# A Few Integrable Couplings of Some Integrable Systems and (2+1)-Dimensional Integrable Hierarchies

Authors: Binlu Feng; Yufeng Zhang; Huanhe Dong
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Two high-dimensional Lie algebras are presented for which four (1+1)-dimensional
expanding integrable couplings of the D-AKNS hierarchy are obtained by using the Tu
scheme; one of them is a united integrable coupling model of the D-AKNS hierarchy
and the AKNS hierarchy. Then (2+1)-dimensional DS hierarchy is derived by using
the TAH scheme; in particular, the integrable couplings of the DS hierarchy are obtained.


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## Body

## 1. Introduction

Seeking new integrable systems has been an important aspect of soliton theory, and the methods for this are also versatile. For example, Magri [1] once proposed the Lax-pair method for generating (1+1)-dimensional integrable Hamiltonian equations. Tu [2] took use of some subalgebras of the Lie algebra A1 to draw a beautiful picture for generating integrable Hamiltonian equation hierarchies with 1+1 dimensions, and in particular, the AKNS hierarchy, the KN hierarchy, the WKI hierarchy, and others were obtained under the frame of zero curvature equations. Ma [3] called the method the Tu scheme, and Ma also obtained a series of interesting results in [4–6]. Hu [7, 8] further generalized the trace identity of Tu scheme to get the supertrace identity for generating superintegrable Hamiltonian systems which has extended the applications of the Tu scheme. Fan [9, 10] applied the Tu scheme not only to have obtained some new integrable hierarchies of equations, but also to have obtained their some geometric properties. Various Lax pairs can be introduced by some reductions of the self-dual Yang-Mills equations. For example, Chakravarty et al. [11] introduced symmetries and gauge transformations as well as isospectral problems by reducing the self-dual Yang-Mills equations to have obtained (1+1)- and (2+1)-dimensional integrable systems. Ablowitz et al. [12] introduced some different Lax pairs by reductions of the self-dual Yang-Mills equations and further obtained some celebrated (1+1)- and (2+1)-dimensional integrable equations, including the KdV equation and the KP equation. It is remarkable that generating (2+1)-dimensional integrable hierarchies is more difficult than (1+1)-dimensional ones. The papers [11, 12] were proceeded by introducing a Lie algebra g={a0+a1(∂/∂y)+a2(∂2/∂y2)}, where ai belong to a ring of n×n complex matrix functions of y,  x, and t, while Tu et al. [13] proposed an efficient and direct approach for generating (2+1)-dimensional equation hierarchies by introducing a residue operator, which was called by us the TAH scheme. Because few people further discussed the scheme and investigated other new (2+1)-dimensional integrable hierarchies, we would like to recall the scheme and apply it to produce new (2+1)-dimensional hierarchies of equations along with multipotential functions. First of all, we recall some related notations.LetA be an associative algebra over the field R. An operator ∂:A→A satisfies that
(1)∂(αf+βg)=α∂f+β∂g,∂(fg)=(∂f)g+f(∂g),
where α,β∈R; f,g∈A.Introduce an associative algebraA[ξ] consisting of the pseudodifferential operator ∑-∞Naiξi, where the coefficients ai∈A, and ξ is an operator given by
(2)ξf=fξ+(∂yf),f∈A.
It can be verified that [13]
(3)ξnf=∑i≥0‍(ni)(∂if)ξn-i,n∈Z,(4)fξn=∑i≥0‍(-1)i(ni)ξn-i(∂if),n∈Z.
Besides, a residue operator is given by
(5)R:A[ξ]⟶A,R(∑‍aiξi)=a-1.
