1 Introduction

The nonlinear σ-models and their supersymmetric extensions have received a lot of attention[1]due to the widely application in gravity theory,[2]the theory of strings[3]and superstrings.[4]The simplest version of the nonlinear σ-model is the Heisenberg ferromagnet(HF)model,which reads as[5]

where S is the spin vector and satisfies the constraint S2=1.It is well known that the HF model is gauge equivalent and geometrical equivalent to the nonlinear Schrödinger equation(NLSE).[6−7]Growing interest has been focused on the extension of the HF model.[8−12]Kundu[13]constructed a deformed HF model and presented that it is gauge equivalent to the mixed derivative NLSE(MDNLSE),which plays an important role in explaining nonlinear propagation of Alfven wave[14]and ultrashot light pulse propagation in the optical fi ber.[15]Recently,Levin et al.[16]investigated the relation between the deformed HF model and the quantum 11-vertex R-matrix.By constructing the classical integrable tops,they establish more complicated integrable systems in terms of the quantum R-matrices.

More motivations come from supersymmetric integrable systems involving the supersymmetric extension of the important integrable systems.[17−22]The Heisenberg supermagnet(HS)model is the super extension of the HF model.It has attracted considerable interest in the study of the HS model and its corresponding gauge equivalent counterpart.[23]It should be noted that the HS model has close relation with the strong electron correlated Hubbard model.Ghose Choudhury and Roy Chowdhury[24]analyzed the nonlocal conservation laws and the corresponding supercharges in the HS model.Lately,the generalization of the integrable HS models attracts a lot of interest.The structure and integrability properties of the generalized HS models have been well discussed.[25−29]The purpose of this paper is to construct an extension of the HS model and to investigate their integrability.

The organization of this paper is as follows.In Sec.2,we recall the HS model and its integrable properties.Section 3 is dedicated to constructing the super generalized HS model and to deriving the Lax representation with two constraints.Using the gauge transformation,we study their gauge equivalent counterparts.In Sec.4,we end this paper with a summary and discussion.

2 HS Model

For later convenience,we shall recall the definitions of the HS model in this section.For a more detailed description we refer the reader to Ref.[23].

Being the superextensions of the HF model,the HS model reads as

where S is the superspin variable and can be represented

Let us decompose the super algebra uspl(2/1)into two orthogonal parts

whereS1,...,S4 arethebosonic componentsand B5,...,B8are the fermionic ones,X1,...,X4are bosonic generators and X5,...,X8are fermionic generators of the superalgebra uspl(2/1).They can be written as

where = σ1,σ2,σ3are Pauli matrices and I2is an identity matrix.

The gauge equivalence is an important concept in the integrable systems.Such interconnections allow us not only to understanding the structure and properties of nonlinear systems already known,but also to conclude about the properties of a system knowing the corresponding properties of its gauge-equivalent counterpart.One can also consider the gauge equivalence of the supersymmetric integrable systems. Under the two constraints(I)S2=S for S∈USPL(2/1)/S(L(1/1)×U(1))and(II)S2=3S−2I for S∈USPL(2/1)/S(U(2)×U(1))Makhankov and Pashaev[23]showed the HS model(2)is gauge equivalent to supersymmetric NLSE and Grassman odd NLSE,respectively

where φ(x,t)is a bosonic filed and ψ, ψ1, ψ2are the fermionic fileds.

3 Generalized Heisenberg Supermagnetic Model

satisfying the condition

where f is the function of x and t,p and q need to be determined later.For the constraint I,it is not difficult to check S[S,Sx]S=0,SSxS=0.We now take

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where λ is the spectral parameter.

The zero-curvature condition equation

where ψ1(x,t),ψ2(x,t)are the fermionic fields.Using the similar methods as shown before,we easily carry out

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Now we will mainly be concerned with Eq.(11)in the frame work of the gauge equivalent.Proceeding the similar procedure as,[23]we suppose

where g(x,t)∈USPL(2/1).

Under the constraint I and II,we take Σ=diag(0,1,1)and Σ=diag(1,1,2),respectively.Then we introduce the currents

Considering the deformation of the HS model under the constraint I.S2=S,one easily gets SStS=0 and S[S,Sxx]S=0.Now we suppose the generalized HS model as follows

甲组和乙组手术时间、术中出血量、骨愈合时间、Harris评分、住院时间比较，差异有统计学意义（P＜0.05），如表1。

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here[L(i),L(j)}⊂L(i+j)mod(2).L(0)is an algebra constructed by means of the generators of the stationary subgroup H.The stationary subgroup H is S(L(1/1)×U(1))and S(U(2)×U(1))for constraint I and II,respectively.

