¯

sion symmetry X1 given by (6.204) is a Hamiltonian symmetry associated to

¯

a linear combination of P2 and P2 we hope to prove the formal construction

and the Lie superalgebra structure of the local and nonlocal symmetries and

the Poisson structure of the associated hierarchies of conserved quantities.

Remark 6.16. The contents of this section clearly indicates how to con-

struct supersymmetric extensions of classical integrable systems, which can

be termed completely integrable by the existence of in¬nite hierarchies of

local and/or nonlocal symmetries and conservation Laws.

CHAPTER 7

Deformations of supersymmetric equations

We shall illustrate the developed theory of deformations of supersym-

metric equations and systems through a number of examples.

First of all we shall continue the theory for the supersymmetric extension

of the KdV equation [35, 72, 74, 87] started in Section 6 of the previous

chapter. We shall construct the recursion operator for symmetries, which is

just realized by the contraction of a symmetry and the deformation. More-

over we construct a new hierarchy of conserved quantities and a hierarchy

of (x, t)-dependent symmetries.

As a second application, we consider the two supersymmetric extensions

of the nonlinear Schr¨dinger equation (Section 2) leading to the recursion

o

operators for symmetries and new hierarchies of odd and even symmetries.

We shall also construct a supersymmetric extension of the Boussinesq

equation, construct deformations for this system and eventually arrive at

the recursion operator for symmetries and at hierarchies of odd and even

symmetries and conservation laws.

Finally, we construct two-dimensional supersymmetric extensions (i.e.,

extensions including two odd dependent variables) of the KdV and study

their symmetries, conservation laws, and deformations, obtaining recursion

operators and hierarchies of symmetries.

1. Supersymmetric KdV equation

We start at the supersymmetric extension of the KdV equation [72, 74]

and restrict our considerations to the case a = 3 in the system

ut = ’u3 + 6uu1 ’ a••2 ,

•t = ’•3 + (6 ’ a)•1 u + a•u1 (7.1)

(see Section 6 of Chapter 6).

Features and properties of the equation were discussed in several papers,

cf. [35, 87].

1.1. Nonlocal variables. In order to construct a deformation of (7.1),

we have to construct an appropriate covering by the introduction of a num-

ber of nonlocal variables. These nonlocal variables, which arise classically

from conserved densities related to conservation laws, have been computed

to be

q 1 =D’1 (•),

2

309

310 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

q 3 =D’1 (p1 •),

2

12

q 5 =D’1 p • ’ u• (7.2)

21

2

and

p1 =D’1 (u),

p0 =D’1 (p1 ),

p1 =D’1 (•q 1 ),

2

’1 2

(u ’ ••1 ),

p3 =D

p3 =D’1 (u2 ’ 2u•q 1 + uq 1 q 3 ), (7.3)

2 2 2

where D = Dx .

Odd nonlocal variables will be denoted by q, while even nonlocal vari-

ables will be denoted by p and p. We mention that, in e¬ect, the total

derivative operator Dx should be lifted to an appropriate covering, where it

is denoted by the same symbol Dx , i.e.,

‚ ‚ ‚ ‚

Dx = + u1 + u2 + u3 + ...

‚x ‚u ‚u1 ‚u2

‚ ‚ ‚

+ (q 1 )x + (q 3 )x + (q 5 )x

‚q 1 ‚q 3 ‚q 5

2 2 2

2 2 2

‚ ‚ ‚

+ (p0 )x+ (p1 )x + (p3 )x

‚p0 ‚p1 ‚p3

‚ ‚

+ (p1 )x + (p3 )x . (7.4)

‚p1 ‚p3

Other odd nonlocal variables, q 7 and q 9 , are given by

2 2

12

q 7 = D’1 p3 • + q 1 p1 u ’ p 1 u 1 + u 2 ’ u 2 ,

2

2 2

q 9 = D’1 6p3 p1 • + q 1 p3 u ’ 3p2 u1 + 6p1 u2 ’ 6p1 u2

1 1

2 2

+ 36uu1 ’ 6u3 . (7.5)

Note that the variables q 3 , q 5 , q 7 , q 9 , p0 , p1 , p3 contain higher nonlocalities.