Based on the above, we recall the TAH scheme as follows.(1)
Fix a matrix operatorU=U(λ+ξ,u)∈A[ξ], where u=(u1,…,up)T.(2)
Solving the matrix-operator equation(6)Vx=[U,V]
leads to a recursion relation among Vn, where V=∑‍Vnλ-n, from which we will obtain a recursion relation among g(n)≡(g1(n),…,gp(n))T, where gi(n) comes from the expansion
(7)〈V,∂U∂ui〉=∑n‍gi(n)λ-n,
where 〈a,b〉=tr⁡R(ab).(3)
Try to find an operatorJ and form the hierarchy
(8)utn=Jg(n).(4)
Employing the trace identity [13]
(9)δδui〈V,∂U∂λ〉=(λ-γ∂∂λλγ)〈V,∂U∂ui〉
could obtain the Hamiltonian structure of the (2+1)-dimensional hierarchy (8).In the paper, we want to employ the TAH scheme to generalize2×2 matrix operators which are presented in [13, 14] to 4×4 cases by using the constructed higher-dimensional Lie algebras so that (2+1)-dimensional equation hierarchies could be generated, whose Hamiltonian structure is also obtained by the trace identity (9). For that sake, we first consider the D-AKNS hierarchy which is derived by our Lie algebra, and then we expand the D-AKNS hierarchy to obtain four different integrable models with 4, 5, and 6 potential functions, respectively. In particular, one of them can reduce to the well-known AKNS hierarchy.


## 2. The D-AKNS Hierarchy and Its Expanding Integrable Models

A Lie subalgebra of the Lie algebraA1 presents that [14]
(10)L=span⁡{h1,h2,h3,h4},
where
(11)h1=(1000),h2=(0100),h3=(0010),h4=(0001).
A loop algebra is defined by
(12)L~=span⁡{h1(n),h2(n),h3(n),h4(n)},
where hi(n)=hiλn, n∈Z, which have the commutative relations
(13)[h1(m),h2(n)]=h2(m+n),[h1(m),h3(n)]=-h3(m+n),[h1(m),h4(n)]=0,[h2(m),h3(n)]=h1(m+n)-h4(m+n),[h2(m),h4(n)]=h2(m+n),[h3(m),h4(n)]=-h3(m+n),mmmmmmmmimmhm,n∈Z.Paper [14] had derived the D-AKNS hierarchy. Now we employ the loop algebra L~ to rederive it again by the Tu scheme to make the paper self-contained by following Tu idea, and then two exlargening Lie algebras of the Lie algebra L can be used to generate the expanding integrable models, specially, containing the (2+1)-dimensional expanding equation hierarchies.Set(14)U=h4(1)+sh4(0)+qh2(0)+rh3(0),V=∑m≥0‍(∑i=14‍Vimhi(-m)).
A solution to (14) for V is given by
(15)V1m,x=qV3m-rV2m=-V4m,x,V2,m+1=-V2m,x-sV2m-qV1m+qV4m,V3,m+1=V3m,x-sV3m-rV1m+rV4m.
Letting
(16)V(n)=∑m=0n‍(∑i=14‍Vimhi(n-m))+(V1n-V4n)h4(0),
a direct calculation yields that
(17)-Vx(n)+[U,V(n)]=(V4n,x-V1n,x)h4(1)+(-V2n,x-sV2n)×h2(0)-(V3n,x-sV3n)h3(0).
According to the Tu scheme, the zero curvature equation
(18)Utn-Vx(n)+[U,V(n)]=0
admits that
(19)qtn=V2n,x+sV2n,rtn=V3n,x-sV3n,stn=-2V4n,x.
Denoting u=(q,r,s)T, then we have
(20)utn=(0∂+s0∂-s0000-2∂)(V3nV2nV4n)=J1(V3nV2nV4n),
which is just right the D-AKNS hierarchy. Obviously, J1 is a Hamiltonian operator. The Hamiltonian structure of (20) can be obtained by the trace identity in [2]:
(21)δδu(4V4)=(λ-γ∂∂λλγ)(V3V2V4).
Comparing the coefficients of λ-n-1 gives
(22)δδu(4V4,n+1)=(-n+γ)(V3nV2nV4n).
