Integrable higher order deformations of Heisenberg

JOURNAL OF MATHEMATICAL PHYSICS 50, 113502 共2009兲

Integrable higher order deformations of Heisenberg supermagnetic model Jia-Feng Guo,1 Shi-Kun Wang,2,3,a兲 Ke Wu,1,3,b兲 Zhao-Wen Yan,1 and Wei-Zhong Zhao1,4,c兲 1

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China College of Mathematics and Information Science, Henan University, Kaifeng 475004, China 3 KLMM, AMSS, Chinese Academy of Sciences, Beijing 100080, China 4 Institute of Mathematics and Interdisciplinary Science, Capital Normal University, Beijing 100048, China 2

共Received 25 June 2009; accepted 18 September 2009; published online 3 November 2009兲

The Heisenberg supermagnet model is an integrable supersymmetric system and has a close relationship with the strong electron correlated Hubbard model. In this paper, we investigate the integrable higher order deformations of Heisenberg supermagnet models with two different constraints: 共i兲 S2 = 3S − 2I for S 苸 USPL共2 / 1兲 / S共U共2兲 ⫻ U共1兲兲 and 共ii兲 S2 = S for S 苸 USPL共2 / 1兲 / S共L共1 / 1兲 ⫻ U共1兲兲. In terms of the gauge transformation, their corresponding gauge equivalent counterparts are derived. © 2009 American Institute of Physics. 关doi:10.1063/1.3251299兴

I. INTRODUCTION

The concept of supersymmetry has attracted a lot of interest from physical and mathematical points of view. In the context of integrable systems, a number of important soliton systems, such as sine-Gorden equation,1 Korteweg-de Vries 共KdV兲 equation,2 Kadomtsev-Petviashvili 共KP兲 hierarchy,3 and nonlinear Schrödinger equation 共NLSE兲,4 have been embedded into their supersymmetric counterparts. Recently, more integrable equations, which have important applications in physics, have been extended to the supersymmetric framework, such as supersymmetric Camassa–Holm equation5 and supersymmetric nonlocal gas equation.6 The Heisenberg ferromagnet 共HF兲 model describes the motion of the magnetization vector of the isotropic ferromagnets. It is an integrable system and gauge equivalent to NLSE.7 Recently, Kazakov et al.8 found that it has a direct relationship with classical integrability in the anti-de Sitter/conformal field theories. The integrable deformations of HF model have attracted considerable interest in the past decades. Mikhailov and Shabat9 constructed the SO共3兲 invariant integrable deformation of the HF model. Then, Porsezian et al.10 showed that it is gauge equivalent to the integrable derivative NLSE, which has important applications in two photon self-induced transparency and ultrashot light pulse propagation in the optical fiber. The higher order integrable deformations of HF model have been investigated by Lakshmanan et al.11 By associating the spin vector with the tangent to a moving curve in Euclidean space, they found that the resultant equivalent equations are the higher order NLSEs. The nonlinear dynamics of inhomogeneous systems has many realistic physics problems. Much interest has arisen in the construction of inhomogeneous integrable equations. The inhomogeneous deformed HF model was first investigated by Lakshmanan and Ganesan.12 Recently, some integrable generalized inhomogeneous HF

a兲

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b兲 c兲

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models have been constructed by means of the prolongation structure theory and the corresponding gauge equivalent generalized inhomogeneous higher order NLSEs have also been presented.13 The supersymmetric generalizations of the HF model, i.e., Heisenberg supermagnet 共HS兲 models, were constructed by Makhankov and Pashaev.14 In terms of the gauge transformation, they established the gauge equivalence between the constructed HS models and the related NLSEs. The relationship of the HS models with the strong electron correlated Hubbard model has also been well discussed there. After that, Ghose Choudhury and Roy Chowdhury15 analyzed the nonlocal conservation laws and the corresponding supercharges in the HS model. They found that the nonlocal supercharges close to form a super-Yangian-type algebra. Not in the case of the HF model, the higher order deformations of the HS model, to our best knowledge, have not been reported so far in the existing literature. The aim of this paper is to investigate the integrable higher order deformations of the HS model and to construct the corresponding gauge equivalent counterparts. The organization of this paper is as follows. In Sec. II, we briefly recall the HS model. Then we investigate the third and fourth order integrable deformations of the HS model and present their gauge equivalent counterparts in Sec. III. We end this paper with a summary and discussion in Sec. IV. II. HS MODEL

Let us start with a short summary of the HS model14 that will be useful in what follows. The HF model is described by the exactly integrable equation, St = S ⫻ Sxx ,

共1兲 16

2

where S denotes the spin vector and satisfies the constraint S = 1. Takhtajan solutions by the inverse scattering method. The HS model is given by iSt = 关S,Sxx兴,

gave its N-soliton

共2兲

where S is the superspin variable 4

8

a=1

a=5

S = 2 兺 S aT a + 2 兺 C aT a .