Based on the gauge transformation,F and G turn toandrespectively,

Taking

where φ(x,t)and ψ(x,t)are the bosonic and fermionic field,respectively.

Using Eqs.(12),(13),and(16),we obtain

Substituting Eq.(17)into Eq.(11),we have

Then we rewrite Eq.(7)as the expression

Equation(18)leads totakes the form

where

Substituting Eqs.(16)and(19)into Eq.(20)and integrating Eq.(20),we have

By mean of Eqs.(20),(21),and J0=+,we obtain J0.In terms of the gauge transformation,F and G in Eq.(8)becomeˆF andˆG,respectively,

Substituting Σ=diag(0,1,1),Eqs.(20)and(21)into Eq.(22),we have

where

where the spectral parameter λ satis fies λt= λx=0.

The zero-curvature condition equation ofˆF andˆG gives the super generalized MDNLSE

Under the reduction f=1,h=0,Eq.(26)reduces to the super MDNLSE.[25]In the bosonic limit and f=1,h=0,Eq.(26)leads to the MDNLSE.[13]

Let us take account of the generalized HS model(11)with the constraint II.It should be noted that the expression of the corresponding integrable extension is also Eq.(11).For this case,the corresponding F and G are

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铁路道岔维修养护工作是一项全面性和复杂性较高的工作，在工作中要明确维修养护的重点内容，对其进行针对性的重点养护，保证铁路道岔的质量。在道岔磨合阶段，针对尖轨、辙叉、基本轨出现的“肥边”，要及时开展打磨，避免剥落掉块等不良状况的产生。对于心轨侧磨问题的产生，检修人员要定期对护轮轨的间隔尺寸进行检查和调整，避免侧磨问题的加剧[5]。针对螺栓失效的问题，工作人员要对螺栓进行定期检查，及时对螺栓进行拧紧、加固，同时还要对螺栓的腐蚀程度进行检查养护，避免腐蚀严重影响道岔整体运行。另外，对于道岔内部的绝缘部件要进行定期检查和养护，保证铁路信号系统的正常运行。

As was done in the constraint I,we consider

基于关联规则挖掘的跨语言译后扩展核心问题是如何计算关联模式支持度.常见支持度计算主要有四种：(1)将关联模式在事务文档中发生的概率作为该模式的支持度[9]；(2)将项目权值总和与无加权支持度的乘积作为加权项集支持度[15]；(3)将特征词项目平均权值与无加权支持度的乘积作为完全加权项集支持度[11,16]；(4)以项集在事务数据库中项集权值总和占事务数据库中所有项目权值总和的百分比作为完全加权项集支持度[10,17].文献[17]表明，方法(4)挖掘效果比方法(3)的好.然而，方法(4)只考虑特征词项目权值对支持度的影响，忽略特征词频度对支持度的作用.

Substituting Eq.(8)into Eq.(9),we obtain

where

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Combining Eqs.(29)and(30),we have

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where Σ=diag(1,1,2).

By virtue of Eqs.(28)and(32),we rewrite Eq.(33)as follows

where the spectral parameter λ satis fies λt= λx=0.

From the zero-curvature condition equation of andwe obtain the generalized fermionic MDNLSE

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Introducing the reduction f=1 and h=0,Eq.(36)leads to the fermionic MDNLSE.[25]

4 Summary and Discussion

We construct the generalization of the HS model with two different constraints.The Lax pairs associated with the generalized models have been deduced.By means of the gauge equivalence,the related super MDNLSEs are derived,which can be regarded as the generalized super MDNLSEs.There has been a considerable interest in the study of the strong electron correlated Hubbard model due to its important applications in physics.Therefore,as to the applications of the integrable generalized HS models presented in this paper,their applications in physics still deserve further study.

2.1 两组患者手术前后生活质量评分比较 治疗前，两组患者社会功能、躯体功能、心理功能、物质生活状态等生活质量变化比较，差异无统计学意义(t=0.24、0.31、0.15、0.23，P>0.05)；治疗后，两组患者社会功能、躯体功能、心理功能、物质生活状态评分均显著上升，组内相比差异有统计学意义(P<0.05)，观察组显著高于对照组，差异有统计学意义(t=9.52、10.27、7.34、10.53，P<0.05)。见表2。

Acknowledgments

The authors thank the valuable suggestions of the referees.

湖北作为中国南方长江流域的一个省份，应该为有这样一处与兵马俑、半坡村齐名的大冶铜绿山古铜矿遗址而骄傲。兵马俑、半坡村遗址是黄河文明的产物，而铜绿山古铜矿遗址是长江文明的产物。二者齐头并进，形成华夏文明的源头。

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