# Three Types GeneralizedZn-Heisenberg Ferromagnet Models

Authors: Yinfei Zhou; Shuchao Wan; Yang Bai; Zhaowen Yan
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

By taking values in a commutative subalgebragln,C, we construct a new generalized Zn-Heisenberg ferromagnet model in (1+1)-dimensions. The corresponding geometrical equivalence between the generalized Zn-Heisenberg ferromagnet model and Zn-mixed derivative nonlinear Schrödinger equation has been investigated. The Lax pairs associated with the generalized systems have been derived. In addition, we construct the generalized Zn-inhomogeneous Heisenberg ferromagnet model and Zn-Ishimori equation in (2+1)-dimensions. We also discuss the integrable properties of the multi-component systems. Meanwhile, the generalized Zn-nonlinear Schrödinger equation, Zn-Davey–Stewartson equation and their Lax representation have been well studied.


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## Body

## 1. Introduction

The Heisenberg ferromagnet (HF) model is one of the most investigated integrable systems which plays an important role in the two-dimensional (2D) gravity theory [1] and anti-de Sitter/conformal field theories [2, 3]. It is proved that the HF model is gauge and geometric equivalent to the nonlinear Schrödinger (NLS) equation [4, 5]. (1+1)-dimensional generalized HF models involving inhomogeneous and higher order deformed HF models have been analyzed [6, 7]. The deformed HF models in (2+1)-dimensions also have been investigated, such as the higher order HF models [8, 9], the HF models with self-consistent potentials [10], the Ishimori equation [11], and inhomogeneous HF models [12, 13].Multi-component version of the integrable systems has deserved much attention due to its wide application in multiple orthogonal polynomials, representation theory, random matrix model, the related Riemann-Hilbert problems, and Brownian motions [14–18]. Many important integrable systems have been extended to their multi-component counterparts, such as multi-component KP [19, 20], multi-component Toda systems [14], and multi-component BKP [21]. After considering commutative subalgebra of diagonal matrices, Bogdanov et al. [22] constructed the generalized multicomponent KP hierarchy which involves N independent generalized scalar KP hierarchies. Starting from the maximal commutative subalgebra of glm,C, one [23, 24] constructed a new Zm-Kadomtsev-Petviashvili (KP) hierarchy and investigated the existence of τ-functions. Meanwhile, the relation between dispersionless reduced Zm-KP hierarchy and Frobenius manifold has been discussed. Recently, Li et al. [25] constructed the extended multi-component Toda hierarchy and extended multi-component bigraded Toda hierarchy. By virtue of taking values in a matrix-valued differential algebra set, they also establish a class of Hirota quadratic equation, which may be useful in Gromov-Witten theory and noncommutative symplectic geometry. In [25], one has defined the new multi-component sinh–Gordon systems by considering commutative subalgebra of gln,C and established their Bäcklund transformations. A natural problem then arises as to how to construct the corresponding extended HF models. With this motivation, this paper will be devoted to constructing three types commutative multi-component generalized HF models by taking values in commutative subalgebra. Furthermore, their corresponding geometrical and gauge equivalent counterparts shall be discussed.This paper is organized as follows. In Section 2, we present a brief review of some elementary facts about theZn-HF model and Zn-NLS equation. Section 3 is devoted to constructing the generalized Zn-HF models and establishing the geometrical equivalence with the Zn-mixed derivative NLSE. In Section 4, we investigate the generalized Zn-inhomogeneous HF models and their structure and integrability. In addition, we deduce the multi-component Ishimori equation and discuss its corresponding gauge equivalent counterpart. The last section will be devoted to a summary and discussion.