共3兲

Here, S1 , . . . , S4 are the bosonic components and C5 , . . . , C8 are the fermionic ones; T1 , . . . , T4 are the bosonic and T5 , . . . , T8 are the fermionic generators of the superalgebra uspl共2 / 1兲. They are given by 1 Tជ = 2

冢冣

1 T4 = 2

冢冣

0 0 −1 i 0 0 0 , T6 = 2 1 0 0

冢冣

0 0 0 i 0 0 −1 . T8 = 2 0 1 0

0

␴ជ 0 , 0 0 0

0 0 1 1 0 0 0 , T5 = 2 1 0 0 0 0 0 1 0 0 1 , T7 = 2 0 1 0

冢冣 0

I2 0 , 0 0 2

冢

冣

冢

冣

共4兲

where ␴ ជ = 兵␴1 , ␴2 , ␴3其 are the Pauli matrices and I2 is a unitary matrix. The spin operator S satisfies constraints 共i兲 S2 = 3S − 2I for S 苸 USPL共2 / 1兲 / S共U共2兲 ⫻ U共1兲兲 and 共ii兲 S2 = S for S


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Integrable higher order deformations

苸 USPL共2 / 1兲 / S共L共1 / 1兲 ⫻ U共1兲兲. Note that if the reductions S4 = Ci = 0 , i = 5 , . . . , 8 are imposed, then Eq. 共2兲 is reduced to the HF model 共1兲 in the matrix extension. The Lax representation of the HS model 共2兲 is given by

␾x = U␾,

␾t = V␾ ,

共5兲

V = i␭2S + ␭关S,Sx兴.

共6兲

where U = i␭S,

The gauge equivalence plays an important role in the integral systems. It allows us not only to group together the large number of nonlinear systems already known but also to conclude about the properties of a system, knowing the corresponding properties of its gauge equivalent counterpart. The gauge equivalence between the HF model and the NLSE has been established. This gauge equivalence can also be generalized to the supersymmetric case. Makhankov and Pashaev14 found that the gauge equivalent counterparts of 共2兲 with constraints 共i兲 and 共ii兲 are the following Grassman odd and supersymmetric NLSE, respectively, i␺1t + ␺1xx + 2¯␺2␺2␺1 = 0, i␺2t + ␺2xx + 2¯␺1␺1␺2 = 0,

共7兲

and ¯ 兲␸ = 0, i␸t + ␸xx + 2共¯␸␸ + ␺␺ i␺t + ␺xx + 2¯␸␸␺ = 0,

共8兲

where ␸共x , t兲 is a Grassman even field, and ␺, ␺1, and ␺2 are the Grassman odd fields. III. INTEGRABLE DEFORMATIONS OF THE HS MODEL

Under constraint 共i兲 S2 = 3S − 2I, we note that St and 关S , Sxx兴 satisfy SStS = 2St and S关S , Sxx兴S = 2关S , Sxx兴, respectively. Thus, the deformation term E added into 共2兲 has to satisfy the transformation condition, 共9兲

SES = 2E.

For the case of the fourth order deformation, it is easy to verify that 关S , Sxxxx兴 satisfies 共9兲. Let us now take account of a possible fourth order deformed HS model with constraint 共i兲, iSt = 关S,Sxx兴 + E, E = ␧关S,Sxxxx兴 + Eˆ共S,Sx,Sxx,Sxxx兲,

共10兲

where ␧ is a deformation parameter, the function Eˆ共S , Sx , Sxx , Sxxx兲 is required to satisfy 共9兲 and need to be determined later. Let us take U = − i␭S, n

V = − ␭关S,Sx兴 + i␭2S − ␭␧关S,Sxxx兴 + ␭␧关Sx,Sxx兴 + 兺 ␭i f i共S,Sx,Sxx兲,

共11兲

i=1

where ␭ is the spectral parameter. Substituting 共11兲 into the zero-curvature condition equation,


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Ut − Vx + 关U,V兴 = 0

共12兲

and equating coefficients in terms with the same powers of ␭, we can determine f i共S , Sx , Sxx兲 and Eˆ via the following path: Eˆ , f 1 → f 2 → f 3 → ¯. After a tedious but straightforward calculation, we obtain Eˆ = 10␧共SxxSx关S,Sx兴 + SxSxx关S,Sx兴 + SxSx关S,Sxx兴兲, f 1 = − 10␧共SxSxSSx − SxSxSxS兲, 3

f 2 = i␧Sxx + i 2 ␧共SSxSx + SxSxS − 2SxSSx兲, f 3 = ␧共SSx − SxS兲, f 4 = − i␧S, f 5 = f 6 = ¯ = f n = 0.

