of evolution equations

Here we give a complete description for C-cohomologies of systems of

evolution equations and consider some examples.

Let E be a system of evolution equations of the form

‚ |σ| u

‚uj j

j = 1, . . . , m, |σ| ¤ k,

= f x, t, u, . . . , ,... , (5.47)

‚t ‚xσ

where x = (x1 , . . . , xn ), u = (u1 , . . . , um ). Then the functions x, t, pj , where

σ

j = 1, . . . , m, σ = (σ1 , . . . , σn ), can be chosen as internal coordinates on E ∞ .

In this coordinate system the element UE is represented in the form

‚

pj i dxi ’ Dσ (f j ) dt — j ,

dpj ’

UE = (5.48)

σ σ+1

‚pσ

j,σ i

3. C-COHOMOLOGIES OF EVOLUTION EQUATIONS 209

σ σ

where Dσ = D1 1 —¦ · · · —¦ Dnn , for σ = (σ1 , . . . , σn ). If

‚

∈ Λ— (E) — D v (E), θ„ ∈ Λ— (E),

j j

θ„ —

˜=

‚pj

„

j,„

then, as it follows from (4.40) on p. 175, the di¬erential ‚C acts in the

following way

j j

dxi § (θ„ +1i ’ Di (θ„ ))

‚C (˜) =

j,„ i

‚

D„ (f j ) θσ ’ Dt (θ„ )

s j

— ‚pj ,

+ dt § (5.49)

„

s

‚pσ

s,σ

where

‚ ‚

Dµ (f j )

Dt = + .

‚pj

‚t µ

j,µ

To proceed with computations consider a direct sum decomposition

Λp (E) — D v (E) = Λp (π) — D v (π) • dt § Λt (π) — D v (π),

p’1

(5.50)

t

where π : Rm — Rn ’ Rn is the natural projection with the coordinates

(u1 , . . . , um ) and (x1 , . . . , xn ) in Rm and Rn respectively, while Λ— (π) denotes

t

the algebra of exterior forms on J ∞ (π) with the variable t ∈ R as a parameter

in their coe¬cients. From (5.49) and due to (4.45) on p. 175 it follows that

if ˜ ∈ Λp (E) — D v (E) and

˜ = ˜p + dt § ˜p’1

is the decomposition corresponding to (5.50), then

‚C (˜) = ‚π (˜p ) + dt § (LE (˜p ) ’ ‚π (˜p’1 )), (5.51)

where

‚ ‚

D„ (f j ) θσ ’ Dt (θ„ ) — j .

s j

LE (˜) = (5.52)

‚ps ‚p„

σ

s,σ

j,„

210 5. DEFORMATIONS AND RECURSION OPERATORS

Consider a diagram

0 0

“

“ i

‚π

’ Λi+1 — Dv

’ Λi — D v ’ ...

... t t

dt § LE dt § LE (5.53)

“

“

’ dt § Λi+1 — Dv

’ dt § Λi — Dv ’ ...

... t t

i

’id § ‚π

“ “

0 0

def

where Λi — Dv = Λi (π) — D v (π). From (5.51) and from the fact that

t t

‚π —¦ ‚π = 0 it follows that (5.53) is a bicomplex whose total di¬erential is

‚C . Thus, from the general theory of bicomplexes (cf. [70]) we see that to

i

calculate HC (E) it is necessary:

(i) To compute cohomologies of the upper and lower lines of (5.53). De-

i i

note them by HC (π) and HL (π) respectively.

(ii) To describe the mappings Li : HC (π) ’ HL (π) induced by dt § LE .

i i

E

Then we have

HC (E) = ker(Li ) • coker(Li’1 ).

i

(5.54)

E E

From Theorem 5.14 it follows that HC (π) = κi (π) and HL (π) = dt §

i i

t

κi’1 (π), where κp (π) is the set of all evolutionary derivations with generating

t t

forms from F(π, π) — C p Λt (π) parameterized by t (we write F(π, π) instead

of F(π, πV ) since π is a vector bundle in the case under consideration). Let

ω = (ω 1 , . . . , ω m ) be such a form. Then, as it is easily seen from (5.52),

Lp ( ω) = ,

(p)

E

E (ω)

where

‚f j ‚

(p)

(Dσ ω s ) ’ Dt (ω j ) — j .

E (ω) = (5.55)

‚ps ‚u

σ

s,σ

j

(p)

Comparing (5.55) with equality (2.23) on p. 71, we see that E is the

extension of the universal linearization operator for the equation (5.47) onto

the module F(π, π) — C p Λt (π).

