Canonical gauge equivalences of the sAKNS and sTB hierarchies

This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 140.113.38.11

This content was downloaded on 28/04/2014 at 11:36

Please note that terms and conditions apply.

Canonical gauge equivalences of the sAKNS and sTB hierarchies

View the table of contents for this issue, or go to the journal homepage for more 1998 J. Phys. A: Math. Gen. 31 6517

(http://iopscience.iop.org/0305-4470/31/30/016)

Canonical gauge equivalences of the sAKNS and sTB

hierarchies

Jiin-Chang Shaw†§ and Ming-Hsien Tu‡k

† Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan ‡ Department of Physics, National Tsing Hua University, Hsinchu, Taiwan

Received 5 January 1998, in final form 18 May 1998

Abstract. We study the gauge transformations between the supersymmetric AKNS (sAKNS) and supersymmetric two-boson (sTB) hierarchies. The Hamiltonian nature of these gauge transformations is investigated, which turns out to be canonical. We also obtain the Darboux–B¨acklund transformations for the sAKNS hierarchy from these gauge transformations.

1. Introduction

During the past ten years, the theory of the soliton [1–3] has played an important role in theoretical and mathematical physics, especially in the explorations of the relationship between integrable models and string theories [4]. On the one hand, several kinds of correlation functions in string theory are governed by the integrable hierarchy equations (e.g. Korteweg–de Vries (KdV), Kadomtsev–Petviashvili (KP) etc) [4]. On the other hand, the idea of the supersymmetric extensions of the integrable systems [5–7] has motivated people to use them to study the theory of superstrings [8].

Recently, several supersymmetric integrable systems have been proposed and studied (see, e.g., [9–17] and references therein). In this paper, we discuss only two of them; the supersymmetric Ablowitz–Kaup–Newell–Segur (sAKNS) hierarchy [13] and the supersymmetric two-boson (sTB) hierarchy [11]. The former was introduced from the study of the reduction scheme in the constrained KP hierarchy [18], and the latter was constructed from the supersymmetric extension of the dispersive long water wave equation [19, 20]. Both of them have supersymmetric Lax representations, being bi-Hamiltonian, and have infinite conserved quantities etc. Besides these properties, these two hierarchies can be related to each other via a gauge transformation [13]. Sometimes, such transformation from one hierarchy to the other is called Miura transformation. However, from our viewpoint, the connection between these two hierarchies has not been totally explored. The purpose of this work is to provide a deeper understanding about the gauge transformations between the sAKNS and the sTB hierarchies.

Our paper is organized as follows: in section 2, we recall the Lax formulation of the sAKNS hierarchy. We then discuss the gauge transformations between the sAKNS and the sTB hierarchies. Section 3 is devoted to the investigation of the canonical property of these gauge transformations from the bi-Hamiltonian viewpoint. Our approach

§ E-mail address: [email protected] k E-mail address: [email protected]

6518 J-C Shaw and M-H Tu

follows very closely that of [21, 22] for other systems. We then show, in section 4, that the Darboux–B¨acklund transformations (DBTs) for the sAKNS hierarchy itself can be constructed from these gauge transformations. Concluding remarks are presented in section 5.

2. sAKNS and sTB hierarchies

The sAKNS hierarchy [13] has the Lax operator of the form

L= ∂ + 8D−19 (2.1)

which satisfies the hierarchy equations

∂L ∂tn

= [Ln

+, L] (2.2)

where D = ∂θ + θ∂ is the supercovariant derivative defined on a (1|1) superspace [23]

with coordinates (x, θ ). D−1 = θ + ∂θ∂−1 is the formal inverse of D, which satisfies

D−1D= D−1D= 1. The multiplication rule for D acting on an arbitrary superfield U is DU = (DU) + (−1)|U|U D. Here, we refer to the parity of a superfield U to be even if |U| = 0 and odd if |U| = 1. The coefficients functions 8 and 9 are superfields with proper parity such that L is a bosonic operator. It can be proved that (2.2) is consistent with the following equations ∂8 ∂tn = (Ln +8) ∂9 ∂tn = −((Ln₎∗ +9) (2.3)

where the conjugate operation ‘∗’ is defined by (AB)∗ = (−1)|A||B|B∗A∗ for the super-pseudo-differential operators A, B and f∗ = f for the arbitrary superfield f . Therefore,

8and 9 are the eigenfunction and adjoint eigenfunction of the hierarchy, respectively. It can be shown [13] that the hierarchy equations (2.2) are invariant under the supersymmetric transformations: δ8 = (D†8) and δ9 = (D†9) where  is an odd constant and

D†≡ ∂θ− θ∂.