## 2.Zn-Heisenberg Ferromagnet Model

The Heisenberg ferromagnet (HF) model in (1+1)-dimensions [4] is an important integrable equation which reads as(1)St=S×Sxx,whereS denotes the spin vector, S = (S1, S2, S3) and satisfies the constraint S2=1.The matrix form of the HF model can be expressed as(2)iSt=12S,Sxx,WhereS=∑i=13Siσi,, S2=I,trS=0 and σii=1,2,3 are Pauli matrices.LetS take values in a commutative subalgebra Zn=CΓ/Γn and Γ=δij,j+1ij∈gln,C. From the equation (2), we obtain(3)iS̃t=12S̃,S̃xx,WhereS̃2=I, I is an identity matrix. Suppose S̃ can be expressed as(4)S̃i=Si0E+Si1Γ+Si2Γ2+…+Sin−1Γn−1.ThenS̃x,t can be divided into n parts(5)S̃=S0+S1+S2+…+Sn−1,where(6)Sk=Sk⋅Xk,Sk=Sk1,Sk2,Sk3,Xk=Xk1,Xk2,Xk3,Xk1=0ΓkΓk0,Xk2=0iΓk−iΓk0,Xk3=Γk00−Γkand whenk=0, Γ0=E, E is a identity matrix. Then we may derive the following theorems.Theorem 1.
The following equation holds(7)iSkt=12∑i+j=kSi,Sjxx,0≤k≤n−1.Proof.
By choosing the coefficient ofΓk for two sides of the identity (3), (3) leads to (7), which will be referred to as the Zn-HF model.The integrability condition of (7) is as the following linear systems(8)Φxx,t,λj=Ux,t,λjΦx,t,λj,Φtx,t,λj=Vx,t,λjΦx,t,λj,where(9)U=i∑l=0n−1∑j+k=lλjSk·Xl,V=2i∑l=0n−1∑t+j+k=lλtλjSk·Xl+12∑l=0n−1∑t+j+k=lλtSj·Skx·Xl.Substituting (5) and (6) into (3), we obtain the following corollary:Corollary 2.
The vector form of the
Z
n
-HF model:
(10)
S
k
t
=
∑
i
+
j
=
k
S
i
×
S
j
x
x
,
0
≤
k
≤
n
−
1
,
here we use the property
(11)
S
i
·
X
i
,
S
j
x
x
·
X
j
=
2
i
S
i
×
S
j
x
x
·
X
i
+
j
,
i
+
j
≤
n
−
1
,
w
h
e
n
i
+
j
>
n
−
1
,
X
i
+
j
=
0
.This proves that theZn-HF is geometrical equivalent to the following Zn-NLSE.Theorem 3.
The following identity holds
(12)
i
φ
k
t
+
φ
k
x
x
+
2
∑
i
+
j
+
l
=
k
φ
i
φ
¯
j
φ
l
=
0
,
0
≤
k
≤
n
−
1
.Proof.
From NLS equation, we obtain(13)iφ̃t+φ̃xx+2φ̃2φ̃=0,
where(14)φ̃=φ0E+φ1Γ+φ2Γ2+…+φn−1Γn−1.
By choosing the coefficients ofΓk for the identity (13), (13) leads to the Zn-NLS equation (12).The Lax pair of (12) can be represented as(15)Φx′x,t,λj=U′x,t,λjΦ′x,t,λj,Φt′x,t,λj=V′x,t,λjΦ′x,t,λj,where(16)U′=∑j=0n−1−iλjΣ+Mjx,tΓj,V′=∑j=0n−1−2i∑j+k=lλjλkΣ·Γl+2∑j+k=lλαMβx,t·Γl−iMlxx,t·Γl−i∑k+t=lMkx,tMtx,t·Γl·Σ,and(17)∑=diagE,−E,Mjx,t=0φj−φ¯j0,whereE is a identity matrix.