共13兲

In this way, we derive the following integrable fourth order deformed HS model with constraint 共i兲: iSt = 关S,Sxx兴 + ␧关S,Sxxxx兴 + 10␧共SxxSx关S,Sx兴 + SxSxx关S,Sx兴 + SxSx关S,Sxx兴兲.

共14兲

For the case of constraint 共ii兲 S2 = S, it is easy to verify that St satisfies SStS = 0. Thus, the deformation term E should satisfy the transformation condition, SES = 0.

共15兲

Proceeding with similar procedure as in the case of constraint 共i兲, we find that the integrable fourth order deformed HS model with constraint 共ii兲 is also given by 共14兲. The corresponding U and V take 共11兲 and 共13兲. In Sec. II, we have pointed out that the gauge equivalent counterparts of 共2兲 with constraints 共i兲 and 共ii兲 are the Grassman odd and super-NLSE, respectively. Let us now construct the gauge equivalent counterpart of 共14兲. As was done in Ref. 14, we take S = g−1共x,t兲⌺g共x,t兲,

共16兲

where g共x , t兲苸 USPL共2 / 1兲, ⌺ takes diag共1,1,2兲 and diag共0,1,1兲 for constraints 共i兲 and 共ii兲, respectively. Then, we introduce the currents J0 = gtg−1 ,

共17兲

⳵tJ1 − ⳵xJ0 + 关J1,J0兴 = 0.

共18兲

J1 = gxg−1, satisfying the condition

˜ and ˜V, respectively, Under the gauge transformation, U and V turn into the following U ˜ = gUg−1 + g g−1 = − i␭⌺ + J , U x 1


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˜V = gVg−1 + g g−1 = − i␭4␧⌺ + ␭3␧J t 1 + i␭2␧共关关⌺,J1兴,J1兴 + 关⌺,J1兴x + 23 共J1关⌺,J1兴 − 关⌺,J1兴J1兲兲 + i␭2⌺ − ␭␧共关⌺,关关关⌺,J1兴,J1兴 + 关⌺,J1兴x,J1兴兴 + 关⌺,关关⌺,J1兴,J1兴x + 关⌺,J1兴xx兴 − 关关⌺,J1兴,关关⌺,J1兴,J1兴兴 − 关关⌺,J1兴,关⌺,J1兴x兴 + 10关⌺,J1兴2J1兲 − ␭J1 + J0 .

共19兲

In order to derive J0 and J1, we decompose algebra uspl共2 / 1兲 into two orthogonal parts, 共20兲

L = L0 丣 L1 , 共i兲

共j兲

共i+j兲mod共2兲

共0兲

and L is an algebra constructed by means of the generators of the where 关L 兴 , 兵L 其傺 L stationary subgroup H. Note that the stationary subgroup H is S共U共2兲 ⫻ U共1兲兲 and S共L共1 / 1兲 ⫻ U共1兲兲 for constraint 共i兲 and 共ii兲, respectively. Let us take

冢

0

0

共I兲 J1 = i 0 ¯␺

0

1

¯␺ 2

冣

␺1 ␺2 苸 L共1兲 0

for S 苸 USPL共2/1兲/S共U共2兲 ⫻ U共1兲兲

共21兲

for S 苸 USPL共2/1兲/S共L共1/1兲 ⫻ U共1兲兲,

共22兲

and

冢冣 0 ␸ ␺

共II兲 J1 = i ¯␸ 0 0 ¯␺ 0 0

苸 L共1兲

where ␸ is the Grassman even field, and ␺1, ␺2, and ␺ are the Grassman odd fields. From 共16兲 and 共17兲, it follows that St = g−1关⌺,J0兴g,

Sx = g−1关⌺,J1兴g.

共23兲

Substituting 共16兲 and 共23兲 into the deformed HS model 共14兲, we have i关⌺,J0兴 = 关⌺,关关⌺,J1兴,J1兴兴 + 关⌺,关⌺,J1兴x兴 + ␧关⌺,关关关关⌺,J1兴,J1兴,J1兴,J1兴兴 + ␧关⌺,关关关⌺,J1兴x,J1兴,J1兴兴 + ␧关⌺,关关关⌺,J1兴,J1兴x,J1兴兴 + ␧关⌺,关关⌺,J1兴xx,J1兴兴 + ␧关⌺,关关关⌺,J1兴,J1兴,J1兴x兴 + ␧关⌺,关关⌺,J1兴x,J1兴x兴 + ␧关⌺,关关⌺,J1兴,J1兴xx兴 + ␧关⌺,关⌺,J1兴xxx兴 + 10␧共共关关⌺,J1兴,J1兴 + 关⌺,J1兴x兲关⌺,J1兴关⌺,关⌺,J1兴兴 + 关⌺,J1兴共关关⌺,J1兴,J1兴 + 关⌺,J1兴x兲关⌺,关⌺,J1兴兴 + 关⌺,J1兴关⌺,J1兴共关⌺,关关⌺,J1兴,J1兴兴 + 关⌺,关⌺,J1兴x兴兲兲.

