Применение суперконформных преобразований в интегрируемых моделях Текст научной статьи по специальности «Математика»

UDC 530.1; 539.1

The Applications of the Superconformal Transformation in the Integrable Models

Z. Popowicz

institute of Theoretical Physics, University of Wroclaw, 50-205 Wroclaw, Poland E-mail: ziemekQift. uni. wroc.pl

We show that superconformal N = 2 transformation could be applied to the construction of the supersymmetric analogon of the reciprocal transformation. Due to it we present the connections between three different N = 2 supersymmetric

N=2

equation.

Keywords: superconformal transformation, Lax representation, supersymmetric Korteweg de Vries and Harry Dvm

equation.

1 Introduction

The reciprocal transformation, also known as hodo-graph transformation, plays an important role when we investigate relations among some nonlinear evolution equations fl] . For instance, the Harry Dym (HD) equation (or hierarchy), which is invariant under a kind of reciprocal transformation, is also reciprocally linked to the Korteweg-de Vries (KdV) equation (or hierarchy).

Very recently, the reciprocal transformation was generalized to N = 1 supersymmetric equations, where a general procedure to construct supersymmetric reciprocal transformation was presented [2] . It appeared that this transformation is connected with the super-conforaml transformation [3] .

As applications, one of the supersymmetric HD equations was shown to be reciprocally linked to the supersymmetric modified KdV equation. This super-symmetric modified KdV equation is connected via the Miura transformation with the supersymmetric KdV equation.

As in the classical case, the supersymmetric reciprocal transformation could be employed to explore integrable properties of supersymmetric equations, which was illustrated by constructing the recursion operators and bi-Hamiltonian structures of the supersymmetric Harry Dym equation.

Besides N = 1 supersymmetric generalizations, the integrable systems also admit N = 2 extended super-symmetric generalizations [4-10]. The idea could almost be traced back to the usage of the supersymmetry in the quantum field theory. As a striking feature, N=2 extended case distinguishes itself from N = 1 nonextended case by the possibility to supply new classical integrable systems. The N = 2 supersymmetric KdV equations were proposed more than twenty years ago and have been studied extensively since then. Various

results for these equations have been obtained, including Lax representations, bi-Hamiltonian structures, bilinear formalism, and so on.

Very recently three different supersymmetric N = 2 of the Harry Dym equation have been also proposed. In this paper we show how these supersymmetric equations are connected each other by teh superconformal transformation.

The paper is organised as follow. In the second section we briefly recapitulate basic facts on the n =1,2 superconformal transformation. In the third section we disccuss the properties of the classical Harry Dym equation and explain how this equation is connected with the KdV equation. Fourth section constains the

N=2

HD equations. In the fifth sections the connection of N=2 N=2

contains concluding remarks.

2 N=1,2 superconformal transformation

The conformal map means a function which preserves angles. It is a general definition of conformal transformation.

The supersymmetric analogon of the conformal map could be constructed using the superconformnal transformations. From the technical reasons we distinguish two cases of the supersymmetry: the N = land N=2

A.) N =1 superconformal transformation.

In order to construct the super covariant operators let us consider the super diflteomorphism N = 1 in which we change [3]

X = (x, 9) ^ Z = (z, y) = (z(x, 9, y(x, 9)) (1)

D = 93x + 3e D = y9z + 07

Due to it one finds

D = (Dy)D+[(Dz) - y(Dy)]dz

(2)

Hence we have

Di = K-1

Diöi)Di +

(13)

We assume that D should transforms covariantly. It means that

D2 — K ^ — (D20i)Di + (D2$i)B2^ where K — (Di^i)2 + (D2^i)2. In fact we obtain

D ^ D — (Dy)-1 D ^ (Dz) — y(Dy)

It is a superconformal transformation. Example: Super Hill operator

d3 + $(x) ^ D3 + ^(Dy)-2(d3 + <^)(Dy)

(3)

D1D2 = K-1

dx = K-1

1 + 2r - Dirö-1©, 1 + 2r - D¿rd-1©.