Since the Lax operator (2.1) is assumed to be homogeneous under Z2-grading, the

gradings of the (adjoint) eigenfunction should satisfy|8| + |9| = 1. There are two cases to be discussed:

(a)|8| = 0 and |9| = 1, (b)|8| = 1 and |9| = 0.

In the following, the sAKNS Lax operators for the case (a) and case (b) will be denoted by La= ∂ + 8aD−19a and Lb = ∂ + 8bD−19b, respectively, and thus|8a| = |9b| = 0

and|9a| = |8b| = 1. For both cases, (2.2) contains the ordinary AKNS hierarchy equations

in the bosonic limit.

Given a sAKNS hierarchy we can construct a non-standard Lax hierarchy via a gauge transformation. For case (a), let us perform the following transformation

Ma: La→ K = 8−1a La8a≡ ∂ − (DJ0)+ D−1J1 (2.4)

where both J0 and J1 are odd superfields which can be expressed in terms of 8aand 9aas

follows

J0 = −(D ln 8a) J1= 8a9a. (2.5)

The hierarchy equations then become

∂K ∂tn

= [Kn

which is the so-called sTB hierarchy [11]. It can be shown [11] that the hierarchy equations (2.6) are invariant under the supersymmetric transformations: δJ0 = (D†J0),

δJ1= (D†J1).

For case (b), we need another gauge transformation to do the job since|8b| = 1 in this

case. Let us consider the following transformation

Mb: Lb→ K = D−19bLb9b−1D≡ ∂ − (DJ0)+ D−1J1 (2.7)

which implies that

J0 = (D ln 9b) J1= 8b9b+ (D3ln 9b) (2.8)

and the Lax operator K still satisfies the hierarchy equations (2.6).

In fact, both gauge transformations Ma and Mb have their inverse transformations Na

and Nb, respectively. In other words, for a given sTB Lax operator K, one can perform the

following transformation to gauge away the constant term and to obtain the Lax operator

La [13] Na: K → La= e− Rx_(DJ 0)_Ke Rx_(DJ 0)≡ ∂ + 8 aD−19a (2.9) where 8a= e− Rx_(DJ 0) ₉ a= J1e Rx_(DJ 0)_{. (2.10)}

It can be proved that Lasatisfies (2.2) if K satisfies (2.6).

Similarly, for case (b), we have

Nb: K→ Lb= e− Rx_(DJ 0)_DKD−1_e Rx_(DJ 0)≡ ∂ + 8 bD−19b (2.11) where 8b= (J1− J0x)e− Rx_(DJ 0) ₉ b= e Rx_(DJ 0)_{. (2.12)}

We would like to mention that the parity of the gauge operator associated with the gauge transformation Ma is even, whereas for Mb is odd. Since Na(Nb) is the inverse of Ma

(Mb) and vice versa, thus we obtain the correspondences between the sAKNS and sTB

hierarchies.

3. Canonical property and Hamiltonian structures

The discussions presented in the previous section establish the gauge equivalences between the sAKNS and the sTB hierarchies at Lax formulation level. In this section, we would like to discuss the Hamiltonian nature of these gauge transformations. Let us start from the sTB hierarchy.

The Lax equation (2.6) of the sTB hierarchy has a bi-Hamiltonian description as follows

∂tn  J0 J1  = 21  δHn+1/δJ0 δHn+1/δJ1  = 22  δHn/δJ0 δHn/δJ1  (3.1) where the first structure 21 and the second structure 22 are given by [11]

21=  0 −D −D 0  (3.2) 22=  2D+ 2D−1J1D−1− D−1J0xD−1 −D3+ D(DJ0)− D−1J1D D3_{+ (DJ} 0)D+ DJ1D−1 J1D2+ D2J1  (3.3)

6520 J-C Shaw and M-H Tu

which have been investigated [11] and found to be compatible by using the prolongation method [24]. The Hamiltonians Hn are defined by

Hn= −1 n strK n_≡−1 n Z dx dθ sresKn (3.4)

where the super-residue (sres) picks up the coefficient of the D−1 term of a super-pseudo-differential operator.