共24兲

共1兲 By solving 共24兲 and using the identity 关⌺ , J共0兲 0 兴 = 0, we obtain the following J0 for constraints 共i兲 and 共ii兲, respectively,

共I兲

J共1兲 0

=

冢

冣

0

0

J13

0

0

J23 , 0

J31 J32

共25兲

where J13 = − ␺1x − ␧␺1xxx − 3␧关␺1¯␺2␺2x + ␺1x¯␺2␺2兴, J23 = − ␺2x − ␧␺2xxx − 3␧关␺2¯␺1␺1x + ␺2x¯␺1␺1兴,


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J31 = ¯␺1x + ␧¯␺1xxx + 3␧关¯␺2x␺2¯␺1 + ¯␺2␺2¯␺1x兴, J32 = ¯␺2x + ␧¯␺2xxx + 3␧关¯␺1x␺1¯␺2 + ¯␺1␺1¯␺2x兴,

共26兲

and

冢

¯J ¯J 12 13

0

¯ 共II兲 J共1兲 0 = J21

0

0

¯J 31

0

0

冣

共27兲

,

where ¯J = − ␸ − ␧␸ − 3␧关共␸ ¯␸ + ␺ ¯␺兲␸ + 共␸␸ + ␺␺ ¯ 兲␸ 兴, 12 x xxx x x x ¯J = − ␺ − ␧␺ − 3␧关共␸ ¯␸ + ␺ ¯␺兲␺ + 共␸␸ ¯ 兲␺ 兴, ¯ + ␺␺ 13 x xxx x x x ¯J = ¯␸ + ␧¯␸ + 3␧关共␸␸ ¯ 兲¯␸ + 共␸␸ ¯ 兲¯␸ 兴, ¯ x + ␺␺ ¯ + ␺␺ 21 x xxx x x ¯J = ¯␺ + ␧¯␺ + 3␧关共␸␸ ¯ 兲¯␺ + 共␸␸ ¯ 兲¯␺ 兴. ¯ x + ␺␺ ¯ + ␺␺ 31 x xxx x x Substituting 共21兲 and 共25兲 into 共18兲, we obtain the expression of

冢

Jˆ11 Jˆ12

0

ˆ ˆ J共0兲 0 = i J21 J22

0

0

0

Jˆ33

冣

J共0兲 0

共28兲

under constraint 共i兲,

,

共29兲

where Jˆ11 = ␺1¯␺1 − 3␧␺1¯␺1␺2¯␺2 + ␧关共␺1¯␺1兲xx − 3␺1x¯␺1x兴, Jˆ12 = ␺1¯␺2 + ␧关共␺1¯␺2兲xx − 3␺1x¯␺2x兴, Jˆ21 = ␺2¯␺1 + ␧关共␺2¯␺1兲xx − 3␺2x¯␺1x兴, Jˆ22 = ␺2¯␺2 − 3␧␺1¯␺1␺2¯␺2 + ␧关共␺2¯␺2兲xx − 3␺2x¯␺2x兴, Jˆ33 = ␺1¯␺1 + ␺2¯␺2 − 6␧␺1¯␺1␺2¯␺2 + ␧关共␺1¯␺1 + ␺2¯␺2兲xx − 3共␺1x¯␺1x + ␺2x¯␺2x兲兴.

共30兲

In a similar way, we may obtain J共0兲 0 under constraint 共ii兲,

冢

˜J 11

J共0兲 0

=i

0

0

冣

0

˜J ˜J 23 , 22

0

˜J ˜J 33 32

共31兲

where ¯ + 3␧共␸␸ ¯ 兲共␸␸ ¯ 兲 + ␧关共␸␸ ¯ 兲 − 3共␸ ¯␸ + ␺ ¯␺ 兲兴, ˜J = ␸␸ ¯ + ␺␺ ¯ + ␺␺ ¯ + ␺␺ ¯ + ␺␺ 11 xx x x x x


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¯ 兲 − ␧关共¯␸␸兲 − 3¯␸ ␸ 兴, ˜J = − ¯␸␸ − 3␧¯␸␸共␸␸ ¯ + ␺␺ 22 xx x x ˜J = − ¯␸␺ − 3␧¯␸␺共␸␸ ¯ 兲 − ␧关共¯␸␺兲 − 3¯␸ ␺ 兴, ¯ + ␺␺ 23 xx x x ˜J = − ¯␺␸ − 3␧¯␺␸共␸␸ ¯ 兲 − ␧关共¯␺␸兲 − 3¯␺ ␸ 兴, ¯ + ␺␺ 32 xx x x ˜J = − ¯␺␺ − 3␧¯␺␺共␸␸ ¯ 兲 − ␧关共¯␺␺兲 − 3¯␺ ␺ 兴. ¯ + ␺␺ 33 xx x x