(14)

(4) where r — log(K)/2. Based on them we can prove

Assuming the infinitesimal form of this transformation as

0-1D1D2 = d-1D1l

(15)

z(x, 0) — x + e(x, 0),

y(x, 0) — 0 + A(x, 0)

we obtain that

(5)

$(Z) = $(X) + A(D$) + (E := e + 9A)$x + ....

Se$(X) = 2 (D5 + 3$d + (D$)D + 2$x) E(X) (6)

From the last equation we obtain the following representation of the N=1 Virasoro algebra

{$(Z), $(£)} = (D5+3$d+(D$)D+2$x)S(x-x)(0-9)

(7)

N=2

The super diflteomorphism between (x, 0i,02) and

ei,^2)-

y ^ x = x(y, ^1,^2), ft ^ 9j = 9j(y, ^1,^2), (8)

where i = 1,2. The superderivatives are

Dfc = , Dk = + 9kdx, (k = 1, 2) (9)

Computing Dk we have

The two last formulas we will use extensively in the next sections.

3 Harry Dym Equation

The Harry Dym equation usually is written in three different but equivalent forms

_ t 1 ^ _1 3

wt — ( /— )xxx,vt — . v vxxx

V w 4

where

v = -21/3^, w

,ut = ( ^)

(16)

(17)

This equation is integrable and possesses the following Bi-Hamiltonian structure

wt = J1 -

JH

1

Jw

J2-

JH

2

Jw

(18)

where

H_1 = 2 j dx H_2 = — j dx w 1/5 w2 (19)

J1 = dx

J2 = dw + wd

(20)

Dfc = ((Dfcx) — 0¿(Dfc0¿)Jdx + (Dk0¿)D.

(10)

where the summation on the repeated index i is assumed.

To ensure that the superderivatives transform covariantly i.e. Dk = (Dk9i)Di we have to assume the constraints

The Harry Dym equation could be obtained from the so called standard or nonstandard Lax representation in a similar manner as for the KdV equation. The Lax representation plays an important role in the theory of integrable systems, because a lot of informations on the properties of the equations follows from these representations. In order to construct such representations for HD equation let us consider first the standard and nonstandard Lax representation of the Korteweg de Vries equation

wt = wxxx + 6wwx

(21)

(Dfc x) = 0j(Dfc 0¿)

(11)

Moreover to ensure that D| — dx, (k — 1, 2) we have assume also

^102) = —(©201), (©202) = (©101)

(12)

The Standard representation of KdV equation is

Ls = w + dxx, Ls,t = LsL (22)

while the nonstandard representation

Lns = d d 1w, Lns,t = [(Lns)>1 ,Lns L (23)

1

d

y

1

x

xx

Now let us establish the reciprocal link between KdV ^ HD. This link could be formulated in three steps: first is to use a gauge transformation of the Lax operator, in the second step we apply the Miura transformation and in the last step we apply the reciprocal transformation

A.) Standard Lax operator:

I.) Gauge transformation

Ls = vLse f v = dxx — 2vdx + u — vx + v2

II.) Miura transformation

u = vx — v

III.) Reciprocal

y = w(x,t),t = t, dx = Wxdy

Ls = w2 dyy + (wx — 2vw)dy

where v = ^

2w

B.) Nonstandard Lax operator:

I.) Gauge transformation

Lns = ^ Lns^ = ^ dx ^ (dxx +

2^x ^ dx + ^xx^ + u)

II.) Analogon of Miura transformation

1

\fWd >— (dxx WxW dx)

\/0J

III.) Reciprocal Transformation Let us notice

^ Lns dT

B = dxxx----------wxw 1dx

21

Wxx — — W

Therefore

dy = wdx + (wxx — — wæw )dT, dT = dt

d dd d 3 2 _ -, d

dy = Wdy,dT = dt + (Wxx — 2WxW )dy

We obtain

Wt — W , Lns —

4 Supersymmetric N=2 extensions of KdV and HD equations

N=2

equation [3,10,11]