3. C-COHOMOLOGIES OF EVOLUTION EQUATIONS 211

Remark 5.9. Note that when the operator ∆ is the sum of monomials

X1 —¦ · · · —¦ Xr , the action

∆(ω) = X1 (X2 (. . . (Xr (ω)) . . . ))

is well de¬ned for any form ω such that Xi ω = 0, i = 1, . . . , r. It is just the

case for formula (5.55), since X ω = 0 for any X ∈ CD(E) and ω ∈ C p Λ(E).

Thus we have the following generalization of Theorem 2.15 (see p. 72).

Theorem 5.17. Let E be a system of evolution equations of the form

(0)

(5.47), E = E be corresponding universal linearization operator restricted

(p)

onto E ∞ and E be the extension of E onto F(π, π) — C p Λt (π). Then

(p) (p’1)

p

HC (E) ≈ ker( • dt § coker(

E) ).

E

Remark 5.10. The result proved is, in fact, valid for all -normal equa-

tions (see De¬nition 2.16). The proof can be found in [98]. Moreover, let

—

us recall that the module HC (E) splits into the direct sum

n

—,q

p,q

—

HC (E) = HC (E) = HC (E),

q=1

i≥0 p+q=i

where the superscripts p and q correspond to the number of Cartan and

horizontal components respectively (see decomposition (4.60) on p. 181).

p,0

As it can be deduced from Proposition 4.29, the component HC (E) always

(p)

coincides with ker E .

As a ¬rst example of application of the above theorem, we shall prove

that evolution equations in one space variable are 2-trivial objects in the

sense of Section 3 of Chapter 4.

Proposition 5.18. For any evolution equation E of the form

‚u ‚f

= f (x, t, u, . . . , uk ), = 0, k > 0,

‚t ‚uk

2,0

one has HC (E) = 0.

Proof. To prove this fact, we need to solve the equation

k

‚f i

Dt ω = D ω, (5.56)

‚ui x

i=1

with ω = ±>β •±β ω± § ωβ , where •±β ∈ F(E) and ω± , ωβ are the Cartan

forms on E ∞ . Let us represent the form ω as

ω = •m,m’1 ωm § ωm’1 + •m,± ωm § ω±

±<m’1

•m’1,β ωm’1 § ωβ + o[m ’ 1],

+ (5.57)

β<m’2

212 5. DEFORMATIONS AND RECURSION OPERATORS

where the term o[m ’ 2] does not contain Cartan forms of degree higher

than m ’ 2.

Note now that for any Cartan form ωi one has

k

‚f ‚f ‚f

i

D t ωi = Dx ω± = ωi+k + iDx ωi+k’1

‚u± ‚uk ‚uk

±=0

and

k

‚f ± ‚f ‚f

ωi+k’1 + o[i + k ’ 2].

D x ωi = ωi+k +

‚ui ‚uk ‚uk’1

±=1

Substituting (5.57) into (5.56) and using the above decompositions, one can

easily see that the coe¬cients •m,± vanish, from where, by induction, it

follows that ω = 0.

Now we shall look more closely at the module

(1) (0)

1

HC (E) ≈ ker( • dt § coker(

E) E)

and describe in¬nitesimal deformations of evolution equations in the form

ready for concrete computations. From the decomposition given by the

previous theorem we see that there are two types of in¬nitesimal defor-

(1)

mations: those ones which lie in ker( E ) and those which originate from

(0)

dt § coker( E ). The latter ones are represented by the elements of the form

‚

g j dt — = dt — θ,

U1 = (5.58)

‚uj

j

where g j ∈ F(E). Deformations corresponding to (5.58) are of the form

U (µ) = UE + U1 µ + . . . (5.59)

But it is easily seen that the ¬rst two summands in (5.59) determine an

equation of the form

uj = f j + µg j , j = 1, . . . , m, (5.60)

t

which is in¬nitesimally equivalent to the initial equation if and only if θ ∈

(0)

im( E ). The deformations (5.60) preserve the class of evolution equations.

(1)

The other ones lie in ker( E ) and we shall deduce explicit formulas for their

computation. For the sake of simplicity we consider the case dim(π) = m =

1, dim(M ) = n = 2 (one space variable).

i

Let ωi = dpi ’ pi+1 dx ’ Dx (f ) dt, i = 0, 1, . . . , be the basis of Cartan

forms on E ∞ , where f = f 1 (x, t, p0 , . . . , pk ), x = x1 , and pi corresponds to

‚ i u/‚xi . Then any form ω ∈ C 1 Λ(E) can be represented as

r

φ i ωi , φi ∈ F(E).