共32兲

共1兲 The matrix J0 may be determined by taking J0 = J共0兲 0 + J0 . Thus, for the case of constraint 共i兲, 共19兲 may be expressed as follows:

˜ =i U

冢

␺1 ␺2

−␭

0

0

−␭

¯␺ 1

¯␺ − 2␭ 2

冢

冣

,

冣

V11 V12 V13 ˜V = i V21 V22 V23 , V31 V32 V33

共33兲

where V11 = − ␭4␧ − ␭2共␧␺1¯␺1 − 1兲 − i␭␧共␺1¯␺1x − ␺1x¯␺1兲 + ␺1¯␺1 − 3␧␺1¯␺1␺2¯␺2 + ␧关共␺1¯␺1兲xx − 3␺1x¯␺1x兴, V12 = − ␭2␧␺1¯␺2 − i␭␧共␺1¯␺2x − ␺1x¯␺2兲 + ␺1¯␺2 + ␧关共␺1¯␺2兲xx − 3␺1x¯␺2x兴, V13 = ␭3␧␺1 − i␭2␧␺1x − ␭␧共2␺1¯␺2␺2 + ␺1xx兲 − ␭␺1 + i␺1x + i␧␺1xxx + 3i␧共␺1¯␺2␺2x + ␺1x¯␺2␺2兲, V21 = − ␭2␧␺2¯␺1 − i␭␧共␺2¯␺1x − ␺2x¯␺1兲 + ␺2¯␺1 + ␧关共␺2¯␺1兲xx − 3␺2x¯␺1x兴, V22 = − ␭2共␧␺2¯␺2 − 1兲 − i␭␧共␺2¯␺2x − ␺2x¯␺2兲 − ␭4␧ + ␺2¯␺2 − 3␧␺1¯␺1␺2¯␺2 + ␧关共␺2¯␺2兲xx − 3␺2x¯␺2x兴, V23 = ␭3␧␺2 − i␭2␧␺2x − ␭␧共2␺2¯␺1␺1 + ␺2xx兲 − ␭␺2 + i␺2x + i␧␺2xxx + 3i␧共␺2¯␺1␺1x + ␺2x¯␺1␺1兲, V31 = ␭3␧¯␺1 + i␭2␧¯␺1x − ␭␧共2¯␺2␺2¯␺1 + ¯␺1xx兲 − ␭¯␺1 − i¯␺1x − i␧¯␺1xxx − 3i␧共¯␺2x␺2¯␺1 + ¯␺2␺2¯␺1x兲, V32 = ␭3␧¯␺2 + i␭2␧¯␺2x − ␭␧共2¯␺1␺1¯␺2 + ¯␺2xx兲 − ␭¯␺2 − i¯␺2x − i␧¯␺2xxx − 3i␧共¯␺1x␺1¯␺2 + ¯␺1␺1¯␺2x兲, V33 = − 2␭4␧ − ␭2关␧共␺1¯␺1 + ␺2¯␺2兲 − 2兴 − i␭␧共␺1¯␺1x − ␺1x¯␺1 + ␺2¯␺2x − ␺2x¯␺2兲 + ␺1¯␺1 + ␺2¯␺2 − 6␧␺1¯␺1␺2¯␺2 + ␧关共␺1¯␺1 + ␺2¯␺2兲xx − 3共␺1x¯␺1x + ␺2x¯␺2x兲兴.

共34兲

˜ and ˜V gives the Grassman odd fourth order Then, the zero-curvature condition equation of U NLSE, i␺1t + ␺1xx + 2␺1¯␺2␺2 + ␧␺1xxxx + 3␧共␺1¯␺2␺2xx + 2␺1x¯␺2␺2x + ␺1xx¯␺2␺2兲 + ␧关共␺1¯␺2兲xx␺2 + ␺1共¯␺2␺2兲xx兴 = 0,


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i␺2t + ␺2xx + 2␺2¯␺1␺1 + ␧␺2xxxx + 3␧共␺2¯␺1␺1xx + 2␺2x¯␺1␺1x + ␺2xx¯␺1␺1兲 + ␧关共␺2¯␺1兲xx␺1 + ␺2共¯␺1␺1兲xx兴 = 0.

