is ful¬lled due to (6.39).

If now

∞ m

‚ ‚

j

θi — ρs —

˜= +

‚uj ‚ws

s

i

i=0 j=1

266 6. SUPER AND GRADED THEORIES

is an element of the module Λp (E• ) — Dv (E• ), then one can easily see that

∞ m

j ˜j

fn

dx § θi+1 ’ Dx (θi )

‚• (˜) = [[U• , ˜]] =

i=0 j=1

∞ m ij

± ‚Dx f ‚

˜j

+ dt § ’ Dt (θi ) —

θβ

‚u± ‚uj

β i

β=0 ±=1

∞ m

‚Xs ‚Xs ˜

±

dx § ’ Dx (ρs )

+ θβ + ργ

‚u± ‚wγ

β

s γ

β=0 ±=1

∞m

± ‚Xs ‚Ts ‚

˜

+ dt § ’ Dt (ρs ) —

θβ ± + ργ . (6.40)

‚uβ ‚wγ ‚ws

γ

β=0 ±=1

Again, con¬ning oneself to the case ˜ ∈ C p Λ(E• ) — Dv (E• ), one gets the

following

Theorem 6.27. Let E be an equation of the form (6.31) and • be its

covering with nonlocal variables w1 , w2 , . . . and extended total derivatives

p,0

given by (6.38). Then the module HC (E• ) consists of the elements

∞ m

‚ ‚

˜i

Dx (θj ) — j + ρs —

= , (6.41)

θ,ρ

‚ws

‚ui s

i=0 j=1

where θ = (θ 1 , . . . , θm ) and ρ = (ρ1 , . . . , ρs , . . . ), θj , ρs ∈ C p Λ(E• ), are

vector-valued forms satisfying the equations

˜(p) (θ) = 0, (6.42)

E

and

∞ m

‚Xs ‚Xs

˜β ˜

Dx (θ± ) ± + ρj = Dx (ρs ),

‚uβ ‚wj

β=0 ±=1 j

∞m

‚Ts ‚Ts

˜β ˜

Dx (θ± ) ± + ρj = Dt (ρs ), (6.43)

‚uβ ‚wj

β=0 ±=1 j

(p) (p)

s = 1, 2, . . . , where ˜E is the natural extension of with Dx and Dt

E

˜ ˜

replaced by Dx and Dt in (6.37).

Similar to Chapter 5, we call (6.42) shadow equations and (6.42) relation

equations for the element (θ, ρ); solutions of (6.42) are called shadow solu-

tions, or simply shadows. Our main concern lies in reconstruction elements

p,0

of the module HC (E• ) from their shadows. Denote the set of such shadows

p,0

by SHC (E• ).

Remark 6.9. Let • be a covering. Consider horizontal one-forms

s

ω• = dh ws = Xs dx + Ts dt, s = 1, 2, . . . ,

3. NONLOCAL THEORY AND THE CASE OF EVOLUTION EQUATIONS 267

where dh is the horizontal de Rham di¬erential associated to •. Then (6.42)

can be rewritten as

s

θ,ρ (ω• ) = d h ρs , s = 1, 2, . . . (6.44)

Remark 6.10. When Xs and Ts do not depend on nonlocal variables,

the conditions of • being a covering is equivalent to

s

dh ω• = 0, s = 1, 2, . . . ,

dh being the horizontal di¬erential on E. In particular, one-dimensional

coverings are identi¬ed with elements of ker(dh ). We say a one-dimensional

covering • to be trivial if corresponding form ω• is exact (for motivations see

Chapter 3). Thus, the set of classes of nontrivial one-dimensional coverings •

with ω• independent of nonlocal variables is identi¬ed with the cohomology

1

group Hh (E), or with the group of nontrivial conservation laws for E.

3.5. The functors K and T . Keeping in mind the problem of recon-

structing recursion operators from their shadows, we introduce two functors

in the category GDE(M ). One of them is known from the classical (non-

graded) theory (cf Chapter 3), the other is speci¬c to graded equations and

is a super counterpart of the Cartan even covering constructed in Chapter

3 (see also [97]).

1

Let (F, F ) be an object of the category GDE(M ) and Hh (F) be the

R-module of its ¬rst horizontal cohomology. Let {w± } be a set of generators

for Hh (F), each w± being the cohomology class of a form ω± ∈ Λ1 (F),

1

h

m i dx . We de¬ne the functor K : GDE(M ) ’ GDE(M ) of

ω± = i=1 X± i

1 (F) as follows.

killing Hh

The algebra KF is a graded commutative algebra freely generated by

i

{w± } over F with gr(w± ) = gr(X± ). The connection KF looks as

‚ ‚ ‚

i

= + X± .

