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

ψ1 of a new symmetry on Nψ . By Theorem 3.11, there exists a covering,

where both ψ and ψ1 are realized as nonlocal symmetries. Thus we can

continue the procedure applying ω to ψ1 and eventually arrive to a covering

in which the whole series {ψk } is realized.

Thus, we can state that classical recursion operators are nonlocal de-

formations of the equation structure. Algorithmically, computation of such

deformations ¬ts the following scheme:

(1)

1. Take an equation E and solve the linear equation E ω = 0, where ω

is an arbitrary Cartan form.

2. If solutions are trivial, take a covering • : N ’ E ∞ and try to ¬nd

shadows of recursion operators. Usually, such a covering is given by

conservation laws of the equation E.

3. If necessary, add another nonlocal variable (perhaps, de¬ned by a

nonlocal conservation law), etc.

4. If you succeeded to ¬nd a nontrivial solution „¦, then the correspond-

ing recursion operator acts by the rule R„¦ : ψ ’ ψ „¦, where ψ is

the generating function of a symmetry.

In the examples below, we shall see how this algorithm works.

Remark 5.12. Let us establish relation between recursion operators in-

troduced in this chapter with their interpretation as B¨cklund transforma-

a

tions given in Section 8 of Chapter 3.

Let „¦ be a shadow of a recursion operator in come covering • : N ’ E ∞ .

Then we can consider the following commutative diagram:

V

„N

N← VN

V„¦