Recursion Operators of Two Supersymmetric Equations

Commun. Theor. Phys. 55 2011) 199 203 Vol. 55, No. 2, February 15, 2011 Recursion Operators of Two Supersymmetric Equations LI Hong-Min Ó ), LI Biao ÓÂ), and LI Yu-Qi Ó ) Department of Mathematics, Ningbo University, Ningbo 315211, China Received May 7, 2010; revised manuscript received June 24, 2010) Abstract From La representations, recursion operators for the supersymmetric KdV and the supersymmetric Kaup Kupershimdt SKK) equations are proposed eplicitly. Under some special conditions, the recursion operator of the supersymmetric Sawada Kotera equation can be recovered by the one of the SKK equation. PACS numbers: 02.30.Ik, 11.30.Pb Key words: supersymmetry, La representation, recursion operator 1 Introduction Supersymmetric integrable systems constitute a very interesting subject and as a consequence a number of well known integrable equations have been generalized into supersymmetric contet. 1 2] The supersymmetric etension of a nonlinear evolution equation KdV for instance) refers to a system of coupled equations for a bosonic field u, t) and a fermionic field φ, t) which reduces to the initial equation in the limit where the fermionic field is zero bosonic limit). In the classical contet, a fermionic field is described by an anticommuting function with values in an infinitely generated Grassmann algebra. However, supersymmetry is not just a coupling of a bosonic field to a fermionic field. It also implies that a transformation supersymmetry invariance) relating these two fields leaves the system invariant. So far, many methods in standard theory have been etended to this framework, such as Bäcklund transformations, 3] prolongation theory, 4 5] hamiltonian formalism, 6] grasmmannian description, 7] τ functions, 8] Darbou transformations, 9] bilinear approach, 10] etc. In order to have a mathematical formulation of these concepts we will consider the space of differential operators on a 1 1) superspace with coordinates, θ). These operators are polynomials in the supercovariant derivative D = θ + θ whose coefficients are superfields. The supercovariant derivative obeys D 2 =, where θ is the Grassmann variable and θ 2 = 0. Recently, Tian and Liu obtained an N = 1 supersymmetric Kaup-Kupershimdt SKK) equation and by a simple Miura-type transformation, they derived a supersymmetric Sawada Kotera SSK) equation and proposed the conserved quantities and recursion operator of it from supersymmetric La representation. 11] In Ref. 12], Popowicz also gave the supersymmetric SK equation from the Bi-Hamiltonian formulation and gave it s odd hamiltonian structure. But to our knowledge, the recursion operator of the SKK equation by Tian and Liu has not been investigated. With regard to the construction of the recursion operator for a given integrable system, there are several works devoted to this subject through some different ways. 138] On the basis of these ideas, Gürses et al. established an etremely simple, effective, and algorithmic method for the construction of recursion operators for nonlinear partial differential equations when the La representation is given. 