In this chapter, we apply the algebraic formalism of Chapter 4 to the

speci¬c case of partial di¬erential equations. Namely, we consider a formally

integrable equation E ‚ J k (π), π : E ’ M , taking the associated triple

(C ∞ (M ), F(E), π∞ ) for the algebra with ¬‚at connection, where F(E) =

—

∞ ∞ ’ M is the

k Fk (E) is the algebra of smooth functions on E , π∞ : E

natural projection and the Cartan connection C plays the role of .

We compute the corresponding cohomology groups for the case E ∞ =

J ∞ (π) and deduce de¬ning equations for a general E. We also establish

relations between in¬nitesimal deformations of the equation structure and

recursion operators for symmetries and consider several illustrative exam-

ples.

We start with repeating some de¬nitions and proofs of the previous

chapter in the geometrical situation.

1. C-cohomologies of partial di¬erential equations

Here we introduce cohomological invariants of partial di¬erential equa-

tions based on the results of Sections 1, 2 of Chapter 4. We call these in-

variants C-cohomologies since they are determined by the Cartan connection

C on E ∞ . We follow the scheme from the classical paper by Nijenhuis and

Richardson [78], especially in interpretation of the cohomology in question.

Let ξ : P ’ M be a ¬ber bundle with a connection , which is considered

as a C ∞ (M )-homomorphism : D(M ) ’ D(P ) taking a ¬eld X ∈ D(M ) to

the ¬eld (X) = X ∈ D(P ) and satisfying the condition X (ξ — f ) = X(f )

for any f ∈ C ∞ (M ).

Let y ∈ P , ξ(y) = x ∈ M , and denote by Py = ξ ’1 (x) the ¬ber of

the projection ξ passing through y. Then determines a linear mapping

y : Tx (M ) ’ Ty (P ) such that ξ—,y ( y (v)) = v for any v ∈ Tx (M ). Thus

with any point y ∈ P a linear subspace y (Tx (M )) ‚ Ty (P ) is associated. It

determines a distribution D on P which is called the horizontal distribution

of the connection . If is ¬‚at, then D is integrable.

As it is well known (see, for example, [46, 47]), the connection form

U = U ∈ Λ1 (P ) — D(P ) can be de¬ned as follows. Let y ∈ P , Y ∈ D(P ),

Yy ∈ Ty (P ) and v = ξ—,y (Yy ). Then we set

U )y = Y y ’

(Y y (v). (5.1)

187

188 5. DEFORMATIONS AND RECURSION OPERATORS

In other words, the value of U at the vector Yy ∈ Ty (P ) is the projection of

Yy onto the tangent plane Ty (P ) along the horizontal plane1 passing through

y ∈ P.

If (x1 , . . . , xn ) are local coordinates in M and (y 1 , . . . , y s ) are coordinates

along the ¬ber of ξ (the case s = ∞ is included), we can de¬ne by the

following equalities

s

‚ ‚ ‚

j

= + = i. (5.2)

i

‚y j

‚xi ‚xi

j=1

Then U is of the form

s n

‚

j

j

dy ’ dxi —

U= , (5.3)

i

‚y j

j=1 i=1

From equality (4.40) on p. 175 it follows that

‚k ‚k ‚

j j

fn ±

dxi § dxj — k .

[[U , U ]] = 2 + (5.4)

i

‚y ±

‚xi ‚y

±

i,j,k

Recall that the curvature form R of the connection is de¬ned by the

equality

’ X, Y ∈ D(M ).

R (X, Y ) = [ X, Y] [X,Y ] ,

We shall express the element [[U , U ]]fn in terms of the form R now

(cf. Proposition 4.24). Let us consider a ¬eld X ∈ D(P ) and represent it in

the form

X = X v + X h, (5.5)

where, by de¬nition,

Xv = X Xh = X ’ Xv

U,

are vertical and horizontal components of X respectively. In the same

manner one can de¬ne vertical and horizontal components of any element

„¦ ∈ Λ— (P ) — D(P ).

Obviously, X v ∈ Dv (P ), where

Dv (P ) = {X ∈ D(P ) | Xξ — (f ) = 0, f ∈ C ∞ (M )},

while the component X h is of the form

fi ∈ C ∞ (P ), Xi ∈ D(M ),

Xh = fi Xi ,

i

and lies in the distribution D .

1

With respect to .