ω= (5.61)

i=0

3. C-COHOMOLOGIES OF EVOLUTION EQUATIONS 213

Thus we have

k

(1) j

φ i ωi ,

fj D x ’ D t

E (ω) = (5.62)

j=0 i

where fj denotes ‚f /‚pj . By de¬nition, we have

j

j

(fj Dx )(φi ωi )

j i

Dx (φi )Dx (ωi ).

j’s s

= fj (Dx (. . . (Dx (φ ωi )) . . . )) = fj

s

s=0

But

Dx (ωi ) = ωi+1 (5.63)

and therefore,

j

j

(fj Dx )(φi ωi ) = fj

j

Dx (φi )ωi+s .

j’s

(5.64)

s

s=0

On the other hand,

Dt (φi ωi ) = Dt (φi )ωi + φi Dt (ωi ). (5.65)

i

Since ωi = Dx (ω0 ) and [Dt , Dx ] = 0, one has

i i

Dt (ωi ) = Dt (Dx (ω0 )) = Dx (Dt (ω0 )). (5.66)

But ω0 = LUE (p0 ) and [Dt , LUE ] = 0. Hence,

k

Dt (ω0 ) = LUE (Dt (p0 )) = LUE (f ) = f j ωj . (5.67)

j=0

(1)

Combining now (5.62)“(5.67), we ¬nd out that the equation E (ω) =0

written in the coordinate form looks as

j

r k

j

Dx (φi )ωi+s

j’s

fj

s

s=0

i=0 j=0

r k i

i

Dt (φi )ωi + φi i’s

= Dx (fj )ωj+s . (5.68)

s

j=0 s=0

i=0

Taking into account that {ωi }i≥0 is the basis in C 1 Λ(E) and equating

the coe¬cients at ωi , we obtain that (5.68) is equivalent to

r

s

φi Dx (fs )

i

E (φ )=

i=0

s r k

i i i’l j

fj Dx (φs’l )

j’l

φ Dx (fs’l ) ’

+ (5.69)

l l

l=1 i=l j=l

where s = 0, 1, . . . , k +r ’1, which is the ¬nal form of (5.62) for the concrete

calculations (we set φi = fj = 0 for i > r and j > k in (5.69)).

214 5. DEFORMATIONS AND RECURSION OPERATORS

Consider some examples now.

Example 5.1. Let E be the heat equation

ut = uxx .

For this equation (5.69) looks as

Dx (φ0 ) = Dt (φ0 ),

2

Dx (φ1 ) + 2Dx (φ0 ) = Dt (φ1 ),

2

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

Dx (φr ) + 2Dx (φr’1 ) = Dt (φr ),

2

Dx (φr ) = 0. (5.70)

Simple but rather cumbersome computations show that the basis of solutions

for (5.70) consists of the functions

s

x2j

0 (j+s)

φ= A ,

(2j)!

j=0

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

s’i 2j

i+s’j (j+s’i) x

2i 2i

φ =2 A ,

2i (2j)!

j=0

s’i 2j+1

i+s’j (j+s’i) x

2i+1 2i+1

φ =2 A ,

2i + 1 (2j + 1)!

j=0

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

φ2s = 22s A

for r = 2s and

s

x2j+1

0 (j+s+1)

φ= A ,

(2j + 1)!

j=0

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

s’i 2j+1

i+s’j (j+s’i+1) x

2i 2i

φ =2 A ,

2i (2j + 1)!

j=0

s’i

i + s ’ j + 1 (j+s’i) x2j

2i+1 2i+1

φ =2 A ,

2i + 1 (2j)!

j=0

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

φ2s+1 = 22s+1 A

for r = 2s + 1.

In both cases A = 1, t, . . . , tr and A(l) denotes dl A/dtl .

3. C-COHOMOLOGIES OF EVOLUTION EQUATIONS 215

j 1

j φ ωj be an element of HC (E) and ψ ∈

Remark 5.11. Let φ =

F(π, π) be a symmetry of the equation E. Then, as it follows from (5.18),

the element Rφ (ψ) is a symmetry of E again. In particular, since the equa-

tion under consideration is linear, it possesses the symmetry ψ = u. Hence,

its symmetries include those of the form

φ j pj ,

Rφ (u) =

j

where φj are given by the formulae above.