## 3. GeneralizedZn-Heisenberg Ferromagnet Model in (1+1)-Dimensions

Let us consider the integrable deformed HF model [28](18)St=S×Sxx+ϵ2Sx·SxSx,whereϵ is a deformation parameter.By expandingS=S0E+S1Γ+…+Sn−1Γn−1, we obtain the generalized Zn-Heisenberg ferromagnet model in (1+1)-dimensions(19)Sjt=∑α+β=jSα×Sβxx+ϵ2∑a+b+c=jSax·Sbx·Scx,0≤j≤n−1,whereϵ is a deformation parameter. When ϵ=0, Eq. (19) reduces to the Zn-HF model (10). The Lax representation of the generalized Zn-HF equation (19) is given by(20)Ũ=ϵ∑k=0n−1∑m+n+p+q=kλmλnSp·Xq+ϵ∑k=0n−1∑m+n+p=kλmSnx·Xp,Ṽ=ϵ∑k=0n−1∑m+n+p+q+l=kλmλnSp×Sqx·Xl−ϵ∑k=0n−1∑m+n+p+q+l+w=kλmλnλpλqSl·Xw+ϵ∑k=0n−1∑m+n+p+q=kλmSn×Spxx·Xq+2ϵ2∑k=0n−1∑m+n+p+q+l+w=kλmλnSpx·SqxSl·Xw+2ϵ2∑k=0n−1∑m+n+p+q+l=kλmSnx·SpxSqx·Xl−ϵ2∑k=0n−1∑m+n+p+q+l=kλmλnλpSqx·Xl−32ϵ2∑k=0n−1∑m+n+p+q+l=kλmλnSpx·Sqx·Xl,whereλj0≤j≤n−1 are spectral parameters.In order to derive the geometrical equivalent counterpart of (19), we introduce the multi-component Serret-Frenet equation(21)tjx=∑m+l=jkmnl,bjx=−∑m+l=jτmnl,njx=∑m+l=jτmbl−kmtl.By introducing the multi-component Hasimoto function(22)φjx=∑α+β=jkαxbβ+i∑p+q+r=jkpτqbr,0≤j≤n−1,where(23)bβ=∑m1·1+m2·2+…+mn−1·n−1=β1m1!m2!…mn−1!A1m1A2m2…An−1mn−1⋅expi∫−∞xτ0dx′,here(24)Ait=expi∫−∞xτitdx′.IdentifyingSj0≤j≤n−1 in (19) with the tangent vector of a curve tj, we obtain(25)tjt=∑α+β=jtα×tβxx+ϵ2∑a+b+c=jtax·tbx·tcx,=∑p+q=jηpnq+ζpbq+ϵ2∑m+n+p+q=jkmknkpnq,0≤j≤n−1.Then we have(26)γj=−∑p+q=jηp+iζqbq=−iφjx−ϵ2∑a+b+c=jφ¯aφbφc.By the equation(27)Rjx=i2∑α+β=jγαφ¯β−γ¯αφβ=12∑a+b=jφ¯aφbx,one finds that the time evolution equation satisfies the following equation(28)φjt+γjx−i∑α+β=jRαφβ=0.Substituting (26) and (27) into (28) and takingφj→2φj, we derive the Zn-mixed derivative NLSE equation(29)iφjt+φjxx+2∑α+β+γ=jφαφ¯βφγ−2iϵ∑α+β+γ=jφαφ¯βφγx=0,0≤j≤n−1.Takingϵ=0, the Zn-mixed derivative NLSE equation (29) degenerates into Zn-NLSE equation (12). Then we obtain the Lax representation of the Zn-mixed derivative NLSE equation(30)U=i∑k=0n−1∑a+b=k2ϵ∑m+n=aλmλn+2λaσ3b−∑k=0n−1∑m+n=k2ϵλm+1An,V=∑k=0n−1∑f+g=k−8iϵ2∑m+n+p+q=fλmλnλpλq−16iϵ∑m+n+p=fλmλnλp+∑m+n+p+q=f−8i+4iϵ2φmφ¯nλpλq+4iϵ∑m+n+p=fλmφnφ¯p+i∑m+n=fφmφ¯nσ3g+∑k=0n−1∑m+n=k8ϵ2∑a+b+c=mλaλbλc+12ϵ∑a+b=mλaλb+4λmAn−∑k=0n−1∑m+n=k2ϵλm+1Bn,where(31)Aj=0φj−φ¯j0,σ3j=Γj00−Γj .and(32)Bj=0iφjx+2ϵ∑a+b+c=jφ¯aφbφciφjx−2ϵ∑a+b+c=jφ¯aφbφ¯c0.