If set V2,0=V3,0=V4,0=0, V1,0=constant, we can get some explicit values from (15) V2,1=-αq,  V3,1=-αr, and  V1,1=0,….In terms of the initial values, we can getγ=0. Thus, we obtain the Hamiltonian structure of (20)
(23)utn=J1δHnδu,
where Hn=-4V4,n+1/n.In what follows, we discuss four different expanding integrable models of the D-AKNS hierarchy. With the help of the Lie algebraL, we introduce an 8-dimensional Lie algebra
(24)G1=span⁡{f1,…,f8},
where
(25)f1=(h100h1),f2=(h400h4),f3=(h200h2),f4=(h300h3),f5=(0h10h1),f6=(0h40h4),f7=(0h20h2),f8=(0h30h3),
which possesses the commutative relations
(26)[f1,f2]=0,[f1,f3]=f3,[f1,f4]=-f4,[f1,f5]=[f1,f6]=0,[f1,f7]=f7,[f1,f8]=-f8,[f2,f3]=-f3,[f2,f4]=f4,[f2,f5]=[f2,f6]=0,[f2,f7]=-f7,[f2,f8]=f8,[f3,f4]=f1-f2,[f3,f5]=-f7,[f3,f6]=f7,[f3,f7]=0,[f3,f8]=f5-f6,[f4,f5]=f8,[f4,f6]=-f8,[f4,f7]=f6-f5,[f4,f8]=[f5,f6]=0,[f5,f7]=f7,[f5,f8]=-f8,[f6,f7]=-f7,[f6,f8]=f8,[f7,f8]=f5-f6.
A loop algebra can be defined as
(27)fi(n)=fiλn,i=1,2,…,8;[fi(m),fj(n)]=[fi,fj]λm+n,1≤i,j≤8;m,n∈Z.
Now we utilize the loop algebra to introduce the following Lax pair:
(28)U=f2(1)+sf2(0)+qf3(0)+rf4(0)+f6(1)+u1f7(0)+u2f8(0)+u3f5(0),V=∑m≥0‍(∑i=58‍V1mf1(-m)+V2mf3(-m)+V3mf4(-m)mmm+V4mf2(-m)+∑i=58‍Vimfi(-m)).
According to the Tu scheme, we first need to solve the matrix equation for V:
(29)Vx=[U,V]
which admits the first part equations (15) and the second part ones as follows:
(30)(V5m)x=u1V3m-u2V2m-(r+u2)V7m+(q+u1)V8m=-(V6m)x,(V7m)x=-2V7,m+1+(-s+u3)V7m+(q+u1)V6m-(q+u1)V5m-u1V1m+u1V4m+u3V2m-V2,m+1,(V8m)x=2V8,m+1+(s-u3)V8m-(r+u2)V6m+(r+u2)V5m+u2V1m-u2V4m-u3V3m+V3,m+1.
Taking
(31)V(n)=∑m=0n‍(∑i=58‍V1mf1(n-m)+V2mf3(n-m)+V3mf4(n-m)mmmh+V4mf2(n-m)+∑i=58‍Vimfi(-m)λn)+(V1n-V4n)f2(0)+(V6n-V5n)f5(0),
one infers that
(32)-Vx(n)+[U,V(n)]=-(V1n-V4n)xf2(0)+[V2,n+1+q(V1n-V4n)]f3(0)-[V3,n+1-r(V6n-V5n)]f4(0)-(V6n-V5n)xf5(0)+[V2,n+1+2V7,n+1+u1(V1n-V4n)mm-(q+u1)(V6n-V5n)]f7(0)+[-2V8,n+1-V3,n+1-u2(V1n-V4n)mm+(r+u2)(V6n-V5n)]f8(0).
Thus, the zero curvature equation
(33)Utn-Vx(n)+[U,V(n)]=0
gives rise to
(34)utn=(qrsu1u2u3)tn=((V2n)x+sV2n(V3n)x-sV3n-2(V4n)x(V7n)x+sV7n-u3(V2n+V7n)(V8n)x-sV8n+u3(V3n+V8n)-2(V5n)x).
When u1=u2=u3=0, the hierarchy (34) reduces to the D-AKNS hierarchy.Whens=u3=0, (34) reduces to the following expanding integrable system of the AKNS hierarchy:
(35)u-tn=(qru1u2)tn=(0∂x00∂x000000∂x00∂x0)(V3nV2nV8nV7n)=J(V3nV2nV8nV7n).