F

KF

‚xi ‚xi ‚w±

±

1

From the fact that Hh is a covariant functor from GDE(M ) into the category

of R-modules it easily follows that K is a functor as well.

To de¬ne the functor T : GDE(M ) ’ GDE(M ), let us set T F = C — Λ(F),

where C — Λ— (F) = p≥0 C p Λ(F) is the module of all Cartan forms on F (see

Subsection 2.7). If F is n-graded, then T F carries an obvious structure of

(n + 1)-graded algebra. The action of vector ¬elds F (X), X ∈ D(M ), on

Λ— (F) by Lie derivatives preserves the submodule C — Λ(F). Since C — Λ(F),

as a graded algebra, is generated by the elements χ and dC ψ, χ, ψ ∈ F, this

action can be written down as

L χ= F (X)χ,

F (X)

L dC ψ = d C F (X)(ψ),

F (X)

(χ) · dC ψ + χL

L (χdC ψ) = L dC ψ.

F (X) F (X)

F

268 6. SUPER AND GRADED THEORIES

Moreover, for any X ∈ D(M ) and ω ∈ C — Λ(F) one has

F (X) ω = 0;

hence, for any θ ∈ C — Λ(F)

(θ § L )(ω)

F (X)

(ω) + (’1)θ1 dθ § (

=θ§L ω) = θ § L

F (X) , (ω),

F (X) F (X)

which means that we have a natural extension of the connection F in F

up to a connection T F in T F. It is easy to see that the correspondence

T : (F, F ) ’ (T F, T F ) is functorial. We call (T F, ∆T F ) the (odd ) Car-

tan covering of (F, F ).

In the case when (F, F ) is an evolution equation E of the form (6.31),

T (F, F ) is again an evolution equation T E with additional dependent vari-

ables v 1 , . . . , v m and additional relations

±

‚f 1 j

1

v = vi ,

t

‚uj

i,j i

(6.45)

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

m

‚f j

m

v =

t vi .

‚uj

i,j i

Note that if a variable uj is of grading (i1 , . . . , in ), then the grading of v j is

(i1 , . . . , in , 1).

3.6. Reconstructing shadows. Computerized computations on non-

local objects, such as symmetries and recursion operators, can be e¬ectively

realized for shadows of these objects (see examples below). Here we describe

a setting which guarantees the existence of symmetries and, in general, el-

p,0

ements of HC (E) corresponding to the shadows computed. Below we still

consider evolution equations only.

Proposition 6.28. Let E be an evolution equation and • be its covering.

p,0

Let θ ∈ SHC (E• ). Then, if the coe¬cients Xs and Ts for the extensions of

total derivatives do not depend on nonlocal variables for all s, then

(i) for any extension θ,ρ of θ up to a vector ¬eld on E• the forms

s def

„¦s

θ,ρ (ω• ) =

(see Remark 6.9 in Subsection 3.4) are dh -closed on E• ;

p,0

(ii) the element θ is extendable up to an element of HC (E• ) if and only

if all „¦s are dh -exact forms.

Proof. To prove the ¬rst statement, note that using Proposition

6.13 (i) one has

‚

2 s

+ d h ρs ) —

0 = ‚• ( θ,ρ ) = ‚• ( θ,ρ (ω• )

‚ωs

s

3. NONLOCAL THEORY AND THE CASE OF EVOLUTION EQUATIONS 269

‚

dh „¦ s —

=’ . (6.46)

‚ωs

s

The second statement immediately follows from (6.43).

Remark 6.11. If Xs , Ts depend on w1 , w2 , . . . , then (6.46) transforms

into

‚Xs ‚Ts ‚

dh „¦s ’ („¦s + dh ρs ) § —

dx + dt = 0. (6.47)

‚ws ‚ws ‚ws

s

p,0 q,0

Let now θ ∈ SHC (E• ) and ¦ ∈ HC (E• ). Then from Proposition

6.13 (iii) it follows that

[i¦ , ‚• ]θ = (’1)q (‚• ¦) θ = 0.