## 4. GeneralizedZn-Heisenberg Ferromagnet Model in (2+1)-Dimensions

Many (2+1)-dimensional integrable inhomogeneous Heisenberg ferromagnet equations have been of interest, for instance, Inhomogeneous M-I equation [13] and the Ishimori equation [11]. The Ishimori equation [11] is a well-known (2+1)-dimensional integrable extension of the HF model, which involves an infinite dimensional symmetry algebra with a loop algebra structure and is solved by the inverse scattering transform approach. There is geometrical and gauge equivalence between the Ishimori equation and Davey-Stewartson equation [29, 30]. In this section, we shall derive the multi-component counterparts of two types deformed HF models in (2+1)-dimensions.


### 4.1.Zn-Inhomogeneous M-I Equation

LetS take values in a commutative algebra, we have S=S0E+S1Γ+…+Sn−1Γn−1. By means of multi-component generalization, we obtain the generalized Zn-inhomogeneous Heisenberg ferromagnet model in (2+1)-dimensions(33)Sjt=∑m+n=jSmx×Sny+Sm×Snxy+umxSn+umSnx+ρmSnx,0≤j≤n−1,where(34)ujx=−∑a+b+c=jSa·Sbx×Scyand the parametersρm satisfy(35)ρm=∑a+b=mμ3axb+ν3m.Whenj=0, Eq. (33) reduced to the integrable inhomogeneous Myrzakulov-I equation [13].The linear problem of the multi-component HF models (33) in (2+1)-dimensions can be expressed as(36)ξxn=i2∑l=0n∑α+β+γ=lλαξγSβσl,ξtn=−∑f+g=nλfξyg+i2∑l=0n∑α+β+p+q=lλαρβ+uβSpξq+∑m+i+j+γ=lλmSi×Sjyξγσl,where(37)Sβ=S1β,S2β,S3β,Xl=σl=σ1l,σ2l,σ3l.and(38)σ1l=0ΓlΓl0,σ2l=0iΓl−iΓl0,σ3l=Γl00−Γl.Then the Lax representation of Eq. (33) is given by(39)F˜=−i2∑l=0n∑α+β=lλαSβσl,G˜=−i2∑l=0n∑p+q+r=lλpρq+uqSr+∑i+j+m=lλmSi×Sjyσl.Now one considers the the geometrical equivalent counterpart of the multi-component Eq. (33). Let us introduce the multi-component Serret-Frenet equation(40)
t
j
y
=
−
∑
p
+
q
+
l
=
j
u
p
x
k
q
b
l
+
∂
x
−
1
∑
p
+
q
=
j
k
p
y
n
q
−
∂
x
−
1
∑
p
+
q
+
l
+
m
=
j
τ
p
u
q
x
k
l
n
m
,
b
j
y
=
−
∑
p
+
q
=
j
u
p
+
∂
x
−
1
τ
p
y
n
q
+
∑
p
+
q
+
l
=
j
u
p
x
k
q
t
l
,
n
j
y
=
∑
p
+
q
=
j
u
p
b
q
+
∂
x
−
1
τ
p
y
b
q
−
∂
x
−
1
k
p
y
t
q
+
∑
p
+
q
+
l
+
m
=
j
τ
p
u
q
x
k
l
t
m
,Then we derive the multi-component Hasimoto function(41)φjy=∑α+β=jkaybβ+i∑p+q+r=j∂x−1kpτqybr,0≤j≤n−1.In order to derive the geometrical equivalent counterpart of (33), we identifySj0≤j≤n−1 in the vector form of the Zn-generalized inhomogeneous HF model in (2+1)-dimensions (33) with the tangent vector of a curve tj. Then we have(42)tjt=∑m+l=jtm×tlxy+∑m+l=jtmx×tly+∑m+l=jumxtl+∑m+l=jumSlx+∑m+l=jρmSlx,=∑p+q=jkpybq+∑p+q+m=j−kp∂x−1τqynm+ρpkqnm,0≤j≤n−1.Thus we obtain(43)γj=−∑p+q=jηp+iζqbq=−iφjy−∑a+b=jρaφb.By the equation(44)Rjx=i2∑α+β=jγαφ¯β−γ¯αφβ=−12∂y∑a+b=jφ¯aφb.Substituting (43) and (44) into (28) and takingRjx→−Rjx, we derive the Zn-NLS equation(45)iφjt−φjxy−i∑α+β=jραφβx−∑a+b=jRaφb=0,0≤j≤n−1,where(46)Rjx=12∂y∑a+b=jφ¯aφb,0≤j≤n−1.Whenj=0, the Zn-NLS equation (45) degrades into the (2+1)-dimensional focusing nonlinear Schrödinger equation equation [13]. The Lax representation of the Zn-HLS equation can be expressed as(47)Ũ=12iλφ−φ¯−iλ,Ṽ=ÃB̃−B̃¯−Ã,where(48)Ã=−12−k=0n−1Rk+i2∑k=0n−1∑m+n=kλmρn,B̃=−12∑k=0n−1φky+i2∑k=0n−1∑m+n=kρmφn,whereλj0≤j≤n−1 are spectral parameters.