N=2

tensions of the KdV equation which could be written in compact for as

(a — 1)

(24)

(25)

(26) (27)

Ut = ( —Uxx + 3UD 1D2U +

-D1D2U + aU 3)x.

where a = 4, —2,1 and

Di

d

D2

(33)

(28)

The bosonic part in which we assume that U — w + 0102u gives us

wt — (— wxx + aw3 + (a + 2)uw)x (34)

— ( uxx + 3u (a 1)wx

(a + 2)wwxx + 3auw2)x

the system of interacted equations. From that reasons the applications of the extended supersymmetry to integrable systems could be considered as the method of the extensions of new integrable systems.

Let us now consider the Lax representations for these three supersymmetric extensions.

1.) a — 4

L4 — —(Di D2 + U )2, ¿4,t — 4[L3/

Ut — (-Uxx — 3(DiU )(D2U) +

6(D1D2U )U + 4U3)

r 3/2 J4,>0

L4]

(29)

2.)

2

L

2 = L4,>1

L-2,t = 4 [L

3/2

2,>0

L-2]

Ut = (—Uxx — 3(D1U )(D2U ) — 2U3)x

3.) a =1

L1 = d + d-1D1D2U, L1,t = [L,L3,>1]

Ut = ( — Uxx — 3U (D1D2U ) + U3)x

35

(30)

(31)

B.) Supersymmetric N=2 Harry Dym equation and its Lax operators a = 4, —2, 1 [5,12,13].

In a similar manner to supersymmetric KdV equa-

N=2

extensions of the Harry Dym equation. We enumera

in the supersymmetric KdV cases. This enumerating system will be explained in the next section. a=4

L4

t =

— (WD1D2)2, L4,t

1

[L3/2

[L4, 2

, L4]

(41)

(32)

ô(2wxxxW — 6(D1D2 Wx)(D1D2 w)w

—3(D1 Wxx)(D1 w) w2 — 3(D2 Wxx)(D2 w)w 2)

a

3

2.) a — —2 where w — e , that L-2 operator generates the super-

1 symmetric MKdV-2 equation via

-(Di w2D + D w2D )d_ ('42')

Wt = -(2w xxx w 3 — 3(D1D2 Wx)(D1D2 w)w 2

1,„ 3 „ w.. \ 2 L-2,t = [L-/22,>1 , L_2] (51)

8( ■ >

—3(D1 wxx)(D1w)w2 + 3(D2wxx)(D2w)w2 and produces the supersymmetnc KdV—2 equation af-

1 xx 1 2 xx 2

3*) a = 1 Miura transformation

L1 = d 1D1D2Vw D1D2 (43) 1 1

1 3 $=-(D1D2r ) + -(D2r )(D1r) (52)

wt = 4(4w Wxxx — 6(D1D2Wx)(D1D2w)w 2 4

—6(D1 Wxx)(D2W)w2 — 6(D1 Wxx)(D1W)w2 III.) a=l.

+3(D2w)(D1w)(D1D2wx)w) If we rewrite the Lax operator

L1 — VW d- 1D1D2V'wD1D2 (53)

in the new variables 13 we obtain

Supersymmetricl reciprocal link between supersymmetric HD and supersymmetric KdV

[2,13] L1 = v/wd-1D1 DM/1 D1D2 + (54)

I.)a=4. 1 Vt1L

Let us rewrite the Lax operator —(D2w)D1---------(D1w)D2 (55)

2w 2w J

L4 = —(WD1D2)2 (44)

in the new variables 13 we obtain L4 = — [D1D2 + (D2r )D1 — (D1r )D2]2 (45) L1 = L ^ = d-1D1

After application of the gauge transformation 1

D1D2 +

(56)

r = log(K)/2, K = w 1 (D1D2w) 3 (D2w)(D1w)