2 2 2 2

1.2. Symmetries. For hierarchies {Y 2n+1 }, {X2n+1 }, n ∈ N, of sym-

2

metries of equation (7.1) we refer to 4 of Chapter 6. Recall that

‚ ‚

Y 1 =•1 +u ,

‚u ‚•

2

‚ ‚

Y 3 =(2q 1 u1 ’ p1 •1 + u• ’ •2 ) + (2q 1 •1 ’ p1 u + u1 ) ,

‚u ‚•

2 2 2

‚ ‚

X1 =u1 + •1 ,

‚u ‚•

1. SUPERSYMMETRIC KDV EQUATION 311

‚ ‚

X3 = ’ u t ’ •t ,

‚u ‚•

X5 = ’ (u5 ’ 10u3 u ’ 20u2 u1 + 30u1 u2 + 5••4 + 5•1 •3

‚

’ 20u••2 ’ 20u1 ••1 )

‚u

‚

’ (•5 ’ 5u•3 ’ 10u1 •2 ’ 10u2 •1 + 10u2 •1 + 20u1 u• ’ 5u3 •) .

‚•

(7.6)

Moreover we found the supersymmetric analogue of the (x, t)-dependent

symmetry which acts as recursion on the even hierarchy {X2n+1 }, n ∈ N,

i.e.,

V2 = ’6tX5 ’ 2xX3 + H2 , (7.7)

where

H2 = ’ q 1 (•2 + p1 •1 ’ •u) + 3q 3 •1 ’ 13••1

2 2

‚

+ 4p1 u1 ’ 2p1 u1 ’ 8u2 + 16u2

‚u

+ ’ q 1 (p1 u ’ u1 ) + 3q 3 u

2 2

‚

+ 2p1 •1 ’ 2p1 •1 ’ 7•2 + 14•u . (7.8)

‚•

It should be noted that the vector ¬elds

‚ ‚ ‚

’ q1 + (p1 q 3 ’ 2q 5 )

Y’ 1 = ,

‚q 1 2 ‚p 2 ‚p

2 2

1 3

2

‚ ‚ ‚ ‚

X ’1 = + q1 + q3 +x ,

‚p1 2 ‚q 3 2 ‚q 5 ‚p0

2 2

‚

X’1 = = (7.9)

‚p1

are symmetries of equation (7.1) in the covering de¬ned by (7.2), (7.3).

These symmetries are vertical in the covering under consideration.

Computation of graded Lie brackets leads to the identities

[Y’ 1 , V2 ] =Y 3 ,

2 2

[X’1 , V2 ] =2Z1 + 4X1 ,

[X ’1 , V2 ] = ’ 2X1 , (7.10)

where Z1 is the nonlocal symmetry of degree 1 (cf. 4 of Chapter 6), which

acts, by its Lie bracket, as a recursion operator on the odd hierarchy {Y n+ 1 },

2

n ∈ N. Recall that

‚ ‚

+ (q 1 u ’ •1 )

Z1 = (q 1 •1 ) + ... (7.11)

‚u ‚•

2 2

312 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

1.3. Deformations. In order to construct a deformation of (7.1), we

formally construct the in¬nite-dimensional Cartan covering (see Subsection

3.5 of Chapter 6) over the in¬nite covering of (7.1) by (7.2), (7.3).

In the setting under consideration, the Cartan covering is described by

the Cartan forms ω0 , . . . , ωk , . . . on the in¬nite prolongation of the super-

symmetric KdV equation together with the forms corresponding to the non-

local variables (7.2), (7.3):

ωq 1 , ωq 3 , ωq 5 , ωp0 , ωp1 , ωp1 , ωp3 , ωp3 , (7.12)

2 2 2

where ωf = LU• (f ) denotes the Cartan form corresponding to the potential

f (see (2.13) on p. 66). According to (7.12), we search for a generalized

vector ¬eld which is linear with respect to the Cartan forms. Applying the

deformation condition on this vector ¬eld and taking into account the grad-

ing of (7.1), (7.2), (7.3), and (7.12), we arrive at the following deformation

U1 = ωu2 + ωu (’4u) + ω•1 (’2•) + ω• (•1 )

+ ωq 1 (q 1 u1 + p1 •1 + •2 ’ u•)

2

2

‚

+ ωp1 (’2u1 ) + ωp1 (u1 ) + ωq 3 (’•1 )

‚u

2

+ ω•2 + ω• (’2u) + ωu (’2•)

+ ωq 1 (’q 1 •1 + p1 u ’ u1 )

2

2

‚

+ ωp1 (’•1 ) + ωp1 (•1 ) + ωq 3 (’u) . (7.13)

‚•

2

Similar to the results of Subsection 2.8 of Chapter 6, the element U1 satis¬es

the identity

[[U1 , U1 ]]fn = 0, (7.14)

which means that U1 is a graded Nijenhuis operator in the sense [49].