### 4.2.Zn-Ishimori Equation

Based on the multi-component generalization, we construct the multi-component Ishimori equation in (2+1)-dimensions(49)iSjt+∑m+n=jiumySnx+iumxSny+12Sm,14Snxx+α2Snyy=0,α2ujyy−14ujxx=∑m+n=jα24itrSmSny,Snx,0≤j≤n−1.The Lax representation of (49) is given by(50)ϕx=Uϕy,ϕt=Wϕxx+Vϕx,where(51)Uj=−αSj,Vj=−iSjxx+ujy+∑m+n=j−iαSnySm−α3umxSn,Wj=−2iSj.In terms of gauge transformation(52)ψ̃j=gjϕj.The functionsgj and Sj can be written as(53)gj=f1j+∑m+n=jf1mS3n,∑m+n=jf1mSn−,∑m+n=jf2mSn+,−f2j−∑m+n=jf2mS3n·ω1j,ω2j,ω3j,ω4j,Sj=∑a+b+c=jga−1σ3bgc,wherefij,ωij,Sj± satisfy the following equations:(54)Sj±=S1j±iS2j,∑m+n=j21+S3mαlnf1ny−12lnf1nx=12S3jx+∑m+n=jS3mS3nx+Smx−Sn+−αS3jy+∑m+n=jS3mS3ny+Smy−Sn+,and(55)∑m+n=j21+S3mαlnf2ny+12lnf2nx=−12S3jx+∑m+n=jS3mS3nx+Smx+Sn−−αS3jy+∑m+n=jS3mS3ny+Smy+Sn−,here(56)ω1j=Γj000,ω2j=0Γj00,ω3j=00Γj0,ω4j=000Γj.Then it follows that(57)αgjy−∑m+n=jB1mgnx=∑m+n=jB0mgn,where(58)B1j=12Γj00−12Γj,B0j=0qjpj0.Thus we obtain the gauge equivalent counterpart of Eq. (49) which can be considered as the Zn-Davey-Stewartson equation(59)iqjt+14qjxx+α2qjyy+∑m+n=jvmqn=0,α2vjyy−14vjxx=∑m+n=j−2α2pmqnyy+14pmqnxx.Its Lax reprensentation is given by(60)αψ̃y=B1̃ψ̃x+B0̃ψ̃,ψ̃t=iC0̃ψ̃xx+C1̃ψ̃x+C0̃ψ̃.with(61)B0̃=B00+B01+B02+…+B0n−1,B1̃=B10+B11+B12+…+B1n−1,where(62)B0j=B0j·aj,B0j=qj,pj,aj=a1j,a2j,a1j=0Γj00,a2j=00Γj0,B1j=12Γj00−12Γj.Similarly,(63)C0̃=C00+C01+C02+…+C0n−1,C1̃=C10+C11+C12+…+C1n−1,C2̃=C20+C21+C22+…+C2n−1,where(64)C0j=C0j·bj,C0j=c11j,c12j,c21j,c22j,bj=b1j,b2j,b3j,b4j.b1j=Γj000,b2j=0Γj00,b3j=00Γj0,b4j=000Γj,with(65)c12j=i2qjx+iαqjy,c21j=i2pjx−iαpjy.Herec11j and c22j are the solution of the following equations(66)12c11jx−αc11jy=i12∑m+n=jpmqnx+α∑m+n=jpmqny,−12c22jx−αc22jy=i12∑m+n=jpmqnx−α∑m+n=jpmqny.C
1
j and C2j are given by(67)C1j=C1j·cj,C1j=qj,pj,cj=c1j,c2j,c1j=0iΓj00,c2j=00iΓj0,C2j=12Γj00−12Γj.