1. C-COHOMOLOGIES OF PARTIAL DIFFERENTIAL EQUATIONS 189

Proposition 5.1. Let : D(M ) ’ D(P ) be a connection in the ¬ber

bundle ξ : P ’ M . Then for any ξ-vertical vector ¬eld X v one has

[[U , U ]]fn = 0.

Xv

fi , gj ∈ C ∞ (P ), Xi , Yj ∈ D(M ),

If X h = i fi Xi , Y h = gj Yj ,

j

are horizontal vector ¬elds, then

[[U , U ]]fn = 2

Yh Xh fi gj R (Xi , Yj ), (5.6)

i,j

or, to be short,

[[U , U ]]fn = 2R .

Proof. Let X ∈ D(P ). Then from equality (4.45) on p. 175 it follows

that

[[U, U ]]fn = 2([[U, X]]fn U ]]fn ),

U ’ [[U, X

X

where U = U . Hence, if X = X v is a vertical ¬eld, then

[[U, U ]]fn = 2([[U, X v ]]fn U ]]fn ) = ’2([[U, X v ]]fn )h .

Xv U ’ [[U, X v

But the left-hand side of this equality is vertical (see (5.4)) and thus vanishes.

This proves the ¬rst part of the proposition.

Let now X = X h be a horizontal vector ¬eld. Then

[[U, U ]]fn = 2[[U, X h ]]fn U = 2([[U, X h ]]fn )v .

Xh

Hence, if Y h is another horizontal ¬eld, then, by (4.31) on p. 173, one has

[[U, U ]]fn ) = 2Y h ([[U, X h ]]fn [[U, X h ]]fn )

Yh (X h U ) = 2(Y h U.

But from (4.45) (see p. 175) it follows that

[[U, X h ]]fn = [[X h , Y h ]]fn

Yh U = [X h , Y h ] U.

Therefore,

[[U, U ]]fn = 2[X h , Y h ]

Yh Xh U = 2([X h , Y h ] ’ [X h , Y h ]h )

h

’[

=2 fi gi ([ Xi , Yj ] Xi , Yj ] )

i,j

But obviously, for any f ∈ C ∞ (M )

for X = i fi and Y = gj Yj .

Xi j

one has

[ Xi , Yj ](f ) = [Xi , Yj ](f )

and, consequently,

h

[ Xi , Yj ] = [Xi ,Yj ] ,

which ¬nishes the proof.

190 5. DEFORMATIONS AND RECURSION OPERATORS

From equality (5.6) and from the considerations in the end of Section 2

of Chapter 4 it follows that if the connection in question is ¬‚at, i.e. R = 0,

then the element U determines a complex

‚0

0 ’ D(P ) ’’ Λ1 (P ) — D(P ) ’ · · ·

‚i

’ Λ (P ) — D(P ) ’’ Λi+1 (P ) — D(P ) ’ · · · ,

i

(5.7)

where ‚ = ‚ i = [[U , · ]]fn .

Remark 5.1. Horizontal vector ¬elds X h are de¬ned by the condition

X h U = 0. Denote the module of such ¬elds by D h (P ):

Dh (P ) = {X ∈ D(P ) | X U = 0}.

Then, by setting ˜ = U = U in (4.31) on p. 173, one can see that

‚ („¦ U ) = ‚ („¦) U

for any „¦ ∈ Λ— (P ) — D(P ). Hence,

‚ (Λ— (P ) — D v (P )) ‚ Λ— (P ) — D v (P )

and

‚ (Λ— (P ) — D h (P )) ‚ Λ— (P ) — D h (P ).

Considering a direct sum decomposition

Λ— (P ) — D(P ) = Λ— (P ) — D v (P ) • Λ— (P ) — D h (P )

one can see that

‚ = ‚v • ‚h ,

where

‚ v = ‚ |Λ— (P )—Dv (P ) , ‚ h = ‚ |Λ— (P )—Dh (P ) .

To proceed further let us compute 0-cohomology of the complex (5.7).

From equality (4.31) on p. 173 it follows that for any two vector ¬elds

Y, Z ∈ D(P ) the equality

‚ 0 Y + [Z, Y ]

Z U = [Z U ,Y ]

holds. Thus Y ∈ ker(‚ 0 ) if and only if

[Z, Y ] U = [Z U ,Y ]

for any Z ∈ D(P ). Using decomposition (5.5) for the ¬elds Y and Z and

substituting it into the last equation, we get that the condition Y ∈ ker(‚ 0 )

is equivalent to the system of equations

[Z v , Y h ] U = [Z v , Y h ], [Z h , Y v ] U = 0. (5.8)

1. C-COHOMOLOGIES OF PARTIAL DIFFERENTIAL EQUATIONS 191

Let Y h = i fi (see above). Then from the ¬rst equality of (5.8) it

Xi

follows that

Z v (fi ) v

= fi [ Xi , Z ].