Example 5.2. The second example we consider is the Burgers equation

ut = uux + uxx . (5.71)

(1)

Theorem 5.19. The only solution of the equation E (ω) = 0 for the

Burgers equation (5.71) is ω = ±ω0 , ± = const.

Proof. Let ω = φ0 ω0 + · · · + φr ωr . Then equations (5.69) transform

into

r

0

Dx (φ0 )

2 0

pj+1 φj ,

p0 Dx (φ ) + = Dt (φ ) +

j=1

r

1

Dx (φ1 )

2 0 1

(j + 1)pj φj ,

p0 Dx (φ ) + + 2Dx (φ ) = Dt (φ ) +

j=2

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

r

j+1

i

Dx (φi )

2 i’1 i

pj’i+1 φj ,

p0 Dx (φ ) + + 2Dx (φ ) = Dt (φ ) +

i

j=i+1

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

p0 Dx (φr ) + Dx (φr ) + 2Dx (φr’1 ) = Dt (φr ) + rp1 φr ,

2

Dx (φr ) = 0.

(5.72)

To prove the theorem we apply the same scheme which was used to

describe the symmetry algebra of the Burgers equation in Chapter 2.

Denote by Kr the set of solutions of (5.72). A direct computation shows

that

K1 = {±ω0 | ± ∈ R} (5.73)

and that any element ω ∈ Kr , r > 1, is of the form

r 1 (1)

p0 ±r + x±r + ±r’1 ωr’1 + „¦(r ’ 2),

ω = ± r ωr + (5.74)

2 2

where ±r = ±r (t), ar’1 = ar’1 (t), a(i) denotes di ±/dti and „¦(s) is an

arbitrary linear combination of ω0 , . . . , ωs with the coe¬cients in F(E).

216 5. DEFORMATIONS AND RECURSION OPERATORS

Lemma 5.20. For any evolution equation E one has

(1) fn (1)

‚ ker(

[[sym(E), ker( E )]] E ).

Proof of Lemma 5.20. In fact, we know that there exists the natural

0 i

action of sym(E) = HC (E) on HC (E). On the other hand, if X = φ ∈

(1)

sym(E) and ˜ = θ ∈ ker( E ) where φ ∈ F(E) and θ ∈ C 1 Λt (π), then

[[X, ˜]]fn = {φ,θ} . But the element

‚ ‚

i i

{φ, θ} = ’ (θ) ’

φ (θ) θ (φ) = Dx (φ) Dx (θ) (φ)

‚pi ‚pi

i i

obviously lies in C 1 Λt (π).

(1) (1)

Thus, if φ ∈ sym(E) and ω ∈ ker( E ) then {φ, ω} lies in ker( E) as

well.

Let φ = p1 . Then we have

‚ ‚

{p1 , ω} = (ω) ’ Dx (ω) = ’ (ω).

pi+1

‚pi ‚x

i

If ω ∈ Kr then, since p1 is a symmetry of E, from (5.74) and from Lemma

5.20 we obtain that

ad(r’1) (ω) = ±r

(r’1)

ω1 + „¦(0) ∈ K1 ,

p1

(r’1)

where adφ = {φ, ·}. Taking into account (5.73) we get that ±r = 0, or

±r = a0 + a1 t + · · · + ar’2 tr’2 , ai ∈ R. (5.75)

Recall now (see Chapter 2) that

¦ = t2 p2 + (t2 p0 + tx)p1 + tp0 + x

is a symmetry of (5.71) and compute {¦, ω} for ω of the form (5.74). To do

this, we shall need another lemma.

Lemma 5.21. For any φ ∈ F(E) the identity

—¦ LU = LU —¦

φ φ

holds, where U = UE .

Proof of Lemma 5.21. In fact,

= LU —¦ ’ —¦ LU .

0 = ‚C ( φ) = [LU , φ] φ φ

Consider the form ω = φs ωs , φs ∈ F(E). Then we have

{¦, φs ωs } = s s

)ωs + φs

ωs ) ’ ’

¦ (φ (φs ωs ) (¦) = ¦ (φ ¦ (ωs ) (φs ωs ) (¦).

But

s

¦ (ωs ) = ¦ LU (ps ) = LU ¦ (ps ) = LU Dx (¦)

= LU t2 ps+2 + (t2 p0 + tx)ps+1 + (s + 1)(t2 p1 + t)ps + „¦(s ’ 1).