In particular, setting u1=u2=0, (35) again reduces to the known AKNS hierarchy
(36)u~tn=(qr)tn=(0∂x∂x0)(V3nV2n).
Therefore, the hierarchy (34) is a united expanding integrable model of the D-AKNS hierarchy, the AKNS hierarchy, and the hierarchy (36). It is easy to see that (34) is different from that presented in [14], which indicates one of the merits for seeking multipotential expanding integrable hierarchies. However, the Hamiltonian structure of (34) cannot be obtained by the variational identity [15], which makes us feel confusions.In order to get the second expanding integrable model of the D-AKNS hierarchy, we first introduce the following Lie algebra:(37)G2=span⁡{g1,…,g8},
where
(38)g1=f1,g2=f2,g3=f3,g4=f4,g5=(0h1h10),g6=(0h4h40),g7=(0h2h20),g8=(0h3h30),
which has the following commutative relations:
(39)[g1,g2]=0,[g1,g3]=g3,[g1,g4]=-g4,[g2,g3]=-g3,[g2,g4]=g4,[g3,g4]=g1-g2,[g1,g5]=[g1,g6]=0,[g1,g7]=g7,[g1,g8]=-g8,[g2,g5]=[g2,g6]=0,[g2,g7]=-g7,[g2,g8]=g8,[g3,g5]=-g7,[g3,g6]=g7,[g3,g7]=0,[g3,g8]=g5-g6,[g4,g5]=g8,[g4,g6]=-g8,[g4,g7]=g6-g5,[g4,g8]=[g5,g6]=0,[g5,g7]=g3,[g5,g8]=-g4,[g6,g7]=-g3,[g6,g8]=g4,[g7,g8]=g1-g2.
Now we let us compare the Lie algebras G1 and G2. Setting
(40)G11=span⁡{f1,f2,f3,f4},G12=span⁡{f5,f6,f7,f8},
we find that
(41)G1=G11⊕G12,[G11,G12]⊂G12.
Condition (41) is a sufficient condition for generating integrable couplings.If we set(42)G21=span⁡{g1,g2,g3,g4},G22=span⁡{g5,g6,g7,g8},
it is easy to see that
(43)G2=G21+G22,[G21,G22]not inG22.
Equation (43) is different from (41) because the subalgebra G22 is a semisimple Lie algebra, while G12 is not. Therefore, if we start the Lie algebra G2 to introduce Lax pairs, some new integrable hierarchies could be generated by employing the Tu scheme, which were not integrable couplings of some known integrable systems. In what follows, we want to make use of a loop algebra of the Lie algebra G2 to introduce a Lax pair so that the second expanding hierarchy of the D-AKNS hierarchy could be derived by the Tu scheme. A loop algebra of the Lie algebra G2 is defined as
(44)G~2=span⁡{g1(n),…,g8(n)},
where
(45)gi(n)=giλn,i=1,2,…,8;[gi(m),gj(n)]=[gi,gj]λm+n,1≤i,j≤8;m,n∈Z.
Consider a Lax pair
(46)U=g2(1)+sg2(0)+qg3(0)+rg4(0)+v1g7(0)+v2g8(0)+v3g5(0),V=∑m≥0‍(∑i=58‍V1mg1(-m)+V2mg3(-m)+V3mg4(-m)mmmh+V4mg2(-m)+∑i=58‍Vimgi(-m)).
Solving the stationary zero curvature equation
(47)Vx=[U,V]
yields that
(48)(V1m)x=qV3m-rV2m+v1V8m-v2V7m=-(V4m)x,V2,m+1=-(V2m)x-sV2m-qV1m+qV4m-v1V5m+v1V6m+v3V7m,(V3m)x=V3,m+1+sV3m+rV1m-rV4m+v2V5m-v2V6m-v3V8m,(V5m)x=qV8m-rV7m+v1V3m-v2V2m=-(V6m)x,V7,m+1=-(V7m)x-qV5m+qV6m-v1V1m+v1V4m+v3V2m-sV7m,(V8m)x=V8,m+1+sV8m+rV5m-rV6m+v2V1m+v2V1m-v2V4m-v3V3m.