Xi

i i

But the left-hand side of this equation is a horizontal vector ¬eld while the

right-hand side is always vertical. Hence,

Z v (fi ) =0

Xi

i

for any vertical ¬eld Z v . Choosing locally independent vector ¬elds Xi , we

see that the functions fi actually lie in C ∞ (M ) (or, strictly speaking, in

ξ — (C ∞ (M )) ‚ C ∞ (P )). It means that, at least locally, Y h is of the form

Yh = X ∈ D(M ).

X,

But since X = X if and only if X = X , the ¬eld X is well de¬ned on

the whole manifold M .

On the other hand, from the second equality of (5.8) we see that Y v ∈

ker(‚ 0 ) if and only if the commutator [Z h , Y v ] is a horizontal ¬eld for any

horizontal Z h . Thus we get the following result:

Proposition 5.2. A direct sum decomposition

ker(‚ 0 ) = Dv (P ) • (D(M ))

: D(M ) ’ D(P )

(D(M )) is the image of the mapping

takes place, where

and

Dv (P ) = {Y ∈ D v (P ) | [Y, D h (P )] ‚ D h (P )}.

One can see now that D v (P ) consists of nontrivial in¬nitesimal sym-

metries of the distribution D while the elements of (D(M )) are trivial

symmetries (in the sense that the corresponding transformations slide inte-

gral manifolds of D along themselves). To skip this trivial part of ker(‚ 0 ),

note that

(i) U ∈ Λ1 (P ) — D v (P ),

and (see Remark 5.1)

(ii) ‚ i Λi (P ) — D v (P ) ‚ Λi+1 (P ) — D v (P ).

Thus we have a vertical complex

‚0

0 ’ D (P ) ’’ Λ1 (P ) — D v (P ) ’ · · ·

v

‚i

’ Λi (P ) — D v (P ) ’’ Λi+1 (P ) — D v (P ) ’ · · · ,

the i-th cohomology of which is denoted by H i (P ). From the above said it

follows that H 0 (P ) coincides with the Lie algebra of nontrivial in¬nitesimal

symmetries for the distribution D .

192 5. DEFORMATIONS AND RECURSION OPERATORS

Consider now an in¬nitely prolonged equation E ∞ ‚ J ∞ (π) and the

Cartan connection C = CE in the ¬ber bundle π∞ : E ∞ ’ M . The corre-

sponding connection form U , where = C, will be denoted by UE in this

case. Knowing the form UE , one can reconstruct the Cartan distribution on

E ∞ . Since this distribution contains all essential information about solutions

of E, one can state that UE determines the equation structure on E ∞ (see

De¬nition 2.4 in Chapter 2).

By rewriting the vertical complex de¬ned above in the case ξ = π∞ , we

get a complex

0

‚C

0 ’ D (E) ’ Λ1 (E) — D v (E) ’ · · ·

v

’

i

‚C

’ Λ (E) — D (E) ’ Λi+1 (E) — D v (E) ’ · · · ,

i v

’ (5.9)

where, for the sake of simplicity, Λi (E) stands for Λi (E ∞ ). The cohomologies

i

of (5.9) are denoted by HC (E) and are called C-cohomologies of the equation

E.

From the de¬nition of the Lie algebra sym(E) and from the previous

considerations we get the following

Theorem 5.3. For any formally integrable equation E one has the iso-

morphism

0

HC (E) = sym(E).

1

To obtain an interpretation of the group HC (E), consider the element

U = UE ∈ Λ1 (E) — D v (E) and its deformation U (µ), U (0) = U , where µ ∈ R

is a small parameter. It is natural to expect this deformation to satisfy the

following conditions:

(i)

U (µ) ∈ Λ1 (E) — D v (E) (verticality)

and

(ii)

[[U (µ), U (µ)]]fn = 0 (integrability). (5.10)

Let us expand U (µ) into a formal series in µ,

U (µ) = U0 + U1 µ + · · · + Ui µi + · · · , (5.11)

and substitute (5.11) into (i) and (ii). Then one can see that U1 ∈ Λ1 (E) —

Dv (E) and

[[U0 , U1 ]]fn = 0.

1 1

Since U0 = U (0) = U , it follows that U1 ∈ ker(‚E ). Thus ker(‚E ) con-

sists of all (vertical) in¬nitesimal deformations of U preserving the natural

conditions (i) and (ii).

1. C-COHOMOLOGIES OF PARTIAL DIFFERENTIAL EQUATIONS 193

0 0

On the other hand, im(‚E ) consists of elements of the form ‚E (X) =

[[U, X]]fn , X ∈ Dv (E). Such elements can be viewed as in¬nitesimal defor-

mations of U originating from transformations of E ∞ which are trivial on

M (i.e., ¬ber-wise transformations of the bundle π∞ : E ∞ ’ M ). In fact,

let P be a manifold and At : P ’ P , t ∈ R, A0 = id, be a one-parameter

group of di¬eomorphisms with

d

(At ) = X ∈ D(P ).

dt t=0

Then for any ˜ ∈ Λ— (P ) — D(P ) one can consider the element At,— (L˜ )

de¬ned by means of the commutative diagram

L˜

Λ— (P ) ’ Λ— (P )

A— A— (5.12)

t t

“ “

At,— (L˜ ) —

—

’ Λ (P )

Λ (P )

Then, obviously, for any homogeneous element ˜ = θ — Y ∈ Λ— (P ) — D(P )

and a form ω ∈ Λ— (P ) we have

d

At,— (L˜ )(ω)

dt t=0

d

A— (θ) § A— Y A— ω + (’1)θ dA— θ § A— (Y A— ω)

= ’t ’t

t t t t

dt t=0

= X(θ) § Y (ω) + θ § [X, Y ](ω) + (’1)θ dX(θ) § (Y ω)

+ (’1)θ dθ § [X, Y ] ω = L([[X,θ—Y ]]fn ) (ω).

— Yi ∈ Λ— (P ) — D(P ) and sets

Thus, if one takes ˜ = i θi

A— (˜) = ˜(t) = A— (θi ) — A— Yi A— , (5.13)

’t

t t t

i

then

˜(t) = ˜ + [[X, ˜]]fn t + o(t).

In other words, [[X, ˜]]fn is the velocity of the transformation of ˜ with re-

spect to At . Taking P = E ∞ and ˜ = U , one can see that the elements

V = [[U, X]]fn are in¬nitesimal transformations of U arising from transfor-

mations At : E ∞ ’ E ∞ . If π∞ —¦ At = π∞ , then X ∈ D v (E) and V ∈ im(‚E ).

0

It is natural to call such deformations of U trivial.

Since, as it was pointed out above, the element U determines the struc-

ture of di¬erential equation on the manifold E ∞ we obtain the following

result.

194 5. DEFORMATIONS AND RECURSION OPERATORS

1

Theorem 5.4. The elements of HC (E) are in one-to-one correspondence

with the classes of nontrivial in¬nitesimal vertical deformations of the equa-

tion E.

Remark 5.2. One can consider deformations of UE not preserving the

verticality condition. Then classes of the corresponding in¬nitesimal defor-

mations are identi¬ed with the elements of the ¬rst cohomology module of

the complex (5.7) (for P = E ∞ and = C). The theory of such deformations

is quite interesting but lies beyond the scope of the present book.

Remark 5.3. Since the operation [[·, ·]]fn de¬ned on HC (E ∞ ) takes its

1

2 2

values in HC (E) the elements of the module HC (E) (or a part of them at least)

can be interpreted as the obstructions for the deformations of E (cf. [78]).

Local coordinate expressions for the element UE and for the di¬erentials

‚C = ‚E in the case E ∞ = J ∞ (π) look as follows.

i

Let (x1 , . . . , xn , u1 , . . . , um ) be local coordinates in J 0 (π) and pj , j =

σ

1, . . . , m, |σ| ≥ 0, be the corresponding canonical coordinates in J ∞ (π).

Then from equality (1.35) on p. 26 and (5.3) it follows that

‚

j

ωσ —

U= , (5.14)

‚pj

σ

j,σ

j

where ωσ are the Cartan forms on J ∞ (π) given by (1.27) (see p. 18).

Consider an element ˜ = j,σ θσ — ‚/‚pj ∈ Λ— (π) — D v (π). Then, due

j

σ

to (5.14) and (4.40), p. 175, we have

n m

‚

j j

dxi § θσ+1i ’ Di (θσ ) —

‚π (˜) = , (5.15)

‚pj

σ

i=1 j=1 |σ|≥0