19] In this paper, we will etend the method in Ref. 19] to obtain the recursion operator of the SSK equation. Then from the recursion operator of SKK equation, we successfully recover the recursion operator of SSK equation in Ref. 11]. At the same time, we also obtain the recursion operator of supersymmetric KdV SKdV) equation by this method. For convenience, the paper is organized as follows. In Sec. 2, we will calculate the recursion operators of the SKdV and SKK equations from their La representations. In Sec. 3, we present conclusions and some interesting open problems. 2 Recursion Operators of MRSKdV and SKK Equations 2.1 MRSKdV Equation From Ref. 20], we know if set L = D 4 +ΦD, from this it follows that L 3/2 + = D6 +3/2)ΦD 3 +3/4)Φ D, we get the SKdV: Φ t = 1 4 Φ + 3 4 ΦDΦ)), 1) Supported by Zhejiang Provincial Natural Science Foundations of China under Grant No. Y6090592, National Natural Science Foundation of China under Grant Nos. 10735030 and 11041003, Ningbo Natural Science Foundation under Grant Nos. 2009B21003, 2010A610103 and 2009B21003, and K.C. Wong Magna Fund in Ningbo University Corresponding author, E-mail: biaolee2000@yahoo.com.cn c 2011 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/ej/journal/ctp http://ctp.itp.ac.cn

200 Communications in Theoretical Physics Vol. 55 which is the supersymmetric etension of the KdV equation found by Manin and Radul 21] by the same La representation. So Eq. 1) usually is also called MRSKdV equation. Furthermore, we find L = D L D, where D is a formal inverse to D, the adjoint operation is defind as D n ) = ) nn+1))/2 D n n Z), fg) = ) f g g f, for any f, g H, where H is the algebra of super pseudodifferentrial operators. According to Ref. 19], we can calculate the recursion operator through the following equality: L tn+4 = LL tn + R n, L], 2) where dφ/dt n = Φ tn = Φ n and so on, R n = LL) n/2) ) +. So we have R n = α + β)d + a + b. By equating the coefficients of powers of D in Eq. 2), we get α = 1 2 Φ n), β = 3 4 Φ n + 1 4 D Φ Φ n)), a = 1 2 Φ Φ n )), b = 0, 3) and the recursion operator for the MRSKdV equation: R = 4 2 + 2ΦD + 2DΦ) + Φ D + DΦ ) + DΦ)D Φ + 2Φ Φ ]. 4) Remark 1 It is necessary to point out that in Ref. 6], the recursion operator 4) had been obtained by the product of two hamiltonian operators. 20] 2.2 SKK Equation In Ref. 11], we know the SKK equation: u t + u + 5 uu + 3 4 u2 + 1 3 u3 + φ Du) + 1 2 φdu ) + 1 2 φφ 3 4 Dφ)2] = 0, φ t + φ + 5 uφ + 1 2 u φ + 1 2 u φ + u 2 φ + 1 2 φdφ ) 1 ] 2 Dφ)φ = 0 5) has the following La pair: L = 3 + u + φd + v, A = 9L 5/3 ) +, 6) where v = 1/2)u Dφ)). The L operator satisfies the reduction L = L, so we use the formula where L tn+6 = L 2 L tn + R n, L], 7) R n = α 5 + β 4 + γ 3 + δ 2 + ξ + η)d + a 5 + b 4 + c 3 + d 2 + e + f. 8) By equating the coefficients of the powers of D in Eq. 7), we obtain α = 0, β = 1 3 φ n ), γ = 5 3 φ n, δ = 1 9 29φ n + 4u φ n ) + uφ n ) + D φ φ n )) + 5φ u n ) + φu n )], ξ = 1 9 {26φ n + 14uφ n φd φ n ) + 4u + Dφ)] φ n) + 15φu n + 5φ u n), η = 1 27{ 28φn + 32uφ n + 2φDφ n ) + 212u Dφ)]φ n + 3D φ φ n ) + 2u 2 + 5u ) φ n) + 