Taking
(49)V(n)=∑m=0n‍(∑i=58‍V1mg1(n-m)+V2mg3(n-m)+V3mg4(n-m)mmmh+V4mg2(n-m)+∑i=58‍Vimgi(n-m))+(V1n-V4n)f2(0)+(V6n-V5n)f5(0),
similar to the above discussion, we obtain the integrable hierarchy
(50)utn=(qrsv1v2v3)tn=((V2n)x+sV2n-v3V7n(V3n)x-sV3n+v3V8n-2(V4n)x(V7n)x+sV7n-v3V2n(V8n)x-sV8n+v3V3n-2(V5n)x).
Obviously, (50) is different from (34). However, when v3=u3=0 in (50) and (34), respectively, they all reduce to the same equation hierarchy (below we will see it). When v1=v2=v3=0, (50) reduces to the D-AKNS hierarchy. Therefore, (50) is the second expanding integrable model of the D-AKNS hierarchy.In the following, we consider the Hamiltonian structure of (50). Through direct calculation, we get that
(51)〈V,Uq〉=2V3,〈V,Ur〉=2V2,〈V,Us〉=2V4,〈V,Uv1〉=2V8,〈V,Uv2〉=2V7,〈V,Uv3〉=2V5,〈V,Uλ〉=2V4,
and here 〈a,b〉=tr⁡(ab),  Vi=∑m≥0Vimλ-m,   i=1,2,…,8.Substituting the above results into the trace identity gives(52)δδu(2V4)=λ-γ∂∂λλγ(2V32V22V42V82V72V5).
Comparing the coefficients of λ-n-1 leads to
(53)δδu(2V4,n+1)=(-n+γ)(2V3n2V2n2V4n2V8n2V7n2V5n).
Similar to the above discussion, we get γ=0. Thus, one gets that
(54)(2V3n2V2n2V4n2V8n2V7n2V5n)=δHnδu,Hn=-2V4,n+1n.
Equation (50) can be writtten as
(55)utn=(qrsv1v2v3)tn=(0∂+s200-v320∂-s200v320000-∂0000-v3200∂+s20v3200∂-s20000000-∂)(2V3n2V2n2V4n2V8n2V7n2V5n)=J2δHnδu.
When v3=0, (55) reduces to
(56)u-tn=(qrsv1v2)tn=((V2n)x+sV2n(V-3n)x-sV3n-2(V4n)x(V7n)x+sV7n(V8n)x-sV8n)=(0∂+s2000∂-s2000000-∂000000∂+s2000∂-s20)(2V3n2V2n2V4n2V8n2V7n)=J3(2V3n2V2n2V4n2V8n2V7n),
which is also an expanding integrable model of the D-AKNS hierarchy.


## 3. The (2+1)-Dimensional D-AKNS Hierarchy and Its Expanding Models

In the section, we will apply the TAH scheme to deduce the (2+1)-dimensional D-AKNS hierarchy and its various expanding models, from which we obtain the united (2+1)-dimensional model of the (2+1)-dimensional D-AKNS hierarchy and the Davey-Stewartson (DS) hierarchy. The key idea presents that 4×4 matrix operators are introduced by our Lie algebras which extend the 2×2 cases in [13]. The work is worth going on because there are few people applying the method to generate new (2+1)-dimensional integrable hierarchies. First of all, we deduce the (2+1)-dimensional D-AKNS hierarchy. Set
(57)U=(0qrλ+ξ+s),V=(ABCD),
where
(58)A=∑m≥0‍Amλ-m,B=∑m≥0‍Bmλ-m,C=∑m≥0‍Cmλ-m,D=∑m≥0‍Dmλ-m.
According to the TAH scheme, we solve the equation of the matrix operators (6) which gives rise to
(59)Ax=qC-Br,Bx=qD-Aq-λB-Bξ-Bs,Cx=rA-Dr+λC+ξC+sC,Dx=rB-Cq+ξD+sD-Dξ-Ds.