We now rede¬ne our hierarchies in the following way. First we put

‚ ‚

Y 1 =•1 +u ,

‚u ‚•

2

‚ ‚

Y 3 =(2q 1 u1 ’ p1 •1 + u• ’ •2 ) + (2q 1 •1 ’ p1 u + u1 ) ,

‚u ‚•

2 2 2

‚ ‚

X1 =u1 + •1 ,

‚u ‚•

‚ ‚

+ (q 1 u ’ •1 )

X 1 =(q 1 •1 ) = Z1 ,

‚u ‚•

2 2

‚ 3 ‚

V0 =(2u + xu1 + 3tut ) + ( • + x•1 + 3t•t ) (7.15)

‚u 2 ‚•

1. SUPERSYMMETRIC KDV EQUATION 313

and de¬ne the odd and even hierarchies of symmetries by

n

Y2n+ 1 = ((. . . (Y 1 U1 ) U1 ) . . . ) U1 ) = Y 1 U1 ,

2 2 2

n times

n

Y2n+ 3 = Y 3 U1 ,

2 2

n

X2n+1 = X1 U1 ,

n

X 2n+1 = X 1 U1 ,

n

V2n = V0 U1 . (7.16)

1.4. Passing from deformations to “classical” recursion oper-

ators. Here we rewrite the main result of the previous subsection in more

conventional terms, i.e., as formal matrix integro-di¬erential operators. We

shall see that this representation is far less “economical” than representation

(7.13). Moreover, if one uses conventional left action of di¬erential opera-

tors, additional parasitic signs arise, which makes this representation even

more cumbersome.

Let X = (F,G) be a nonlocal symmetry of (7.1) in the covering de¬ned

by (7.2), (7.3) with 2-component generating function (F, G) and let |X| be

the degree of X; then one has |F | = |X| and |G| = |X| + 1.

It means that X is of the form

∞

‚ ‚

Di (F )) + Di (G)

X= , (7.17)

‚ui ‚•i

i=0

where F and G satisfy the shadow equation for the covering in question and

D denotes the extension of the total derivative Dx onto the covering. Then

one has

iX (ωui ) = Di (F ),

iX (ω•i ) = Di (G) (7.18)

for all i = 0, 1, . . . From the de¬nition of nonlocal variables (see (7.2) and

(7.3)) one also has

iX (ωp1 ) =D’1 (F ),

iX (ωq 1 ) =D’1 (G),

2

iX (ωp0 ) =D’1 (D’1 (F )),

iX (ωq 3 ) =D’1 (D’1 (F )• + Gp1 ),

2

iX (ωp1 ) =D’1 (Gq 1 ’ D’1 (G)•),

2

1

iX (ωq 5 ) =D’1 (D’1 (F )p1 • + Gp2 ’ F • ’ Gu),

21

2

iX (ωp3 ) =D’1 (2uF ’ G•1 + D(G)•), (7.19)

while

314 7. DEFORMATIONS OF SUPERSYMMETRIC EQUATIONS

iX (ωp3 ) = D’1 2F u ’ 2F •q 1 ’ 2Guq 1 + 2D ’1 (G)u•

2 2

+ F q 1 q 3 + D’1 (G)uq 3 ’ D’1 D’1 (F )• + Gp1 uq 1 (7.20)

2 2 2 2

(the last equality is given for reasons of completeness only and will not be

used below).

Then the recursion operator R corresponding to the deformation U 1 ,

(7.13) acts as

R(X) = iX (U1 ) (7.21)

and is of the form

R(F, G) = (F1 , G1 ), (7.22)

where

F1 = D2 (F ) + F (’4u) + D(G)(’2•) + G(•1 )

+ D’1 (G)(’q 1 u1 + p1 •1 + •2 ’ u•) + D ’1 (F )(’2u1 )

2

+ D’1 (Gq 1 ’ D’1 (G)•)u1 + D’1 (D’1 (F )• + Gp1 )(’•1 ),

2

G1 = D2 (G) + G(’2u) + F (’2•)

+ D’1 (G)(’q 1 •1 + p1 u ’ u1 ) + D’1 (F )(’•1 )

2

’1

(Gq 1 ’ D’1 (G)•)(•1 ) + D’1 (D’1 (F )• + Gp1 )(’u).