## 4.1.Zn-Inhomogeneous M-I Equation

LetS take values in a commutative algebra, we have S=S0E+S1Γ+…+Sn−1Γn−1. By means of multi-component generalization, we obtain the generalized Zn-inhomogeneous Heisenberg ferromagnet model in (2+1)-dimensions(33)Sjt=∑m+n=jSmx×Sny+Sm×Snxy+umxSn+umSnx+ρmSnx,0≤j≤n−1,where(34)ujx=−∑a+b+c=jSa·Sbx×Scyand the parametersρm satisfy(35)ρm=∑a+b=mμ3axb+ν3m.Whenj=0, Eq. (33) reduced to the integrable inhomogeneous Myrzakulov-I equation [13].The linear problem of the multi-component HF models (33) in (2+1)-dimensions can be expressed as(36)ξxn=i2∑l=0n∑α+β+γ=lλαξγSβσl,ξtn=−∑f+g=nλfξyg+i2∑l=0n∑α+β+p+q=lλαρβ+uβSpξq+∑m+i+j+γ=lλmSi×Sjyξγσl,where(37)Sβ=S1β,S2β,S3β,Xl=σl=σ1l,σ2l,σ3l.and(38)σ1l=0ΓlΓl0,σ2l=0iΓl−iΓl0,σ3l=Γl00−Γl.Then the Lax representation of Eq. (33) is given by(39)F˜=−i2∑l=0n∑α+β=lλαSβσl,G˜=−i2∑l=0n∑p+q+r=lλpρq+uqSr+∑i+j+m=lλmSi×Sjyσl.Now one considers the the geometrical equivalent counterpart of the multi-component Eq. (33). Let us introduce the multi-component Serret-Frenet equation(40)
t
j
y
=
−
∑
p
+
q
+
l
=
j
u
p
x
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q
b
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−
1
∑
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+
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=
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+
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t
m
,Then we derive the multi-component Hasimoto function(41)φjy=∑α+β=jkaybβ+i∑p+q+r=j∂x−1kpτqybr,0≤j≤n−1.In order to derive the geometrical equivalent counterpart of (33), we identifySj0≤j≤n−1 in the vector form of the Zn-generalized inhomogeneous HF model in (2+1)-dimensions (33) with the tangent vector of a curve tj. Then we have(42)tjt=∑m+l=jtm×tlxy+∑m+l=jtmx×tly+∑m+l=jumxtl+∑m+l=jumSlx+∑m+l=jρmSlx,=∑p+q=jkpybq+∑p+q+m=j−kp∂x−1τqynm+ρpkqnm,0≤j≤n−1.Thus we obtain(43)γj=−∑p+q=jηp+iζqbq=−iφjy−∑a+b=jρaφb.By the equation(44)Rjx=i2∑α+β=jγαφ¯β−γ¯αφβ=−12∂y∑a+b=jφ¯aφb.Substituting (43) and (44) into (28) and takingRjx→−Rjx, we derive the Zn-NLS equation(45)iφjt−φjxy−i∑α+β=jραφβx−∑a+b=jRaφb=0,0≤j≤n−1,where(46)Rjx=12∂y∑a+b=jφ¯aφb,0≤j≤n−1.Whenj=0, the Zn-NLS equation (45) degrades into the (2+1)-dimensional focusing nonlinear Schrödinger equation equation [13]. The Lax representation of the Zn-HLS equation can be expressed as(47)Ũ=12iλφ−φ¯−iλ,Ṽ=ÃB̃−B̃¯−Ã,where(48)Ã=−12−k=0n−1Rk+i2∑k=0n−1∑m+n=kλmρn,B̃=−12∑k=0n−1φky+i2∑k=0n−1∑m+n=kρmφn,whereλj0≤j≤n−1 are spectral parameters.