Inserting (58) into (59) reads
(60)Anx=qCn-Bnr,Bn+1=-Bnx+qDn-Anq-Bnξ-Dns,Cn+1=Cnx-rAn+Dnr-ξCn-sCn,Dnx-Dny=rBn-Cnq+sDn-Dns.
Taking a few initial values as follows
(61)B0=C0=D0=0,A0=-ξ-1,
one infers from (60) that
(62)B1=qξ-1-qyξ-2+qyyξ-3+O(ξ-4),C1=rξ-1,A1=A-1ξ-2+A~1ξ-3+O(ξ-4),
where ∂xA-1=(qr)y,  ∂xA~1=-qyy, and
(63)D1x-D1y+[D1,s]=0⟹D1=0,B2=-q+(-qx+qy-qs)ξ-1+O(ξ-2),C2=-r+(rx-ry-sr)ξ-1+O(ξ-2),D2x-D2y+[D2,s]=[-(rq)x+(rq)y+[-rq,s]]ξ-1mmmmmmmhmm+O(ξ-2)⟹D2=-(rq)ξ-1+O(ξ-2).
Thus, in terms of step (3) of the TAH scheme and (20), we obtain the (2+1)-dimensional D-AKNS hierarchy
(64)utn=(qrs)tn=((∂+s)R(V2n)(∂-s)R(V3n)-2R(V4n,x)).
When s=0, (64) reduces to the DS hierarchy.Whenn=2, t2=t, (64) reduces to
(65)qt=-qxx+qxy+(qs)x-s(qx+qy+qs),rt=rxx-rxy-(sr)x-s(rx-ry-sr),st=2(rq)x.
Now we consider the Hamiltonian structure of (64). A direct computation gives
(66)∂U∂q=Uq=(0100),Ur=(0010),Us=(0001),Uλ=(0001),〈V,Uq〉=R(V3),〈V,Ur〉=R(V2),〈V,Us〉=R(V4),〈V,Uλ〉=4R(V4).
Substituting the above results into the trace identity (9) yields that
(67)δδu(4R(V4))=(λ-γ∂∂λλγ)(R(V3)R(V2)R(V4)).
Comparing the coefficients of λ-n-1 reads
(68)δδu(4R(V4,n+1))=(-n+γ)(R(V3n)R(V2n)R(V4n)).
Inserting the initial values in (60) gives γ=0. Thus, we have
(69)(R(V3n)R(V2n)R(V4n))=δδu(-4R(V4,n+1)n)≡δHnδu.
Therefore, the (2+1)-dimensional D-AKNS hierarchy (64) can be written as the Hamiltonian form
(70)utn=(qrs)tn=JδHnδu,
where
(71)J=(0∂+s0∂-s0000-2∂).
In what follows, we want to discuss the (2+1)-dimensional expanding models of the (2+1)-dimensional D-AKNS hierarchy (64). Employing the loop algebra of the Lie algebra G2 introduces the following Lax pair:
(72)U=(0q0v1rλ+ξ+sv200v10qv20rλ+ξ+s),V=(ABBA),
where
(73)A=(V1V2V3V4),B=(V5V7V8V6),Vi=∑m≥0‍Vimλ-m,i=1,2,…,8.
Solving the matrix-operator equation (6) yields that
(74)V1x=qV3+v3V5+v1V8-V2r-V5v3-V7v2,V2x=qV4+v3V7+v1V6-V1q-λV2-V2ξ-V2s-V5v1,V5x=qV8+v3V1+v1V3-V1v3-V2v2-V7r,V7x=qV6+v3V2+v1V4-V1v1-V5q-V7λ-V7ξ-V7s,V3x=rV1+λV3+ξV3+sV3+v2V5-V4r-V8v3-V6v2,V4x=rV2+ξV4+sV4+v1V7-V3q-V4ξ-V4s-V8v1,V8x=rV5+λV8+ξV8+sV8+v2V1-V3v3-V4v2-V6r,V6x=rV7+ξV6+sV6+v2V2-V3v1-V8q-V6ξ-V6s.