共35兲

For the case of constraint 共ii兲, 共19兲 takes the form

冢

0

␸

␺ 0

冣

˜ = i ¯␸ − ␭ , U ¯␺ 0 − ␭

冢

Vˆ11 ˜V = i Vˆ 21

Vˆ12 Vˆ13

冣

Vˆ22 Vˆ23 ,

Vˆ31 Vˆ32 Vˆ33

共36兲

where ¯ 兲 − i␭␧共␸␸ ¯ − ␺ ¯␺兲 + ␸␸ ¯ ¯ + ␺␺ ¯ x − ␸x¯␸ + ␺␺ ¯ + ␺␺ Vˆ11 = − ␭2␧共␸␸ x x ¯ 兲共␸␸ ¯ 兲 + ␧关共␸␸ ¯ 兲 − 3共␸ ¯␸ + ␺ ¯␺ 兲兴, ¯ + ␺␺ ¯ + ␺␺ ¯ + ␺␺ + 3␧共␸␸ xx x x x x ¯ 兲␸ + ␸ 兴 − ␭␸ ¯ + ␺␺ Vˆ12 = ␭3␧␸ − i␭2␧␸x − ␭␧关2共␸␸ xx ¯ 兲␸ 兴, ¯ + ␺␺ + i␸x + i␧␸xxx + 3i␧关共␸x¯␸ + ␺x¯␺兲␸ + 共␸␸ x ¯ 兲␺ + ␺ 兴 − ␭␺ ¯ + ␺␺ Vˆ13 = ␭3␧␺ − i␭2␧␺x − ␭␧关2共␸␸ xx ¯ 兲␺ 兴, + i␺x + i␧␺xxx + 3i␧关共␸x¯␸ + ␺x¯␺兲␺ + 共␸␸ + ␺␺ x ¯ 兲 + ¯␸ 兴 − ␭¯␸ ¯ + ␺␺ Vˆ21 = ␭3␧¯␸ + i␭2␧¯␸x − ␭␧关2¯␸共␸␸ xx ¯ 兲 + ¯␸ 共␸␸ ¯ ¯ x + ␺␺ − i¯␸x − i␧¯␸xxx − 3i␧关¯␸共␸␸ x x ¯ + ␺␺兲兴, Vˆ22 = − ␭4␧ + ␭2共1 + ␧¯␸␸兲 − i␭␧共¯␸␸x − ¯␸x␸兲 ¯ 兲 − ␧关共¯␸␸兲 − 3¯␸ ␸ 兴, ¯ + ␺␺ − ¯␸␸ − 3␧¯␸␸共␸␸ xx x x Vˆ23 = ␭2␧¯␸␺ − i␭␧共¯␸␺x − ¯␸x␺兲 − ¯␸␺ ¯ 兲 − ␧关共¯␸␺兲 − 3¯␸ ␺ 兴, ¯ + ␺␺ − 3␧¯␸␺共␸␸ xx x x ¯ 兲 + ¯␺ 兴 − ␭¯␺ ¯ + ␺␺ Vˆ31 = ␭3␧¯␺ + i␭2␧¯␺x − ␭␧关2¯␺共␸␸ xx ¯ ¯ 兲 + ¯␺ 共␸␸ ¯ x + ␺␺ − i¯␺x − i␧¯␺xxx − 3i␧关¯␺共␸␸ x ¯ + ␺␺兲兴, x Vˆ32 = ␭2␧¯␺␸ − i␭␧共¯␺␸x − ¯␺x␸兲 − ¯␺␸ ¯ 兲 − ␧关共¯␺␸兲 − 3¯␺ ␸ 兴, ¯ + ␺␺ − 3␧¯␺␸共␸␸ xx x x


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Vˆ33 = − ␭4␧ + ␭2共1 + ␧¯␺␺兲 − i␭␧共¯␺␺x − ¯␺x␺兲 − ¯␺␺ ¯ 兲 − ␧关共¯␺␺兲 − 3¯␺ ␺ 兴. ¯ + ␺␺ − 3␧¯␺␺共␸␸ xx x x

共37兲

˜ and ˜V, it gives the supersymmetric fourth From the zero-curvature condition equation of U order NLSE, ¯ 兲␸ + ␧␸ ¯ 兲 ␸兴 ¯ + ␺␺ ¯ + ␺␺ ␸␸兲xx + ␺共¯␺␸兲xx + 共␸␸ i␸t + ␸xx + 2共␸␸ xxxx + ␧关␸共¯ xx ¯ 兲␸ 兴 + 6␧共␸␸ ¯ 兲共␸␸ ¯ 兲␸ = 0, ¯ + ␺␺ ¯ + ␺␺ ¯ + ␺␺ + 3␧关共␸xx¯␸ + ␺xx¯␺兲␸ + 2共␸x¯␸ + ␺x¯␺兲␸x + 共␸␸ xx ¯ 兲 ␺兴 ¯ ␺ + ␧␺xxxx + ␧关␸共¯␸␺兲xx + ␺共¯␺␺兲xx + 共␸␸ ¯ + ␺␺ i␺t + ␺xx + 2␸␸ xx ¯ 兲␺ 兴 + 6␧共␸␸ ¯ 兲共␸␸ ¯ 兲␺ = 0. ¯ + ␺␺ ¯ + ␺␺ ¯ + ␺␺ + 3␧关共␸xx¯␸ + ␺xx¯␺兲␺ + 2␸x¯␸␺x + 共␸␸ xx