## 4.2.Zn-Ishimori Equation

Based on the multi-component generalization, we construct the multi-component Ishimori equation in (2+1)-dimensions(49)iSjt+∑m+n=jiumySnx+iumxSny+12Sm,14Snxx+α2Snyy=0,α2ujyy−14ujxx=∑m+n=jα24itrSmSny,Snx,0≤j≤n−1.The Lax representation of (49) is given by(50)ϕx=Uϕy,ϕt=Wϕxx+Vϕx,where(51)Uj=−αSj,Vj=−iSjxx+ujy+∑m+n=j−iαSnySm−α3umxSn,Wj=−2iSj.In terms of gauge transformation(52)ψ̃j=gjϕj.The functionsgj and Sj can be written as(53)gj=f1j+∑m+n=jf1mS3n,∑m+n=jf1mSn−,∑m+n=jf2mSn+,−f2j−∑m+n=jf2mS3n·ω1j,ω2j,ω3j,ω4j,Sj=∑a+b+c=jga−1σ3bgc,wherefij,ωij,Sj± satisfy the following equations:(54)Sj±=S1j±iS2j,∑m+n=j21+S3mαlnf1ny−12lnf1nx=12S3jx+∑m+n=jS3mS3nx+Smx−Sn+−αS3jy+∑m+n=jS3mS3ny+Smy−Sn+,and(55)∑m+n=j21+S3mαlnf2ny+12lnf2nx=−12S3jx+∑m+n=jS3mS3nx+Smx+Sn−−αS3jy+∑m+n=jS3mS3ny+Smy+Sn−,here(56)ω1j=Γj000,ω2j=0Γj00,ω3j=00Γj0,ω4j=000Γj.Then it follows that(57)αgjy−∑m+n=jB1mgnx=∑m+n=jB0mgn,where(58)B1j=12Γj00−12Γj,B0j=0qjpj0.Thus we obtain the gauge equivalent counterpart of Eq. (49) which can be considered as the Zn-Davey-Stewartson equation(59)iqjt+14qjxx+α2qjyy+∑m+n=jvmqn=0,α2vjyy−14vjxx=∑m+n=j−2α2pmqnyy+14pmqnxx.Its Lax reprensentation is given by(60)αψ̃y=B1̃ψ̃x+B0̃ψ̃,ψ̃t=iC0̃ψ̃xx+C1̃ψ̃x+C0̃ψ̃.with(61)B0̃=B00+B01+B02+…+B0n−1,B1̃=B10+B11+B12+…+B1n−1,where(62)B0j=B0j·aj,B0j=qj,pj,aj=a1j,a2j,a1j=0Γj00,a2j=00Γj0,B1j=12Γj00−12Γj.Similarly,(63)C0̃=C00+C01+C02+…+C0n−1,C1̃=C10+C11+C12+…+C1n−1,C2̃=C20+C21+C22+…+C2n−1,where(64)C0j=C0j·bj,C0j=c11j,c12j,c21j,c22j,bj=b1j,b2j,b3j,b4j.b1j=Γj000,b2j=0Γj00,b3j=00Γj0,b4j=000Γj,with(65)c12j=i2qjx+iαqjy,c21j=i2pjx−iαpjy.Herec11j and c22j are the solution of the following equations(66)12c11jx−αc11jy=i12∑m+n=jpmqnx+α∑m+n=jpmqny,−12c22jx−αc22jy=i12∑m+n=jpmqnx−α∑m+n=jpmqny.C
1
j and C2j are given by(67)C1j=C1j·cj,C1j=qj,pj,cj=c1j,c2j,c1j=0iΓj00,c2j=00iΓj0,C2j=12Γj00−12Γj.


## 5. Summary and Discussion

Considering the commutative subalgebragln,C, we have constructed three types generalized Zn-HF models in (1+1) and (2+1)-dimensions. From the geometrical and gauge equivalence point of view, we also establish the corresponding equivalent counterparts of three types generalized Zn-Heisenberg ferromagnet models. The introduction of new degrees of freedom may emerge from multiscale procedures or regularizations of gradient catastrophes. Their physical meaning and application should be of interest. The methods in the paper may clearly be applied to the other generalized Heisenberg supermagnetic models. Therefore, other types of generalized Zn-HF models still deserve further study.



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