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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

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