<< . .

. 40
( : 58)



. . >>

[[U• , U• ]]fn = 0
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)

<< . .

. 40
( : 58)



. . >>