Applying the gauge transformation 2 w 4 w2

L = e-rL4er = — + (DDr) + (D2r)(Dr)] 2 After the identification e.g using the supersymmetrical

Miura transformation

(46)

1 (D1©2w) 3 (D2w)(D1w)

we obtained the Lax operator which gives us the su- $ = ----------------+ --------2----

persymmetrical MKdV4 equation via

: u : n3/2 : i A is a L^ operator for the supersymmetrical KdV1

Am = [(L4)>1, A] (47)

The obtained equation is connected with the supersymmetric KdV4 by the following supersymmetric Miura transformation

6 Conclusion

$ = (D1D2r) + (D2r)(D1r) In this paper we showed how one can adopt the su-

perconformal transformation to the supersymmetrical

II.) a=-2.

integrable models. We presented it on the connections

we rewrite t e ax operator between the supersymmetrical Harry Dym equations

L =1 /p w2p p w2p z^o-v and the supersymmetrical KdV equation. This link

-2 2' 1 1 2 2' x appeared very usefull because it simplifies the investi-

in the new variables 13 and using the gauge transfor- gations of the properties of the new integrable equa-

mation we obtain tions.

L—2 = e-r L4er = (49)

1

— — 2(D2 )(D1r )D1D2 +

1 The author would like to thanks prof. J. Buch-

-( — 2(D2ry) + (D2ry) — (D1r)(D1D2r^D2 + binder for the kind invitation to the conference. I

1 would like to thanks organizing committee for the nice

-( — 2(D1ry) + (D1ry) — (D2r)(D1D2r^D1 (50) scientific atmosphere during the conference.

References

[1] Calogero F. and Degasperis A. Spectral transformation and solitons. Amsterdam: Elsevier. 1982.

[2] Liu Q. P, Popowicz Z. and Tian K. 2010 JMP 51 093511.

[3] Mathieu P. Integrable and superintegrable systems ed. Kupershmidt B. A. 1990. Singapore: World Scientific. P. 352.

[4] Bourque S. and Mathieu P. JMP 2001. 42 3517-3539.

[5] Bruneili J. C., Das A. and Popowicz Z. JMP. 2003. 44 4756-4767.

[6] Kersten P. H. M. and Sorin A. S. Phys. Letts. A. 2002. 300 397-406.

[7] Labelle P. and Mathieu P. JMP. 1991. 32 923-927.

[8] Laberge C. A. and Mathieu P. Phys. Letts. B. 1988. 215 718-722.

[9] Mathieu P. JMP. 1988. 29 2499-2506.

[10] Oevel W. and Popowicz Z. Comm. Math. Phys.1991. 139 441-460.

[11] Popowicz Z. Phys. Letts. A. 1993. 174 411-415.

[12] Tian K. and Liu Q. P. JMP. 2011. 2-12 53 Issue: 5 , 053503. Preprint arXiv:1108.2110.

[13] Tian K., Popowicz Z„ Liu Q. J. Phys. A: Math.Theor. 2012. 45 122001.

Received 01.10.2012

3. Попович

ПРИМЕНЕНИЕ С УIIЕ Р К О Н Ф О Р М НЫХ ПРЕОБРАЗОВАНИЙ В ИНТЕГРИРУЕМЫХ

МОДЕЛЯХ

Показано, что N=2 суперконформные преобразования могут быть применимы для построения суперсимметричного аналога обратных преобразований. С их помощью устанавливается связь между тремя различными N=2 суперсимметричными расширениями уравнений Кортевега- де Фриза и тремя различными N = 2 суперсимметричными расширениями уравнений Гарри - Дыма.

Ключевые слова: суперконформные преобразования, представление Лакса, суперсимметричные уравнения Кортевега - де Фриза и Гарри - Дыма.

Попович 3., профессор.

Вроцлавский университет.

Институт теоретической физики, 50-205 Вроцлав, Польша.

E-mail: [email protected]