共38兲

Note that 共38兲 is reduced to the fourth order NLSE derived by Lakshmanan et al.11 in the bosonic limit. We have constructed the fourth order deformed HS models. Let us turn to consider the case of the third order deformation. Under constraint 共ii兲 S2 = S, we note that 关S , Sxxx兴 and Sxxx + 23 共SSxxSx + SSxSxx + SxxSxS + SxSxxS兲 satisfy the transformation relation 共15兲. Thus, we may take the following deformation for constraint 共ii兲: iSt = 关S,Sxx兴 + ␦关S,Sxxx兴 + i␧Sxxx + 23 i␧共SSxxSx + SSxSxx + SxxSxS + SxSxxS兲 + ¯E .

共39兲

where ␦ and ␧ are the deformation parameters, the function ¯E共S , Sx , Sxx兲 satisfies the transformation relation 共15兲 and need to be determined. Proceeding with similar procedure as in the case of fourth order deformation, we finally obtain the integrable third order deformed HS model with constraint 共ii兲 as follows: iSt = 关S,Sxx兴 + i␧Sxxx − 3i␧共SxxSSx + SxSSxx兲 + 23 i␧共SSxxSx + SSxSxx + SxxSxS + SxSxxS兲.

共40兲

The corresponding U and V are U = − i␭S, V = − i␭␧Sxx + 共i␭3␧S + i␭2S − 23 i␭␧SxSx兲S + 共3i␭␧SxS − 23 i␭⑀SSx兲Sx + 共− ␭2␧ − ␭兲关S,Sx兴. 共41兲 Performing the gauge transformation of linear problem 共41兲 generated by g from 共16兲, we obtain the supersymmetric third order NLSE, ¯ 兲␸ − i␧␸ − 3i␧关共␸ ¯␸ + ␺ ¯␺兲␸ + 共␸␸ ¯ 兲␸ 兴 = 0, ¯ + ␺␺ ¯ + ␺␺ i␸t + ␸xx + 2共␸␸ xxx x x x ¯ 兲␺ 兴 = 0. ¯ ␺ − i␧␺xxx − 3i␧关共␸x¯␸ + ␺x¯␺兲␺ + 共␸␸ ¯ + ␺␺ i␺t + ␺xx + 2␸␸ x

共42兲

˜ and ˜V are The corresponding U

冢

冣

0 ␸ ␺ ˜ U = i ¯␸ − ␭ 0 , ¯␺ 0 − ␭


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冢

˜ ˜V 11 V12 ˜V = i ˜V ˜ 21 V22

冣

˜V 13 ˜V 23 ,

˜V ˜ ˜ 31 V32 V33

共43兲

where ˜V = i␧共␸␸ ¯ − ␺ ¯␺兲 + 共1 + ␭␧兲共␸␸ ¯ 兲, ¯ x − ␸x¯␸ + ␺␺ ¯ + ␺␺ 11 x x ˜V = − ␭2␧␸ + i␭␧␸ − ␭␸ + 2␧共␸2¯␸ + ␺␺ ¯ ␸兲 + ␧␸ + i␸ , x xx x 12 ˜V = − ␭2␧␺ + i␭␧␺ − ␭␺ + 2␧␸␸ ¯ ␺ + ␧␺xx + i␺x , 13 x ˜V = − ␭2␧¯␸ − i␭␧¯␸ − ␭¯␸ + 2␧共¯␸2␸ + ¯␸␺␺ ¯ 兲 + ␧¯␸ − i¯␸ , 21 x xx x ˜V = ␭3␧ + ␭2 − ␭␧¯␸␸ + i␧共¯␸␸ − ¯␸ ␸兲 − ¯␸␸ , 22 x x ˜V = − ␭␧¯␸␺ + i␧共¯␸␺ − ¯␸ ␺兲 − ¯␸␺ , 23 x x ˜V = − ␭2␧¯␺ − ␭共i␧¯␺ + ¯␺兲 + 2␧¯␺␸␸ ¯ + ␧¯␺xx − i¯␺x , 31 x ˜V = − ␭␧¯␺␸ + i␧共¯␺␸ − ¯␺ ␸兲 − ¯␺␸ , 32 x x ˜V = ␭3␧ + ␭2 − ␭␧¯␺␺ + i␧共¯␺␺ − ¯␺ ␺兲 − ¯␺␺ . 33 x x

共44兲

Under the reduction ␺ = 0, 共42兲 is reduced to the Hirota equation derived by Lamb.17 Therefore, 共42兲 can be regarded as the supersymmetric Hirota equation. For constraint 共i兲 S2 = 3S − 2I, we find that the deformation term in 共40兲 does not satisfy the transformation relation 共9兲. It implies that 共40兲 is not an integrable deformation with constraint 共i兲. We have tried to constructed its third order deformation. Unfortunately, we did not succeed in finding any integrable third order deformed HS model for this case. IV. SUMMARY AND DISCUSSION

