<< . .

. 41
( : 58)



. . >>

algebra generated by •, . . . , •5 The symmetry condition (6.37) for p = 0
reads in this case to the system
Dt (¦u ) = ’ u3 + 3••2 ),
¦ (6uu1
Dt (¦• ) = + 6u•1 ’ 4•3 ),
¦ (3u1 • (6.56)
which results in equations
Dt (¦u ) ’ 6¦u u1 ’ 6uDx (¦u ) + Dx (¦u ) ’ 3¦• •2 ’ 3•Dx (¦• ) = 0,
3 2

Dt (¦• ) ’ 3Dx (¦u )• ’ 3u1 ¦• ’ 6¦u •1 ’ 6uDx (¦• ) + 4Dx (¦• ) = 0.
3

(6.57)
Substitution of the representation(6.55) of ¦ = (¦u , ¦• )), leads to an
overdetermined system of classical partial di¬erential equations for the co-
e¬cients f0 , . . . , f26 , g1 , . . . , g32 , which are, as mentioned above, functions
depending on the variables x, t, u, u1 , . . . , u5 .
The general solution of equations (6.57) and (6.55) is generated by the
functions
¦1 = (u1 , •1 );
¦2 = (6uu1 ’ u3 + 3••2 , 3u1 • + 6u•1 ’ 4•3 );
¦3 = (6tu1 + 1, 6t•1 );
¦4 = (3t(6uu1 ’ u3 + 3••2 ) + x(u1 ) + 2u,
3
3t(3u1 • + 6u•1 ’ 4•3 ) + x•1 + •);
2
¦5 = (u5 ’ 10u3 u ’ 20u2 u1 + 30u1 u2 ’ 15••4 ’ 10•1 •3
+ 30u1 ••1 + 30u••2 ,
4. THE KUPERSHMIDT SUPER KDV EQUATION 273

16•5 ’ 40u•3 ’ 60u1 •2 ’ 50u2 •1 + 30u2 •1 + 30u1 u• ’ 15u3 •).
(6.58)
We note that the vector ¬elds ¦1 , ¦2 , ¦3 , ¦4 are equivalent to the
classical symmetries

S1 = ,
‚x

S2 = ,
‚t
‚ 1‚

S3 = t ,
‚x 6 ‚u
‚ ‚ ‚ 3‚
S4 = ’x ’ 3t + 2u +• . (6.59)
‚x ‚t ‚u 2 ‚•
In (6.59) S1 , S2 re¬‚ect space and time translation, S3 re¬‚ects Galilean
invariance, while S4 re¬‚ects the scaling as mentioned already. In (6.50), the
evolutionary vector ¬eld ¦5 is the ¬rst higher symmetry of the super KdV
equation and reduces to

(u5 ’ 10u3 u ’ 20u2 u1 + 30u1 u2 ) + ..., (6.60)
‚u
in the absence of odd variables •, •1 , . . . , being then just the classical ¬rst
higher symmetry of the KdV equation
ut = 6uu1 ’ u3 . (6.61)
4.2. A nonlocal symmetry. In this subsection we demonstrate the
existence and construction of nonlocal higher symmetries for the super KdV
equation (6.48). The construction runs exactly along the same lines as it is
for the classical equations.
So we start at the construction of conservation laws, conserved densities
and conserved quantities as discussed in Section 2. According to this con-
struction we arrive, amongst others, at the following two conservation laws,
i.e.,
Dt (u) = Dx (3u2 ’ u2 + 3••1 ),
Dt (u2 + 3••1 ) = Dx (4u3 + u2 ’ 2uu2 + 12u••1 + 8•1 •2 ’ 4••3 ), (6.62)
1
from which we obtain the nonlocal variables
x
p1 = u dx,
’∞
x
(u2 + 3••1 ) dx.
p3 = (6.63)
’∞
Now using these new nonlocal variables p1 , p3 , we de¬ne the augmented
system E of partial di¬erential equations for the variables u, p1 , p3 , •,
where u, p1 , p3 are even and • is odd,
ut = 6uux ’ uxxx + 3••xx ,
274 6. SUPER AND GRADED THEORIES

•t = 3ux • + 6u•x ’ 4•xxx ,
(p1 )x = u,
(p1 )t = 3u2 ’ u2 + 3••1 ,
(p3 )x = u2 + 3••1 ,
(p3 )t = 4u3 + u2 ’ 2uu2 + 12u••1 + 8•1 •2 ’ 4••3 . (6.64)
1