Hence, since by the de¬nition of shadows ‚• θ is a •-vertical element,

i¦ ‚• θ is vertical too. It means that ‚• i¦ θ is a •-vertical element, i.e.,

p+q’1

i• θ ∈ SHC (E• ). It proves the following result (cf. similar results of

Chapter 5):

p,0 q,0

Proposition 6.29. For any θ ∈ SHC (E• ) and ¦ ∈ HC (E• ) the ele-

p+q’1

ment ¦ θ lies in SHC (E• ). In particular, when applying a shadow of

a recursion operator to a symmetry, one gets a shadow of a symmetry.

The next result follows directly from the previous ones.

Theorem 6.30. Let E be an evolution equation of the form (6.31) and

E• be its covering constructed by in¬nite application of the functor K : E • =

K (∞) E, where

K (∞) E = inj lim(K n E), K n E = (K —¦ · · · —¦ K) E.

n’∞

n times

Then for any shadow R of a recursion operator in E• and a symmetry ¦ ∈

sym E• the shadow R(¦) can be extended up to a symmetry of E• . Thus, an

1,0

action of SHC (E• ) on sym(E• ) is de¬ned modulo “shadowless” symmetries.

1,0

To be sure that elements of SHC (E• ) can be extended up to recursion

operators in an appropriate setting, we prove the following two results.

Proposition 6.31. Let E be an equation and E• be its covering by means

of T E. Then there exists a natural embedding

—,0

Tsym : HC (E) ’ sym(T E)

of graded Lie algebras.

—,0

Proof. Let ¦ ∈ HC (E). Then L¦ acts on Λ— (E) and this action pre-

serves the submodule C — Λ(E) ‚ Λ— (E), since

[L¦ , dC ] = L[[¦,UE ]]fn = 0.

270 6. SUPER AND GRADED THEORIES

Let X ∈ CD(E). Then, due to (6.11), [[X, ¦]]fn UE = 0. But, using

(6.11) again, one can see that [[X, ¦]]fn is a vertical element. Hence,

[[X, ¦]]fn UE = [[X, ¦]]fn = 0.

—,0

Proposition 6.31 allows one to compute elements of HC (E) as nonlocal

symmetries in E• = T E. This is the base of computational technology used

in applications below.

The last result of this subsection follows from the previous ones.

Theorem 6.32. Let E be an evolution equation and E• be its covering

constructed by in¬nite application of the functor K —¦ T . Then any shadow

—,0 —,0

¦ ∈ SHC (E• ) can be extended up to an element of HC (E• ). In particular,

1,0

to any shadow SHC (E• ) a recursion operator corresponds in E• .

Remark 6.12. For “¬ne obstructions” to shadows reconstruction one

should use corresponding term of A.M. Vinogradov™s C-spectral sequence

([102], cf. [58]).

4. The Kupershmidt super KdV equation

As a ¬rst application of the graded calculus for symmetries of graded par-

tial di¬erential equations we discuss the symmetry structure of the so-called

Kupershmidt super KdV equation, which is an extension of the classical

KdV equation to the graded setting [24].

At this point we have already to make a remark. The equation under

consideration will be a super equation but not a supersymmetric equation

in the sense of Mathieu, Manin“Radul, where a supersymmetric equation

is an equation admitting and odd, or supersymmetry [74], [72]. The super

KdV equation is given as the following system of graded partial di¬erential

equations E for an even function u and an odd function • in J 3 (π; •), where

J 3 (π; •) is the space J 3 (π) for the bundle π : R — R2 ’ R2 , (u, x, t) ’ (x, t),

extended by the odd variable •:

ut = 6uux ’ uxxx + 3••xx ,

•t = 3ux • + 6u•x ’ 4•xxx , (6.48)

where subscripts denote partial derivatives with respect to x and t. As

usual, t is the time variable and x is the space variable. Here u, x, t, u, ux ,

ut , uxx , uxxx are even (commuting) variables, while •, •x , •xx , •xxx are

odd (anticommuting) variables. In the sequel we shall often use the term

“graded” instead of “super”.

We introduce the total derivative operators Dx and Dt on the space

∞ (π; •), by

J

‚ ‚ ‚ ‚ ‚

+ ··· ,

Dx = + ux + •x + uxx + •xx

‚x ‚u ‚• ‚ux ‚•x

4. THE KUPERSHMIDT SUPER KDV EQUATION 271

‚ ‚ ‚ ‚ ‚

+ ···

Dt = + ut + •t + utx + •tx (6.49)

‚t ‚u ‚• ‚ux ‚•x

The in¬nite prolongation E ∞ is the submanifold of J ∞ (π; •) de¬ned by

the graded system of partial di¬erential equations

nm

Dx Dt (ut ’ 6uux + uxxx ’ 3••xx ) = 0,

nm

Dx Dt (•t ’ 3ux • ’ 6u•x + 4•xxx ) = 0, (6.50)

where n, m ∈ N.