4. FROM DEFORMATIONS TO RECURSION OPERATORS 217

On the other hand,

= t2 φs ωs+2 + 2t2 Dx (φs ) + (t2 p0 + tx)φs ωs+1

(φs ωs ) (¦)

+ t2 Dx (φs ) + (t2 p0 + tx)Dx (φs ) + (t2 p1 + t)φs ωs .

2

Thus, we ¬nally obtain

{¦, φs ωs } = {¦, φs }ωs + (s + 1)(t2 p1 + t)ωs

’ 2t2 Dx (φs )ωs+1 + „¦(s ’ 1). (5.76)

Applying (5.76) to (5.74), we get

ad¦ (ω) = {¦, ω} = (rt±r ’ t2 ±r )ωr + „¦(r ’ 1).

(1)

(5.77)

Let now ω ∈ Kr and suppose that ω has a nontrivial coe¬cient ±r of the

form (5.75) and ai is the ¬rst nontrivial coe¬cient in ±r . Then, by (5.77),

r’i

ad¦ (ω) = ±r ωr + „¦(r ’ 1) ∈ Kr ,

where ±r is a polynomial of the degree r ’ 1. This contradicts to (5.75) and

thus ¬nishes the proof.

4. From deformations to recursion operators

The last example of the previous section shows that our theory is not

complete so far. In fact, it is well known that the Burgers equation pos-

sesses a recursion operator. On the other hand, in Chapter 4 we identi¬ed

1,0

the elements of the group HC (E) with the algebra of recursion operators.

Consequently, the result of Theorem 5.19 contradicts to practical knowledge.

The reason is that almost all known recursion operators contain “nonlocal

’1

terms” like Dx . To introduce terms of such a type into our theory, we

need to combine it with the theory of coverings (Chapter 3), introducing

necessary nonlocal variables

Let us do this. Namely, let E be an equation and • : N ’ E ∞ be a

covering over its in¬nite prolongation. Then, due to Proposition 3.1 on

p. 102, the triad F(N ), C ∞ (M ), (π∞ —¦ •)— is an algebra with the ¬‚at

connection C • . Hence, we can apply the whole machinery of Chapter 4 to

this situation. To stress the fact that we are working over the covering •, we

shall add the symbol • to all notations introduced in this chapter. Denote

•

by UC the connection form of the connection C • (the structural element of

the covering •).

In particular, on N we have the C • -di¬erential

• •

‚C = [[UE , ·]]fn : Dv (Λi (N )) ’ D v (Λi+1 (N )),

0

whose 0-cohomology HC (E, •) coincides with the Lie algebra sym• E of non-

1,0

local •-symmetries, while the module HC (E, •) identi¬es with recursion

operators acting on these symmetries and is denoted by R(E, •). We also

have the horizontal and the Cartan di¬erential d• and d• on N and the

C

h

p Λp (N ) — Λq (N ).

i (N ) =

p+q=i C

splitting Λ h

218 5. DEFORMATIONS AND RECURSION OPERATORS

Choose a trivialization of the bundle • : N ’ E ∞ and nonlocal coordi-

nates w 1 , w2 , . . . in the ¬ber. Then any derivation X ∈ D v (Λi (N )) splits to

the sum X = XE + X v , where XE (wj ) = 0 and X v is a •-vertical derivation.

Lemma 5.22. Let • : E ∞ — RN ’ E ∞ , N ¤ ∞, be a covering. Then

p,0 • p,0

HC (E, •) = ker ‚C C p Λ(N ). Thus the module HC (E, •) consists of

derivations „¦ : F(N ) ’ C p Λ(N ) such that

v

• •

[[UE , „¦]]fn = 0, [[UE , „¦]]fn = 0. (5.78)

E

Proof. In fact, due to equality (4.55) on p. 179, any element lying in

•

the image of ‚C contains at least one horizontal component, i.e.,

•

‚C Dv (C p Λ(N )) ‚ Dv (C p Λ(N ) — Λ1 (N )).

h

Thus, equations (5.78) should hold.

We call the ¬rst equation in (5.78) the shadow equation while the second

one is called the relation equation. This is explained by the following result

(cf. Theorem 3.7).