+D (7.23)

2

Due to the relations

D’1 (Gq 1 ) = D’1 (G)q 1 ’ D’1 (D’1 (G)•)

2 2

= ’(’1)|X| q 1 D’1 (G) + (’1)|X| D’1 (•D’1 (G)),

2

D’1 (Gp1 ) = p1 D’1 (G) ’ D ’1 (uD’1 (G)), (7.24)

we rewrite F1 , G1 in a left action notation as

F1 = D2 (F ) ’ 4uF + (’1)|X| 2•D(G) ’ (’1)|X| •1 G

’ (’1)|X| (’q 1 u1 + p1 •1 + •2 ’ u•)D ’1 (G) ’ 2u1 D’1 (F )

2

|X|

u1 q 1 D’1 (G) + (’1)|X| u1 D’1 (•D’1 (G))

’ (’1)

2

+ (’1)|X| u1 D’1 (•D’1 (G)) + •1 D’1 (•D’1 (F )

+ (’1)|X| •1 p1 D’1 (G) ’ (’1)|X| •1 D’1 (uD’1 (G))),

G1 = D2 (G) ’ 2uG ’ (’1)|X| 2•F

+ (’q 1 •1 + p1 u ’ u1 )D’1 (G) ’ (’1)|X| •1 D’1 (F )

2

+ (’1)|X| •1 ((’1)|X|+1 q 1 D’1 (G) ’ (’1)|X|+1 D’1 (•D’1 (G)))

2

’1 ’1

(G)) ’ (’1)|X| uD’1 (•D’1 (F ))

2|X|

+ (’1) •1 D (•D

’ up1 D’1 (G) + uD ’1 (uD’1 (G)). (7.25)

2. SUPERSYMMETRIC EXTENSIONS OF NLS 315

From this we ¬nally arrive at

F1 = D2 (F ) ’ 4uF ’ 2u1 D’1 (F ) + •1 D’1 (•D’1 (F ))

+ (’1)|X| 2•D(G) ’ (’1)|X| •1 G ’ (’1)|X| (•2 ’ u•)D ’1 (G)

+ (’1)|X| 2u1 D’1 (•D’1 (G))

’ (’1)|X| •1 D’1 (uD’1 (G)),

G1 = ’(’1)|X| 2•F ’ (’1)|X| •1 D’1 (F )

’ (’1)|X| uD’1 (•D’1 (F ))

+ D2 (G) ’ 2uG ’ u1 D’1 (G) + 2•1 D’1 (•D’1 (G))

+ uD’1 (uD’1 (G)), (7.26)

or

F1 = D2 (F ) ’ 4uF ’ 2u1 D’1 (F ) + •1 D’1 (•D’1 (F ))

+ (’1)|X| 2•D(G) ’ •1 G + (’•2 + u•)D ’1 (G)

+ 2u1 D’1 (•D’1 (G)) ’ •1 D’1 (uD’1 (G)) ,

G1 = (’1)|X| ’ 2•F ’ •1 D’1 (F ) ’ uD ’1 (•D’1 (F ))

D2 (G) ’ 2uG ’ u1 D’1 (G) + 2•1 D’1 (•D’1 (G)) + uD ’1 (uD’1 (G)),

(7.27)

leading to the recursion operator R = Rij , where

R11 = D2 ’ 4u ’ 2u1 D’1 + •1 D’1 •D’1 ,

R12 = (’1)|X| (2•D ’ •1 ’ •2 D’1 + u•D ’1 + 2u1 D’1 •D’1

’ •1 D’1 uD’1 ),

R21 = (’1)|X| (’2• ’ •1 D’1 ’ uD’1 •D’1 ),

R22 = D2 ’ 2u ’ u1 D’1 + 2•1 D’1 •D’1 + uD’1 uD’1 . (7.28)

Note that the classical recursion operator for the KdV equation is just the

•-independent part of R11 :

R0 = D2 ’ 4u ’ 2u1 D’1 . (7.29)

From the above representation it becomes clear that the action of the re-

cursion operator considered as action from the left, requires introduction of

the sign (’1)|X| , which makes the operation not natural. Therefore we shall

restrict ourselves to representations similar to (7.13).

2. Supersymmetric extensions of the NLS equation

In this section, we shall discuss deformations and recursion operators for

the two supersymmetric extensions of the nonlinear Schr¨dinger equation