Since the bi-Hamiltonian structure is one of the most important properties of an integrable system, it is quite natural to ask whether the gauge transformations discussed here are canonical or not. To see this, from the gauge transformation Na, we can obtain the

linearized map N_a0 and its transposed map Na0† as follows

N_a0=  −8aD−1 0 9aD−1 8−1a  N_a0†=  D−18a −D−19a 0 8−1_a  (3.5) where 8a and 9a are related to J0 and J1 via equation (2.5) (or equation (2.10)). A

straightforward calculation shows that

N_a021Na0† =  0 1 −1 0  ≡ Pa (3.6) N_a022Na0† =     −8aD−28aD− D8aD−28a D2+ D8aD−29a+ 8aD−2(D9a) −28aD−28a9aD−28a +28aD−28a9aD−29a D2_{+ 9} aD−28aD+ (D9a)D−28a −9aD−2(D9a)− (D9a)D−29a +29aD−28a9aD−28a −29aD−28a9aD−29a     ≡ Qa (3.7)

where Pa and Qa are just the first and the second Hamiltonian structures obtained in [14].

Moreover, it has been shown [14] that Pa and Qa are compatible through the method of

prolongation and describe the hierarchy equations (2.2) as follows

∂tn  8a 9a  = Pa  δHn₊₁/δ8a δHn+1/δ9a  = Qa  δHn/δ8a δHn/δ9a  (3.8) where the Hamiltonians Hn are defined by Hn = −(1/n)strLna. Hence, the gauge

transformation Na (or Ma) is a canonical map.

Next, let us turn to the gauge transformation Nb. From (2.11), the linearized map Nb0

and its transposed map N_b0† can be constructed as follows

N_b0=  −8bD−1− 9b−1∂ 9b−1 9bD−1 0  N_b0†=  ∂9_b−1+ D−18b −D−19b 9_b−1 0  (3.9) where 8b and 9b are related to J0 and J1 via (2.8) (or (2.12)). Using (3.9), we can obtain

two Poisson structures of the sAKNS hierarchy for the case (b). After some algebras, we have N_b021N_b0†=  0 1 −1 0  ≡ −Pb (3.10) N_b022Nb0†=     −8bD−2(D8b)− (D8b)D−28b D2+ 8bD−29bD+ (D8b)D−29b −28bD−28b9bD−28b +28bD−28b9bD−29b D2_{+ D9} bD−28b+ 9bD−2(D8b) −9bD−19bD− D9bD−29b +29bD−28b9bD−28b −29bD−28b9bD−29b     ≡ − Qb (3.11)

which imply that the hierarchy equations (2.2) for case (b) can be written as ∂tn  8b 9b  = Pb  δHn+1/δ8b δHn+1/δ9b  = Qb  δHn/δ8b δHn/δ9b  . (3.12)

Note that the parity of the gauge operator of the gauge transformation Nb is odd. Hence,

from (3.4), (2.11) and the identity strAB = (−1)|A||B|strBA, the Hamiltonians in (3.12) can be expressed in terms of Lb as (1/n)strLnb which are just the Hamiltonians of the

sAKNS hierarchy defined earlier with a minus sign. Therefore, the minus sign appearing in the front of Pb and Qb in (3.10) and (3.11) is used to compensate the sign from the

Hamiltonians. We follow the same line in [14] to investigate the Jacobi identity for Pb

and Qb by using the prolongation method. It turns out that Pb and Qb are compatible and

indeed define a bi-Hamiltonian structure of the associated hierarchy. Hence, just like Na,

the gauge transformation Nb is canonical as well.