Internal coordinates for the in¬nite prolongation E ∞ of this augmented
system (6.64) are given as x, t, u, p1 , p3 , •, u1 , •1 , . . . . The total derivative
operators Dx and Dt on E ∞ are given by
‚ ‚
+ (u2 + 3••1 )
Dx = D x + u ,
‚p1 ‚p3

Dt = Dt + (3u2 ’ u2 + 3••1 )
‚p1

+ (4u3 + u2 ’ 2uu2 + 12u••1 + 8•1 •2 ’ 4••3 ) . (6.65)
1
‚p3
We are motivated by the result for the classical KdV equation (see Sec-
tion 5 of Chapter 3) and our search is for a nonlocal vector ¬eld ¦ of the
following form
¦ = C1 t¦4 + C2 x¦2 + C3 p1 ¦1 + p3 ¦— + ¦—— , (6.66)
whereas in (6.66) C1 , C2 , C3 are constants and ¦— = (¦—u , ¦—• ), ¦—— =
(¦——u , ¦——• ) are functions to be determined.
We now apply the symmetry condition resulting from the augmented
system (6.64), compare with (6.57)
Dt (¦u ) ’ 6¦u u1 ’ 6uDx (¦u ) + Dx (¦u ) ’ 3¦• •2 ’ 3•Dx (¦• ) = 0,
3 2

Dt (¦• ) ’ 3Dx (¦u )• ’ 3u1 ¦• ’ 6¦u •1 ’ 6uDx (¦• ) + 4Dx (¦• ) = 0.
3

(6.67)
Condition (6.67) leads to an overdetermined system of partial di¬erential
equations for the functions ¦—u , ¦—• , ¦——u , ¦——• , whose dependency on the
internal variables is induced by the scaling of the super KdV equation, which
means that we are in e¬ect searching for a vector ¬eld ¦ , where ¦u , ¦•
are of degree 4 and 3 1 respectively. Solving the overdetermined system of
2
equations leads to the following result.
The vector ¬eld ¦ with ¦ de¬ned by
3 1 1
¦ = ’ t¦5 ’ x¦2 ’ p1 ¦1 + ¦—— , (6.68)
4 4 2
where ¦5 , ¦2 , ¦1 are de¬ned by (6.58) and
3 7
¦—— = (u2 ’ 2u2 ’ ••1 , •2 ’ 3u•), (6.69)
2 2
is a nonlocal higher symmetry of the super KdV equation (6.48). In e¬ect,
the function ¦ is the shadow of the associated symmetry of (6.64).
5. THE KUPERSHMIDT SUPER MKDV EQUATION 275

The ‚/‚p1 - and ‚/‚p3 -components of the symmetry can be com-
¦
puted from the invariance of the equations (6.70),
(p1 )x = u,
(p3 )x = u2 + 3••1 , (6.70)
but considered in a once more augmented setting. The reader is referred to
the construction of nonlocal symmetries for the classical KdV equation for
the details of this calculation.
It would be possible to describe the recursion here, but we prefer to
postpone it to the chapter devoted to the deformations of the equation
structure (see Chapter 7), from which the recursion operator can be obtained
rather easily and straightforwardly.

5. The Kupershmidt super mKdV equation
As a second application of the graded calculus for symmetries of graded
partial di¬erential equations, we discuss the symmetry structure of the so-
called Kupershmidt super mKdV equation, which is an extension of the
classical mKdV equation to the graded setting [24].
The super mKdV equation is given as the following system of graded
partial di¬erential equations E for an even function v and an odd function
ψ on J 3 (π; ψ) (see the notation in the previous section),
3 3 3 3
vt = 6v 2 vx ’ vxxx + ψx ψxx + ψψxxx + vx ψψx + vψψxx ,
4 4 2 2
ψt = (6v 2 ’ 6vx )ψx + (6vvx ’ 3vxx )ψ ’ 4ψxxx , (6.71)
where subscripts denote partial derivatives with respect to x, t. Here t is the
time variable and x is the space variable, v, x, t, v, vx , vt , vxx , vxxx are even
(commuting) variables, while ψ, ψx , ψxx , ψxxx are odd (anticommuting)
variables.
We introduce the total derivative operators Dx , Dt on J ∞ (π; ψ) by
‚ ‚ ‚ ‚ ‚
+ ··· ,
Dx = + vx + ψx + vxx + ψxx
‚x ‚v ‚ψ ‚vx ‚ψx
‚ ‚ ‚ ‚ ‚
+ ···
Dt = + vt + ψt + vtx + ψtx (6.72)
‚t ‚v ‚ψ ‚vx ‚ψx
The in¬nite prolongation E ∞ is the submanifold of J ∞ (π; ψ) de¬ned by
the graded system of partial di¬erential equations
3 3 3 3
Dx Dt (vt ’ 6v 2 vx + vxxx ’ ψx ψxx ’ ψψxxx ’ vx ψψx ’ vψψxx ) = 0,
nm
4 4 2 2
Dx Dt (ψt ’ (6v 2 ’ 6vx )ψx ’ (6vvx ’ 3vxx )ψ + 4ψxxx ) = 0,
nm
(6.73)
where n, m ∈ N.
We choose internal coordinates on E ∞ as x, t, v, ψ, v1 , ψ1 , . . . , where
we use a notation
vx = v 1 , ψx = ψ 1 , vxx = v2 , ψxx = ψ2 , . . . (6.74)
276 6. SUPER AND GRADED THEORIES