Proposition 5.23. Let E be an evolution equation of the form

‚ku

ut = f (x, t, u, . . . , k )

‚u

and • : N = E ∞ — RN ’ E ∞ be a covering given by the vector ¬elds2

˜ ˜

Dx = Dx + X, Dt = Dt + T,

˜˜

where [Dx , Dt ] = 0 and

‚ ‚

Xs Ts

X= , T= ,

‚ws ‚ws

s s

p,0

w1 , . . . , ws , . . . being nonlocal variables in •. Then the group HC (E, •)

consists of elements

‚ ‚

ψ s s ∈ Dv (C p Λ(N ))

Ψi —

Ψ= +

‚ui ‚w

s

i

˜i

such that Ψi = Dx Ψ0 and

˜(p) (Ψ0 ) = 0, (5.79)

E

‚X s ˜ ± ‚X s β ˜

ψ = Dx (ψ s ),

D (Ψ0 ) + (5.80)

‚u± x β

‚w

± β

‚T s ˜ ± ‚T s β ˜

ψ = Dt (ψ s ),

D (Ψ0 ) + (5.81)

‚u± x β

‚w

± β

(p) (p)

s = 1, 2, . . . , where ˜E is the natural extension of the operator to N .

E

2

To simplify the notations of Chapter 4, we denote the lifting of a C-di¬erential

˜

operator ∆ to N by ∆.

4. FROM DEFORMATIONS TO RECURSION OPERATORS 219

Proof. Consider the Cartan forms

i

θ s = dws ’ X s dx ’ T s dt

ωi = dui ’ ui+1 dx ’ Dx (f ) dt,

on N . Then the derivation

‚ ‚

•

θs —

ωi —

UE = +

‚ws

‚ui s

i

is the structural element of the covering •. Then, using representation (4.40)

on p. 175, we obtain

‚

• ˜

‚C Ψ = dx § Ψi+1 ’ Dx (Ψi ) —

‚ui

i

i

‚(Dx f ) ‚

˜

+ dt § Ψ ± ’ Dt Ψ i —

‚u± ‚ui

±

i

s ‚X s β

‚X ‚

˜

ψ ’ Dx (ψ s ) —

+ dx § Ψ± +

‚wβ ‚ws

‚u±

s ± β

‚T s ‚T s β ‚

˜

ψ ’ Dt (ψ s ) —

+ dt § Ψ± + ,

‚ws

‚wβ

‚u±

s ± β

which gives the needed result.

˜i

Note that relations Ψi = Dx (Ψ0 ) together with equation (5.79) are

equivalent to the shadow equations. In the case p = 1, we call the solu-

tions of equation (5.79) the shadows of recursion operators in the covering

•. Equations (5.80) and (5.81) are exactly the relation equations on the

case under consideration.

1,0

Thus, any element of the group HC (E, •) is of the form

‚ ‚

˜i ψs —

Dx (ψ) —

Ψ= + , (5.82)

‚ws

‚ui s

i

where the forms ψ = Ψ0 , ψ s ∈ C 1 Λ(N ) satisfy the system of equations

(5.79)“(5.81).

As a direct consequence of the above said, we obtain the following

Corollary 5.24. Let Ψ be a derivation of the form (5.82) with ψ, ψ s ∈

C p Λ(N ). Then ψ is a solution of equation (5.79) in the covering • if and

•

only if ‚C (Ψ) is a •-vertical derivation.

We can now formulate the main result of this subsection.

Theorem 5.25. Let • : N ’ E ∞ be a covering, S ∈ sym• E be a •-

symmetry, and ψ ∈ C 1 Λ(N ) be a shadow of a recursion operator in the

covering •. Then ψ = iS ψ is a shadow of a symmetry in •, i.e., ˜E (ψ ) = 0.

220 5. DEFORMATIONS AND RECURSION OPERATORS

Proof. In fact, let Ψ be a derivation of the form (5.82). Then, due to

identity (4.54) on p. 179, one has

• • •

‚C (iS Ψ) = i‚ • S ’ iS (‚C Ψ) = ’iS (‚C Ψ),

C

•

since S is a symmetry. But, by Corollary 5.24, ‚C Ψ is a •-vertical derivation

• •

and consequently ‚C (iS Ψ) = ’iS (‚C Ψ) is •-vertical as well. Hence, iS Ψ is

a •-shadow by the same corollary.

Using the last result together with Theorem 3.11, we can describe the

process of generating a series of symmetries by shadows of recursion op-

erators. Namely, let ψ be a symmetry and ω ∈ C 1 Λ(N ) be a shadow of a

recursion operator in a covering • : N ’ E ∞ . In particular, ψ is a •-shadow.

•

Then, by Theorem 3.9, there exists a covering •ψ : Nψ ’ N ’ E ∞ where

’

ψ can be lifted to as a •ψ -symmetry. Obviously, ω still remains a shadow

in this new covering. Therefore, we can act by ω on ψ and obtain a shadow