To sum up, the canonical property of the gauge transformations between the sAKNS and sTB hierarchies can be summarized as follows

N_i021Ni0†= (−1)|Ni|Pi Ni022Ni0†= (−1)|Ni|Qi i= a, b. (3.13)

4. Darboux–B¨acklund transformations

Having constructed the canonical gauge transformations between the sAKNS and sTB hierarchies, now we would like to use these gauge transformations to derive the Darboux–B¨acklund transformations (DBTs) for the sAKNS hierarchy itself. Given a sAKNS Lax operator, say La, we can perform the gauge transformation Mafollowed by Nbto obtain

the Lax operator Lb as follows

La

Ma → K Nb

→ Lb. (4.1)

That is, using (2.4) and (2.11), we can define the gauge operator T (8a)= 8aD8−1a such

that

La→ Lb = T LaT−1≡ ∂ + 8bD−19b (4.2)

where the (adjoint) eigenfunctions are related by

8b= 8a(8a9a+ (D3ln 8a)) (4.3)

9b= 8−1a . (4.4)

Notice that although the gauge transformation (4.2) preserves the form of the Lax operator and the Lax formulations, the parity of the transformed (adjoint) eigenfunction has been changed due to the fact that the parity of the gauge operator T is odd. Thus, strictly speaking, the gauge transformation (4.2) is not a DBT but a ‘quasi-DBT’. On the other hand, we can construct another quasi-DBT from Lb to Laas follows

Lb

Mb → K Na

→ La (4.5)

which is triggered by the gauge operator S(9b)= 9b−1D−19b such that

Lb→ La= SLbS−1 ≡ ∂ + 8aD−19a. (4.6)

Here

8a= 9b−1 (4.7)

6522 J-C Shaw and M-H Tu

Note that both quasi-DBTs (4.2) and (4.6) are canonical since they are constructed out from the canonical transformations Mi and Ni. We also remark that the form of the gauge

operator T was first considered in [25] for studying the DBT for the Manin–Radul super KdV equation [5]. Motivated by the above discussions, we may have true DBTs by considering the hierarchy equations (2.2) associated with the Lax operator

L= ∂ + 81D−191+ 82D−192 (4.9)

with parity|81| = |92| = 0 and |91| = |82| = 1. Let us consider the DBT triggered by

the eigenfunction 81 as follows

L→ ˆL = T LT−1 T (81)≡ 81D8−11

≡ ∂ + ˆ81D−1ˆ91+ ˆ82D−1ˆ92 (4.10)

where the transformed (adjoint) eigenfunctions are given by ˆ81= 81(D8−11 82)= (T (81)82)

ˆ91 = 8−11 (D−18192)= (S(81)92)

ˆ82= 81(8191− 8292+ (D3ln 81)+ (D8−11 82)(D−18192))= (T (81)L81)

ˆ92 = 8−11 (4.11)

with parity| ˆ81| = | ˆ92| = 0 and | ˆ91| = | ˆ82| = 1. On the other hand, we can consider the

DBT triggered by the adjoint eigenfunction 92 as follows

L→ ˆL = SLS−1 S(92)≡ 92−1D−192≡ ∂ + ˆ81D−1ˆ91+ ˆ82D−1ˆ92 (4.12) where ˆ81= 92−1 ˆ91 = (8292− 8191+ (D3ln 92)+ (D−19281)(D919₂−1))92= −(T (92)L∗92) ˆ82= 92−1(D−19281)= (S(92)81) ˆ92 = 92(D92−191)= (T (92)91) (4.13)

with parity| ˆ81| = | ˆ92| = 0 and | ˆ91| = | ˆ82| = 1. Finally, we would like to mention that

the above scheme can be generalized to a class of supersymmetric hierarchies which have Lax operators of the form

L= ∂ +

n

X

i₌₁

(82i−1D−192i−1+ 82iD−192i) (n> 1) (4.14)

with parity|82i−1| = |92i| = 0 and |82i| = |92i−1| = 1. The gauge operators of the DBTs

then can be constructed from the even (adjoint) eigenfunctions as Ti = 82i−1D8−12i−1 or

Si = 92i−1D−192i which not only preserve the Lax formulations but also the parity content

of the (adjoint) eigenfunctions in the Lax operator.