2u uφ n) 3φ Du) 2φ) φ n) + D 3uφ + 5φ ) φ n) + φ uφ n) + 2Du) + φ)d φ φ n))] + 30φu n + 25φ u n + 10φ u n) + D φ φu n)) 3φ φd u n )), a = 1 3 u n ), b = 1 6 11u n D φ n )], c = 1 18 {73u n + 10u u n) 2 φd u n )) 15Dφ n ) 2 Du) 2φ] φ n), d = 1 18 {81u n + 33uu n D φu n ) + 2φD u n ) + 53u Dφ)] u n ) 29Dφ n ) 4uD φ n ) D uφ n ) 7Du) φ n), e = 1 { 134un + 106uu n + 46φDu n ) + 92u 45Dφ)]u n + 3D φ u n ) 53Dφ ) 2u 2 7u ] u n ) 6u φd u n )) + u 2u 2 3Dφ ))u n 22φ + uφ)d u n ) + 2φD φd u n )) + 22φ Du)) φu n )] 78Dφ n ) 42uDφ n ) + φ 32Du)]φ n

No. 2 Communications in Theoretical Physics 201 9D Dφ)φ n ) 12u D φ n ) + 44φ 25Du )) φ n) + 7φ + Du ))φ n + 6Dφ )D φ n ) 2Du) 2φ) uφ n ) + 2φD Du) φ n )) 2Du)D φ φ n )] + 6u 2φ Du)) φ n )), f = 1 { 28un + 32uu n + 32φDu n ) + 68u 30Dφ)]u n + 38φ Du n ) + 63u + 4u 2 55Dφ )]u n + 8φ + uφ)du n ) + 37u + 18uu 35Dφ ) 12uDφ) + 13φDu)]u n + 2φ + 6uφ + φdφ) + 3u φ]d u n ) 2u D φu n ) 2φ D φd u n )) + 10φ Du) + 45u + 10u 2 + 10uu 10Dφ ) 10u Dφ) + 10φDu ) 10uDφ )] u n) + φ 3Du )] φu n) 3u Dφ )] φd u n )) 28Dφ n ) 32uDφ n ) 62Du) + 5φ]φ n + 49Dφ) 14u ]Dφ n ) 34Du ) + φ ]φ n 29u 39Dφ ) + 4u 2 ]Dφ n ) 78φ 9φu + 32Du ) + 12uDu)]φ n 5u 3Dφ ) Du)φ + 4uu ]D φ n ) 2u D uφ n ) 10Du ) 8uφ + 10u Du) 4φDφ) + 3u φ + 2φ + 7uDu )] φ n) + φ 3Du )] uφ n ) + 10φ 3Du )]D φ φ n )) 2φ D Du) φ n )) 3u Dφ )] Du) 2φ) φ n )). The recursion operator of SKK equation is found as R11 R 12 R = R 21 R 22 with R 11 = 1 + ), 9) { 2 6 12u 4 36u 3 49u + 18u 2 ) 2 35u + 60uu ) 118u 41uu 69 2 u2 8u 3) u u + 2u 2 ) 107u + 10uu + 25u u + 10u 2 u ) 6φD 3 21φ D 2 23φ + 6φu)D 6φDu) 11φ + 15uφ + 9φu + 6φDφ)]D + 198Dφ ) 12φDu ) 9φφ + 27 2 Dφ)2 + 21Du)φ ] 2φ + φ Dφ) φdφ ) + 5u φ + 6uφ + 3u φ + 2u 2 φ]d + 9Dφ )D φ + 3u D φ + 3φD Dφ ) + φd u + 2φD u 2 + 5φDu ) 5φφ 15φ Du ) 10φ Du) + 9Dφ)Dφ ) 10φuDu)] + Du ) + 2uDu) + 11φ 4uφ] φ 3u Dφ )) + 4φ + uφ)d φd + 2φD 2φ + uφ)d 2Du)D φ φ 4u φ + 2uφ)D + 2φD Du) φ φd φd ] + u 22φ Du) φ + 2φD φd ], R 12 = 1 { 60Du) 3 + 24Du ) 2 + 12uDu) + 19Du )] 60uu D + 7Du ) + 12uDu ) + 12u Du) + Du ) + 4uDu ) + 8u Du ) + 5u Du) + 2u 2 Du)] Du ) + 2uDu)] u + u Du ) 2u Du) u 215φ 3 18Dφ)D 2 + 57φ 2 45Dφ )D + 310φ 12φu) + 3 7Dφ ) + 12uDφ) 2φDu)]D + 280φ 57uφ + 27φu + 6Du)Dφ) 12φDφ)] + 4φDu ) + 9φφ + 3Dφ ) 7φ Du)

202 Communications in Theoretical Physics Vol. 55 R 21 = 1 + 6uDφ ) 6φDφ)]D + 16φ 32uφ + 31u φ + 4Dφ)Du ) 33φ Dφ) 2u φ 4u 2 φ + 4Du)Dφ ) 24φDφ )] 9u D Dφ) + 3Dφ )D u φd Du ) + 4Du) 9φ]D φ + 11φ + 4uφ) u + 7u φ + Du ) 18φ + 2uDu) 12uφ]D φ + 22φ uφ)d Du) + φd 12uφ 2Du)D 2Du) + φ]d φ φd Du)] 2Du)D φ u + 3uφ ) + 2Du) u 6Dφ )D ] + 6u Dφ )D + 2u + 2φD 2φD φ + 5Du)D φ 2φ Du)] u 24φDu), φ + 2u φd Du) Du)D φ] { 6φ 4 15φ 3 213φ + 3φu) 2 24φ 23uφ 27φu ) + 120φφ D + 12φ Dφ) + 29uφ + 92u φ 11φDφ ) 11φ + 45u φ] + 14φφ D + 15u φ 5Dφ )φ + 50u φ + 5φ Dφ) 2φ + 10uφ + 10u 2 φ + 45u φ ] φ 2uφ)D φ + 3φ D φ + 3u + Dφ ) 2uDφ)] φ + φ 2u 2 u 3Dφ )] + 2φ φd φ 2φ 2φ + uφ)d + 2Dφ)D φ φ + 2φ 2φ Du)] φ + 2Dφ) Dφ) φ, R 22 = 1 { 2 6 12u 4 24u 3 25u + 18u 2 ) 2 16u + 3uu ) 6u + 29uu + 45 2 u2 + 8u3) u + 6uu + 12u u + 8u 2 u ) u + 4uu ) u 6φD 3 + 6Dφ) 3 + φ D 2 Dφ ) 2 + 22φ 5uφ)D 22Dφ ) 3uDφ)] + 4φ 5uφ 7φu + 4φ φ + 6φDφ)]D 4Dφ ) + 3uDφ ) φdu ) 14φφ 15 2 Dφ)2 ] 3u Dφ) 82φ Du) + 3φu + φdφ ) + φ + 4u φ + 4uφ 2Dφ)φ + 2u 2 φ]d + Dφ ) 4uDφ ) 4u 7Dφ))Dφ ) + 3u 2u 2 )Dφ) + Du)φ + φ Du ) 2φDu ) + 14φφ ] + 4Dφ)D φ 15φ D Dφ) 2uφ + φ )D u] + Dφ ) + 2uDφ) 4φDu)] u + φ Du ) + 13φ ) + u + Dφ ) 4uu 2uDφ) 4φDu)]D φ + 2Dφ)D 2Du) + φ]d φ + 2Dφ)D φ u + 6Dφ)D uφ 2φ Du)D φ + 2φ φd Du) 2φ Du) 2φ] u. 10) Remark 2 In Eq. 9), if setting {u = DΦ), φ = Φ, the recursion operator of SSK equation 11] can be reproduced easily. Note that the classical recursion operator for the KK equation 19] is just the φ-independent part of R 11 : R 0 = 1 { 2 6 12u 4 36u 3 49u + 18u 2 ) 2 35u + 60uu ) 3 Conclusion + 118u 41uu 69 2 u2 8u 3) u u + 2u 2 ) 107u + 10uu + 25u u + 10u 2 u ). 11) In this work, the recursion operators of supersymmetic KdV and supersymmetric Kaup-Kupershimdt SKK)

No. 2 Communications in Theoretical Physics 203 equations are proposed eplicitly by their La representations. From the operator of SKK equation, the operator of supersymmetric Sawada-Kotera equation can be easily obtained by a simple transformation. However, when the order of recursion operator is high, it is a very hard and tedious work to compute the operator of supersymmetric equation by its La representations. Therefore, on the basis of symbolic computation system, it is a very useful work to establish a simple, effective, and algorithmic method for computing the recursion operator of supersymmetric equations. References 1] I.S. Krasil shchik and P.H.M. Kersten, Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations, Kluwer Acad. Publ., Dordrecht/Boston/London 2000). 2] Q.P. Liu and M. Manas, Supersymmetry and Integrable Systems, Springer, Berlin 1998); Q.P. Liu, Phys. Lett. A 235 1997) 335; Q.P. Liu, Commun. Theor. Phys. 25 1996) 505. 3] M. Chaichain and P. Kulish, Phys. Lett. B 78 1978) 413. 4] G.H. Roelofs and P.H.M. Kersten, J. Math. Phys. 29 1992) 2499. 5] X.B. Hu, J. Phys. A: Math. Gen. 30 1997) 619. 6] W. Oevel and Z. Popowicz, Commun. Math. Phys. 139 1991) 441. 7] K. Ueno and H. Yamada, Adv. Stud. in Pure Math. 16 1988) 373. 8] L.A. Ibort, L. Martinez Alonso, and E. Medina, J. Math. Phys. 37 1996) 6157. 9] Q.P. Liu and M. Manas, Phys. Lett. B 394 1997) 337. 10] Y.X. Yu, Commun. Theor. Phys. 49 2008) 686. 11] K. Tian and Q.P. Liu, Phys. Lett. A 373 2009) 1808. 12] Z. Popowicz, Phys. Lett. A 373 2009) 3318. 13] L.A. Dickey, Soliton Equations and Hamiltonian Systems, 2nd ed. World Scientific, Singapore 2003). 14] A.S. Fokas and R.L. Anderson, J. Math. Phys. 23 1982) 1066. 15] A.S. Fokas, Stud. Appl. Math. 77 1987) 253. 16] A.P. Fordy and J. Gibbons, J. Math. Phys. 22 1980) 1170. 17] P.M. Santini and A.S. Fokas, Commun. Math. Phys. 115 1988) 375. 18] A.S. Fokas and P.M. Santini, Commun. Math. Phys. 116 1988) 449. 19] M. Gürses, A. Karasu, and V.V. Sokolov, J. Math. Phys. 40 1999) 12. 20] J.M. Figueroa-OºFarrill, J. Mas, and E. Ramos, Rev. Math. Phys. 3 1991) 479; Integrability and Bihamiltonian Structure of the Even Order SKdV Hierarchies, Leuven Preprint KUL-TF-91/17 April 1991). 21] Yu.I. Manin and A.O. Radul, Commun. Math. Phys. 98 1985) 65.