The restriction of the total derivative operators Dx , Dt to E ∞ , again denoted
by the same symbols, are then given by
‚ ‚ ‚
Dx = + vn+1 + ψn+1 ,
‚x ‚vn ‚ψn
n≥0
‚ ‚ ‚
Dt = + (vn )t + (ψn )t . (6.75)
‚t ‚vn ‚ψn
n≥0

We note that (6.71) admits a scaling symmetry, which leads to the assigning
a degree to each variable,
deg(x) = ’1, deg(t) = ’3,
deg(v) = 1, deg(v1 ) = 2, . . . ,
1 3
deg(ψ) = , deg(ψ1 ) = , . . . (6.76)
2 2
From this we see that each term in (6.71) is of degree 4 and 3 1 respectively.
2

5.1. Higher symmetries. We start the discussion of searching for
(higher) symmetries at the representation of vertical vector ¬elds,
‚ ‚ ‚ ‚
= ¦v + ¦ψ Dx (¦v )
n
+ Dx (¦ψ )
n
+ , (6.77)
¦
‚v ‚ψ ‚vn ‚ψn
n>0

where ¦ = (¦v , ¦ψ ) is the generating function of the vertical vector ¬eld ¦ .
We restrict our search for higher symmetries to even vector ¬elds, meaning
that ¦v is even, while ¦ψ is odd. Moreover we restrict our search for higher
symmetries to vector ¬elds ¦ whose generating function ¦ = (¦v , ¦ψ )
depends on the variables x, t, v, ψ, . . . , v5 , ψ5 . The above mentioned re-
quirements lead to a representation of the function ¦ = (¦v , ¦ψ ) in the
following form
¦v = f0 + f1 ψψ1 + f2 ψψ2 + f3 ψψ3 + f4 ψψ4 + f5 ψψ5 + f6 ψ1 ψ2
+ f7 ψ1 ψ3 + f8 ψ1 ψ4 + f9 ψ1 ψ5 + f10 ψ2 ψ3 + f11 ψ2 ψ4 + f12 ψ2 ψ5
+ f13 ψ3 ψ4 + f14 ψ3 ψ5 + f15 ψ4 ψ5 + f16 ψψ1 ψ2 ψ3 + f17 ψψ1 ψ2 ψ4
+ f18 ψψ1 ψ2 ψ5 + f19 ψψ1 ψ3 ψ4 + f20 ψψ1 ψ3 ψ5 + f21 ψψ1 ψ4 ψ5
+ f22 ψψ2 ψ3 ψ4 + f23 ψψ2 ψ3 ψ5 + f24 ψψ2 ψ4 ψ5 + f25 ψψ3 ψ4 ψ5
+ f26 ψ1 ψ2 ψ3 ψ4 + f27 ψ1 ψ2 ψ3 ψ5 + f28 ψ2 ψ3 ψ4 ψ5
+ f29 ψψ1 ψ2 ψ3 ψ4 ψ5 ,