We have investigated the integrable higher order deformations of the HS model with two different constraints: 共i兲 S2 = 3S − 2I for S 苸 USPL共2 / 1兲 / S共U共2兲 ⫻ U共1兲兲 and 共ii兲 S2 = S, for S 苸 USPL共2 / 1兲 / S共L共1 / 1兲 ⫻ U共1兲兲. We found that the expressions of the fourth order deformed HS models are the same for constraints 共i兲 and 共ii兲. In terms of the gauge transformation, the resultant gauge equivalent counterparts were the Grassman odd and supersymmetric fourth order NLSE, respectively. Moreover, we constructed the third order deformed HS model with constraint 共ii兲 and its gauge equivalent counterpart was found to be the supersymmetric Hirota equation. Unfortunately, we did not succeed in finding the integrable third order deformed HS model with constraint 共i兲. Whether there exists this kind of deformed HS model still deserves further study. It should be pointed out that our approach can be applied to construct more integrable higher order deformed HS models. The strong electron correlated Hubbard model has attracted a lot of interest in physics. The relationship of the HS models with the strong electron correlated Hubbard model has been well investigated. As a future study, it should be interesting to find out the applications of these deformed HS models.


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ACKNOWLEDGMENTS

We would like to thank the referee for his/her helpful comments. This work was partially supported by NKBRPC 共Grant Nos. 2004CB318000 and 2006CB805905兲 and Beijing Jiao-Wei Key project 共Grant No. KZ200810028013兲. P. Di Vecchia and S. Ferrara, Nucl. Phys. B 130, 93 共1977兲; M. Chaichian and P. Kulish, Phys. Lett. B 78, 413 共1978兲. P. Mathieu, J. Math. Phys. 29, 2499 共1988兲; S. Bellucci, E. Ivanov, S. Krivonos, and A. Pichugin, Phys. Lett. B 312, 463 共1993兲; F. Toppan, Int. J. Mod. Phys. A 11, 3257 共1996兲; D. Sarma, Nucl. Phys. B 681, 351 共2004兲. 3 Yu. Manin and A. Radul, Commun. Math. Phys. 98, 65 共1985兲; V. Kac and E. Medina, Lett. Math. Phys. 37, 435 共1996兲; A. LeClair, Nucl. Phys. B 314, 425 共1989兲; L. Martinez Alonso and E. Medina Reus, J. Math. Phys. 36, 4898 共1995兲. 4 G. H. M. Roelofs and P. H. M. Kersten, J. Math. Phys. 33, 2185 共1992兲; Z. Popowicz, Phys. Lett. A 194, 375 共1994兲. 5 Z. Popowicz, Phys. Lett. A 354, 110 共2006兲. 6 A. Das and Z. Popowicz, J. Math. Phys. 46, 082702 共2005兲; X. X. Chen, X. Y. Jia, K. Wu, and W. Z. Zhao, Phys. Lett. A 373, 430 共2009兲. 7 V. E. Zakharov and L. A. Takhtadzhyan, Theor. Math. Phys. 38, 17 共1979兲. 8 V. A. Kazakov, A. Marshakov, J. A. Minahan and K. Zarembo, J. High Energy Phys. 5, 024 共2004兲. 9 A. V. Mikhailov and A. B. Shabat, Phys. Lett. A 116, 191 共1986兲. 10 K. Porsezian, K. M. Tamizhmani, and M. Lakshmanan, Phys. Lett. A 124, 159 共1987兲. 11 M. Lakshmanan, K. Porsezian, and M. Daniel, Phys. Lett. A 133, 483 共1988兲; D. G. Zhang and G. X. Yang, J. Phys. A 23, 2133 共1990兲; K. Porsezian, M. Daniel, and M. Lakshmanan, J. Math. Phys. 33, 1807 共1992兲; K. Porsezian, Chaos, Solitons Fractals 9, 1709 共1998兲. 12 M. Lakshmanan and S. Ganesan, J. Phys. Soc. Jpn. 52, 4031 共1983兲. 13 W. Z. Zhao, Y. Q. Bai, and K. Wu, Phys. Lett. A 352, 64 共2006兲. 14 V. G. Makhankov and O. K. Pashaev, J. Math. Phys. 33, 2923 共1992兲. 15 A. Ghose Choudhury and A. Roy Chowdhury, Int. J. Theor. Phys. 33, 2031 共1994兲. 16 L. A. Takhtajan, Phys. Lett. A 64, 235 共1977兲. 17 G. L. Lamb, J. Math. Phys. 18, 1654 共1977兲. 1 2