5. Concluding remarks

We have established the gauge equivalences between the sAKNS and sTB hierarchies. We have also shown that the gauge transformations connecting these two hierarchies are canonical, in the sense that the bi-Hamiltonian structure of the sAKNS hierarchy is mapped to the bi-Hamiltonian structure of the sTB hierarchy according to equation (3.13). Using these gauge transformations, the (quasi) DBTs for the sAKNS hierarchy and its generalizations can be constructed, which turns out to be canonical as well. Some other topics such as iterated DBTs, soliton solutions and non-local conserved charges of these hierarchies are worth further investigation [26]. We leave this work to a future publication.

Acknowledgments

We would like to thank Professor W-J Huang for helpful discussions. This work was supported by the National Science Council of Taiwan under grant No NSC-87-2811-M-007-0025. MHT also wishes to thank the National Centre for Theoretical Sciences of the National Science Council of Taiwan for partial support.

References

[1] Faddeev L D and Takhtajan L A 1987 Hamiltonian Methods in the Theory of Solitons (Berlin: Springer) [2] Das A 1989 Integrable Models (Singapore: World Scientific)

[3] Dickey L A 1991 Soliton Equations and Hamiltonian Systems (Singapore: World Scientific) [4] Nissimov E and Pacheva S 1993 String theory and integrable systems Preprint hep-th/9310113

Marshakov A 1994 String theory and classical integrable systems Preprint hep-th/9404126 and references therein

[5] Manin Y I and Radul A O 1985 Commun. Math. Phys. 98 65

[6] Kupershmidt B A 1987 Elements of Super Integrable Systems (Dordrecht: Kluwer) [7] Mathieu P 1988 J. Math. Phys. 29 2499

Mathieu P 1988 Phys. Lett. 203B 287

[8] Alvarez-Gaum´e L and Man˜es J L 1991 Mod. Phys. Lett. A 6 2039

Alvarez-Gaum´e L, Itoyama H, Man˜es J L and Zadra A 1992 Int. J. Mod. Phys. A 7 5337 Becker M 1994 PhD Thesis Bonn University hep-th/9403129

Stanciu S 1994 PhD Thesis Bonn University hep-th/9403129

[9] Figueroa-O’Farrill J M, Mas J and Ramos E 1991 Rev. Math. Phys. 3 479 [10] Oevel W and Popowicz Z 1991 Commun. Math. Phys. 139 441

[11] Brunelli J C and Das A 1994 Phys. Lett. 337B 303 Brunelli J C and Das A 1995 Phys. Lett. 354B 307 Brunelli J C and Das A 1995 Int. J. Mod. Phys. A 10 4563 [12] Bonora L, Krivonos S and Sorin A 1996 Nucl. Phys. B 477 835 [13] Aratyn H and Rasinariu C 1997 Phys. Lett. 391B 99

[14] Aratyn H and Das A 1998 Mod. Phys. Lett. A 13 1185 [15] Delduc F and Gallot L 1997 Commun. Math. Phys. 190 395 [16] Popowicz Z 1996 J. Phys. A: Math. Gen. 29 1281

[17] Krivonos S, Sorin A and Toppan F 1995 Phys. Lett. 206A 146 [18] Cheng Y 1992 J. Math. Phys. 33 3774

Xu B and Li Y 1992 J. Phys. A: Math. Gen. 25 2957 Sidorenko J and Strampp W 1993 Commun. Math. Phys. 157 1

Oevel W and Strampp W 1993 Commun. Math. Phys. 157 51 and references therein [19] Kaup D J 1975 Prog. Theor. Phys. 54 396

[20] Broer L J F 1975 Appl. Sci. Res. 31 377

[21] Morosi C and Pizzocchero L 1994 J. Math. Phys. 35 2397 [22] Shaw J C and Tu M H 1998 J. Phys. A: Math. Gen. 31 4805

[23] Dewitt B 1992 Supermanifolds (Cambridge: Cambridge University Press)

[24] Olver P J 1986 Applications of Lie Group to Differential Equations (Graduate Texts in Mathematics 117) (New York: Springer)

[25] Liu Q P 1995 Lett. Math. Phys. 35 115 [26] Shaw J C and Tu M H, in preparation