Substituting (73) into (74) gets that
(75)V1n,x=qV3n+v1V8n-V2nr-V7nv2,V2,n+1=-V2n,x+qV4n+v1V6n-V1nq-V2nξ-V2ns-V5nv1,V5n,x=qV8n+v1V3n-V2nv2-V7nr,V7,n+1=-V7n,x+qV6n+v1V4n-V1nv1-V5nq-V7nξ-V7ns,V3,n+1=∂-V3n-rV1n-V3nξ-sV3n-v2V5n+V4nr+V6nv2,V4n,x=rV2n+V4n,y+sV4n+v2V7n-V3nq-V4ns-V8nv1,V8,n+1=∂-V8n-rV5n-V8nξ-sV8n-v2V1n+V4nv2+V6nr,V6n,x=rV7n+V6n,y+sV6n+v2V2n-V3nv1-V8nq-V6ns,
where ∂-=∂x-∂y.Taking(76)V2,0=V3,0=V4,0=V6,0=V7,0=V8,0=0,V1,0=V5,0=-ξ-1,
then one infers from (60) that
(77)V2,1=(q+v1)ξ-1-(v1y+qy)ξ-2+(qyy+v1yy)ξ-3m+O(ξ-4),V3,1=V8,1=(r+v2)ξ-1,V7,1=(q+v1)ξ-1-(v1y+qy)ξ-2+(v1yy+qyy)ξ-3m+O(ξ-4),V1,1=V-1,1ξ-2+O(ξ-3),
where ∂xV-1,1=(qr+v1r+v1v2+qv2)y,
(78)V5,1=V-5,1ξ-2+O(ξ-3),
where ∂xV-5,1=(v1v2+v1r+qr+qv2)y,
(79)(V4,1)x+[V4,1,s]=0⟹V4,1=0,(V6,1)x+[V6,1,s]=0⟹V6,1=0,V2,2=-q-v1-(qx-qy+v1x-v1y-qs-v1s)ξ-1+O(ξ-2),V7,2=(-v1x-qx+v1y+qy-v1s-qs)ξ-1+O(ξ-2),V3,2=V8,2=-r-v2+(rx+v2x-ry-v2y-sr-sv2)ξ-1+O(ξ-2),(V4,2)x-(V4,2)y+[V4,2,s]=-(rq+rv1+v2v1+v2q)xξ-1+(rq+rv1+v2v1+v2q)yξ-1+[(rq+rv1+v1v1+v2q)smm-s(rq+v2q+rv1+v2v1)]ξ-1+O(ξ-2)⟹V4,2=-(rq+rv1+v2v1+v2q)ξ-1.
According to step (3) of the TAH scheme and (56), we obtain the (2+1)-dimensional equation hierarchy
(80)utn=(qrsv1v2)tn=J3(2R(V3n)2R(V2n)2R(V4n)2R(V8n)2R(V7n)),
where J3 appears in (56).Whenn=2,  t2=t, (80) reduces to the following equations:
(81)qt=-qxx+qxy-v1xx+v1xy+(qs+v1s)x-sqx-sqy-sv1x+sv1y+sqs+sv1s,rt=rxx-rxy+v2xx-v2xy-(sr+sv2)x-srx-sv2x+sry+sv2y+s2r+s2v2,st=2(rq+rv1+v2v1+v2q)x,(v1)t=-qxx+qxy-v1xx+v1xy-(v1s+qs)x-sv1x-sqx+sv1y+sqy-sv1s-sqs,(v2)t=rxx-rxy+v2xx-v2xy-(sr+sv2)x-srx-sv2x+sry+sv2y+s2r+s2v2.
We can regard (81) as a kind of expanding model of (65). That is, when letting s=v1=v2=0, the previous three equations of (81) are just right (65). Therefore, when s=v1=v2=0, (80) can be regarded as a united model of the (2+1)-dimensional D-AKNS hierarchy and the DS hierarchy.Remark.Seeking the Hamiltonian structure of (80) is our ineresting problem. However, if we apply the trace identity (9) to deduce its Hamiltonian structure, we find that it is the same with that in (70). We guess this is not a right result. In forthcoming days, we will go on to consider the problem.



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*Source: 932672-2014-08-14.xml*