We choose internal coordinates on E ∞ as x, t, u, •, u1 , •1 , . . . , where we

introduced a further notation

ux = u 1 , •x = • 1 , uxx = u2 , •xx = •2 , . . . (6.51)

The restriction of the total derivative operators Dx and Dt to E ∞ , again

denoted by the same symbols, are then given by

‚ ‚ ‚

Dx = + (un+1 + •n+1 ),

‚x ‚un ‚•n

n≥0

‚ ‚ ‚

Dt = + ((un )t + (•n )t ). (6.52)

‚t ‚un ‚•n

n≥0

We note that (6.48) admits a scaling symmetry, which leads to the in-

troduction of a degree to each variable,

deg(x) = ’1, deg(t) = ’3,

deg(u) = 2, deg(u1 ) = 3, . . . ,

3 5

deg(•) = , deg(•1 ) = , . . . (6.53)

2 2

From this we see that each term in (6.48) is of degree 5 and 4 1 respectively.

2

4.1. Higher symmetries. We start the discussion of searching for

(higher) symmetries at the representation of vertical vector ¬elds,

‚ ‚ ‚ ‚

= ¦u + ¦• Dx (¦u )

n

+ Dx (¦• )

n

+ , (6.54)

¦

‚u ‚• ‚un ‚•n

n>0

where ¦ = (¦u , ¦• ) is the generating function of the vertical vector ¬eld ¦ .

We restrict our search for higher symmetries to even vector ¬elds, meaning

that ¦u is even, while ¦• is odd.

Moreover we restrict our search for higher symmetries to vector ¬elds

u •

¦ whose generating function ¦ = (¦ , ¦ ) depends on the variables x, t,

u, •, . . . , u5 , •5 . These requirements lead to a representation of the func-

tion ¦ = (¦u , ¦• ), ¦u , ¦• ∈ C ∞ (x, t, u, u1 , . . . , u5 ) — Λ(•, . . . , •5 ) in the

following form

¦u = f0 + f1 ••1 + f2 ••2 + f3 ••3 + f4 ••4 + f5 ••5 + f6 •1 •2

+ f7 •1 •3 + f8 •1 •4 + f9 •1 •5 + f10 •2 •3 + f11 •2 •4 + f12 •2 •5

+ f13 •3 •4 + f14 •3 •5 + f15 •4 •5 + f16 ••1 •2 •3 + f17 ••1 •2 •4

272 6. SUPER AND GRADED THEORIES

+ f18 ••1 •2 •5 + f19 ••1 •3 •4 + f20 ••1 •3 •5 + f21 ••1 •4 •5

+ f22 ••2 •3 •4 + f23 ••2 •3 •5 + f24 ••2 •4 •5 + f25 ••3 •4 •5

+ f26 •1 •2 •3 •4 + f27 •1 •2 •3 •5 + f28 •2 •3 •4 •5

+ f29 ••1 •2 •3 •4 •5 ,

¦ • = g 1 • + g 2 •1 + g 3 •2 + g 4 •3 + g 5 •4 + g 6 •5

+ g7 ••1 •2 + g8 ••1 •3 + g9 ••1 •4 + g10 ••1 •5 + g11 ••2 •3

+ g12 ••2 •4 + g13 ••2 •5 + g14 ••3 •4 + g15 ••3 •5 + g16 ••4 •5

+ g17 •1 •2 •3 + g18 •1 •2 •4 + g19 •1 •2 •5 + g20 •1 •3 •4 + g21 •1 •3 •5

+ g22 •1 •4 •5 + g23 •2 •3 •4 + g24 •2 •3 •5 + g25 •2 •4 •5 + g26 •3 •4 •5

+ g27 ••1 •2 •3 •4 + g28 ••1 •2 •3 •5 + g29 ••1 •2 •4 •5 + g30 ••1 •3 •4 •5

+ g31 ••2 •3 •4 •5 + g32 •1 •2 •3 •4 •5 , (6.55)

where f0 , . . . , f29 , g1 , . . . , g32 are functions depending on the even variables

x, t, u, u1 , . . . , u5 . We have to mention here that we are constructing

generic elements, even and odd explicitly, of the following exterior alge-

bra C ∞ (x, t, u, . . . , u5 ) — Λ(•, . . . , •5 ), where Λ(•, . . . , •5 ) is the (exterior)