¦ψ = g 1 ψ + g 2 ψ1 + g 3 ψ2 + g 4 ψ3 + g 5 ψ4 + g 6 ψ5
+ g7 ψψ1 ψ2 + g8 ψψ1 ψ3 + g9 ψψ1 ψ4 + g10 ψψ1 ψ5 + g11 ψψ2 ψ3
+ g12 ψψ2 ψ4 + g13 ψψ2 ψ5 + g14 ψψ3 ψ4 + g15 ψψ3 ψ5 + g16 ψψ4 ψ5
+ g17 ψ1 ψ2 ψ3 + g18 ψ1 ψ2 ψ4 + g19 ψ1 ψ2 ψ5 + g20 ψ1 ψ3 ψ4 + g21 ψ1 ψ3 ψ5
+ g22 ψ1 ψ4 ψ5 + g23 ψ2 ψ3 ψ4 + g24 ψ2 ψ3 ψ5 + g25 ψ2 ψ4 ψ5 + g26 ψ3 ψ4 ψ5
5. THE KUPERSHMIDT SUPER MKDV EQUATION 277

+ g27 ψψ1 ψ2 ψ3 ψ4 + g28 ψψ1 ψ2 ψ3 ψ5 + g29 ψψ1 ψ2 ψ4 ψ5 + g30 ψψ1 ψ3 ψ4 ψ5
+ g31 ψψ2 ψ3 ψ4 ψ5 + g32 ψ1 ψ2 ψ3 ψ4 ψ5 , (6.78)
where f0 , . . . , f29 , g1 , . . . , g32 are functions depending on the even variables
x, t, v, v1 , . . . , v5 . We have to mention here that we are constructing generic
elements, even and odd explicitly, of the exterior algebra C ∞ (x, t, v, . . . , v5 )—
Λ(ψ, . . . , ψ5 ), where Λ(ψ, . . . , ψ5 ) is the exterior algebra generated by the
elements ψ, . . . , ψ5 . The symmetry condition (6.37) reads in this case
3 3 3 3
Dt (¦v ) = ¦ (6v 2 v1 ’ v3 + ψ1 ψ2 + ψψ3 + v1 ψψ1 + vψψ2 ),
4 4 2 2
ψ 2
Dt (¦ ) = ¦ ((6v ’ 6v1 )ψ1 + (6vv1 ’ 3v2 )ψ ’ 4ψ3 ), (6.79)
which results in equations
3
Dt (¦v ) ’ 12¦v vv1 ’ 6v 2 Dx (¦v ) + Dx (¦v ) ’ Dx (¦ψ )ψ2
3
4
3 3 3
’ ψ1 Dx (¦ψ ) ’ ¦ψ ψ3 ’ ψDx (¦ψ )
2 3
4 4 4
3 3 3
’ Dx (¦v )ψψ1 ’ v1 ¦ψ ψ1 ’ v1 ψDx (¦ψ )
2 2 2
3 3 3
’ ¦v ψψ2 ’ v¦ψ ψ2 ’ vψDx (¦ψ ) = 0,
2
2 2 2
Dt (¦ψ ) ’ (12v¦v ’ 6Dx (¦v ))ψ1 ’ (6v 2 ’ 6v1 )Dx (¦ψ )
’ (6¦v v1 + 6vDx (¦v ) ’ 3Dx (¦v ))ψ
2

’ (6vv1 ’ 3v2 )¦ψ + 4Dx (¦ψ ) = 0.
3
(6.80)
Substitution of the representation (6.78) for ¦ = (¦v , ¦ψ )), leads to
an overdetermined system of classical partial di¬erential equations for the
coe¬cients f0 , . . . , f26 , g1 , . . . , g32 which are as mentioned above functions
depending on the variables x, t, v, v1 , . . . , v5 .
The general solution of equations (6.80) and (6.78) is generated by the
functions
¦1 = (v1 , ψ1 ),
3 3 3 3
¦2 = (’v3 + 6v 2 v1 + v1 ψψ1 + vψψ2 + ψψ3 + ψ1 ψ2 ,
2 2 4 4
2
’ 4ψ3 + (6v ’ 6v1 )ψ1 + (6vv1 ’ 3v2 )ψ),
¦3 = ’2x¦1 ’ 6t¦2 + (’2v, ’ψ),
15 25 5
¦v = v5 ’ 10v3 v 2 ’ 40v2 v1 v ’ 10v1 + 30v1 v 4 ’
3
ψψ5 ’ ψ1 ψ4 ’ ψ2 ψ3
4
4 4 2
15 15 15
’ vψψ4 ’ 5vψ1 ψ3 + ( v 2 ’ 15v1 )ψψ3 + ( v 2 ’ 5v1 )ψ1 ψ2
2 2 2
15
+ (15v 3 + 15v1 v ’ 15v2 )ψψ2 + (45v1 v 2 ’ v3 )ψψ1 ,
2
¦ψ = 16ψ5 + (40v1 ’ 40v 2 )ψ3 + (60v2 ’ 120v1 v)ψ2
4
278 6. SUPER AND GRADED THEORIES

+ (50v3 ’ 100v2 v ’ 60v1 v 2 ’ 70v1 + 30v 4 )ψ1
2

+ (15v4 ’ 30v3 v ’ 30v2 v 2 ’ 60v2 v1 ’ 60v1 v + 60v1 v 3 )ψ.
2
(6.81)
We note that the vector ¬elds ¦1 , ¦2 , ¦3 are equivalent to the classical
symmetries

S1 = ,
‚x

S2 = ,
‚t
‚ ‚ ‚ ‚
+ 6t ’ 2v ’ψ
S4 = 2x . (6.82)
‚x ‚t ‚v ‚ψ
In (6.82), S1 , S2 re¬‚ect space and time translation, while S3 re¬‚ects the scal-
ing as mentioned already, (6.73). The ¬eld ¦4 is the ¬rst higher symmetry
of the super mKdV equation and reduces to the evolutionary vector ¬eld

(v5 ’ 10v3 v 2 ’ 40v2 v1 v ’ 10v1 + 30v1 v 4 )
3
+ ..., (6.83)
‚v
in the absence of odd variables ψ, ψ1 , . . . , being then just the classical ¬rst
higher symmetry of the mKdV equation.
vt = 6v 2 vx ’ vxxx . (6.84)
Remark 6.13. It should be noted that this section is just a copy of
the previous one concerning the Kupershmidt super KdV equation, except
for the speci¬c results! This demonstrates the algorithmic structure of the
symmetry computations.
5.2. A nonlocal symmetry. In this subsection we demonstrate the
existence and construction of nonlocal higher symmetries for the super
mKdV equation (6.71). The construction runs exactly along the same lines
as it is for the classical equations.
So we start at the construction of conservation laws, conserved densities
and conserved quantities as discussed in Section 2. According to this con-
struction, we arrive, amongst others, at the following two conservation laws,
i.e.,
3 3
Dt (v) = Dx (2v 3 ’ v2 + ψψ2 + vψψ1 ),
4 2
1
Dt (v 2 + ψψ1 ) = Dx (3v 4 ’ 2v2 v + v1 ’ ψψ3 + 2ψ1 ψ2
2
4
3 9
+ vψψ2 ’ 3v1 ψψ1 + v 2 ψψ1 ), (6.85)
2 2
from which we obtain the nonlocal variables
x
p0 = v dx,
’∞
x
1
(v 2 + ψψ1 ) dx.
p1 = (6.86)
4
’∞
5. THE KUPERSHMIDT SUPER MKDV EQUATION 279

Now using these new nonlocal variables p0 , p1 we de¬ne the augmented
system E of partial di¬erential equations for the variables v, p0 , p1 , ψ,
where v, p0 , p1 are even and ψ is odd,
3 3 3 3
vt = 6v 2 vx ’ vxxx + ψx ψxx + ψψxxx + vx ψψx + vψψxx ,
4 4 2 2
2
ψt = (6v ’ 6vx )ψx + (6vvx ’ 3vxx )ψ ’ 4ψxxx ,
(p0 )x = v,
3 3
(p0 )t = 2v 3 ’ v2 + ψψ2 + vψψ1 ,
4 2
1
(p1 )x = v 2 + ψψ1 ,
4
3 9
(p1 )t = 3v 4 ’ 2v2 v + v1 ’ ψψ3 + 2ψ1 ψ2 + vψψ2 ’ 3v1 ψψ1 + v 2 ψψ1 .
2
2 2
(6.87)

Internal coordinates for the in¬nite prolongation E ∞ of this augmented
system (6.87)) are given as x, t, v, p0 , p1 , ψ, v1 , ψ1 , . . . . The total derivative
operators Dx and Dt on E ∞ are given by
‚ 1 ‚
+ (v 2 + ψψ1 )
Dx = D x + v ,
‚p0 4 ‚p1
3 3 ‚
Dt = Dt + (2v 3 ’ v2 + ψψ2 + vψψ1 )
4 2 ‚p0
3 9 ‚
+ (3v 4 ’ 2v2 v + v1 ’ ψψ3 + 2ψ1 ψ2 + vψψ2 ’ 3v1 ψψ1 + v 2 ψψ1 )
2
.
2 2 ‚p1
(6.88)

<< . .

. 41
( : 58)



. . >>