# Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated withSO(3)

Authors: Jian Zhang; Chiping Zhang; Yunan Cui
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

Based on the three-dimensional real special orthogonal Lie algebraSO(3), by zero curvature equation, we present bi-integrable and tri-integrable couplings associated with SO(3) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by applying the variational identities.


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## Body

## 1. Introduction

Among the well-known soliton hierarchies are the KdV hierarchy, the AKNS hierarchy, and the Kaup-Newell hierarchy [1]. The trace identity is used for constructing Hamiltonian structures of soliton equations, which is proposed by Tu [2, 3]. In the case of non-semi-simple Lie algebras, integrable couplings of soliton equations are generated by zero curvature equations [4, 5] and the corresponding Hamiltonian structures are obtained by the variational identity [6–8].An integrable coupling equation(1)ut=Ku=Kx,t,u,ux,ut,uxx,uxt,utt,…is a triangular integrable system of the following form [9]: (2)ut=Ku,vt=Su,v,where u is a function of variables t and x, ux=∂u/∂x, and ut=∂u/∂t. If S is nonlinear with respect to the second dependent variable v, the integrable coupling is called nonlinear.An integrable system of the following form [10] (3)ut=Ku,u1,t=S1u,u1,u2,t=S2u,u1,u2is called a bi-integrable of (1).Similarly, an integrable system of the following form [10] (4)ut=Ku,u1,t=S1u,u1,u2,t=S2u,u1,u2,u3,t=S3u,u1,u2,u3is called a tri-integrable of (1).Integrable couplings correspond to non-semi-simple Lie algebrasg¯, and such Lie algebras can be written as semidirect sums [11]: (5)g¯=g⊎gc,g-semisimple,gc-solvable.The notion of semidirect sums g¯=g⊎gc means that g and gc satisfy [g,gc]⊆gc, where [g,gc]={[A,B]∣A∈g,B∈gc}, with [·,·] denoting the Lie bracket of g¯. Obviously, gc is an ideal of g¯. The subscript c indicates a contribution to the construction of coupling systems. We also require the closure property between g and gc under the matrix multiplication: ggc,gcg⊆gc, where ggc={AB∣A∈g,B∈gc}.Integrable couplings are useful tools for describing and explaining nonlinear phenomena of new evaluation equations. There are very rich mathematical structures behind integrable couplings. In particular, integrable couplings generalize the symmetry problem and describe other integrable properties of integrable equations. In order to enrich multicomponent integrable equations, it has been an important task to explore more integrable properties from multi-integrable couplings. For example, one can find work on the integrable couplings [12, 13]. It is always interesting to explore any new procedure for generating integrable couplings for different soliton hierarchies, even from existing non-semi-simple Lie algebras.Recently, seeking new integrable systems including soliton hierarchies and integrable couplings forms a pretty important and interesting area of research in mathematical physics. To generate integrable couplings, bi-integrable couplings and tri-integrable couplings of soliton hierarchies, Ma proposed a new way to generate integrable couplings through a few classes of matrix Lie algebras consisting of block matrices [10]. Recently, bi-integrable couplings and tri-integrable couplings for the KdV hierarchy and the AKNS hierarchy have been studied considerably [14, 15]. From [16, 17], bi-integrable couplings of a new soliton hierarchy associated with SO(3) and bi-integrable couplings of a new soliton hierarchy associated with SO(4) have been studied.In this paper, we will construct bi-integrable and tri-integrable couplings associated withSO(3) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Our work is essentially motivated by [17–19].


## 2. Bi-Integrable Couplings and Hamiltonian Structures

### 2.1. Bi-Integrable Couplings Associated withSO(3)

So as to generate bi-integrable couplings, we introduce a kind of block matrices:(6)M2A,A1,A2=AA1A20A+αA1A1+αA200A+αA1,where α is an arbitrary nonzero constant and A, A1, and A2 are square matrices of the same order. In the following, we define the corresponding non-semi-simple Lie algebra g¯ by a semidirect sum: (7)g¯λ=g⊎gc,with(8)g=M2A,0,0∣A∈SO3~,gc=M20,A1,A2∣A1,A2∈SO3~,where the loop algebra SO(3)~ is defined by (9)SO3~=Aλ∈SO3∣entries  ofAλ-Laurent  series  inλ.Obviously, we have the matrix commutator relation: (10)M2A,A1,A2,M2B,B1,B2=M2C,C1,C2,with C, C1, and C2 being defined by (11)C=A,B,C1=A,B1+A1,B+αA1,B1,C2=A,B2+A2,B+A1,B1+αA1,B2+αA2,B1.Let us consider the Lie algebraSO(3). It has a basis (12)e1=001000-100,e2=00000-1010,e3=010-100000,with which the structure equations of SO(3) are [e1,e2]=e3, [e2,e3]=e1,  [e3,e1]=e2.The soliton hierarchy introduced in [18] has a spectral problem (13)ϕx=Uϕ=Uu,λϕ,u=p,qT,with the spectral matrix U being chosen as(14)U=Uu,λ=λe1+pe2+qe3=0qλ-q0-p-λp0.Based on this special non-semi-simple Lie algebrag¯(λ), we begin with the corresponding enlarged spectral matrix U¯1=M2(U,U1,U2) and let supplementary spectral matrices be(15)U1=U1u1,λ=0q′0-q′0-p′0p′0,u1=p′q′,U2=U2u2,λ=0q′′0-q′′0-p′′0p′′0,u2=p′′q′′.For purpose of solving the enlarged stationary zero curvature equationV¯1x=[U¯1,V¯1], we take V¯1=M2(V,V1,V2), where V is defined as in [18](16)V=Vu,λ=0ca-c0-b-ab0=∑i≥00ciai-ci0-bi-aibi0λ-i,and the supplementary spectral matrices V1 and V2 read(17)V1=V1u,u1,λ=0c′a′-c′0-b′-a′b′0=∑i≥00ci′ai′-ci′0-bi′-ai′bi′0λ-i,V2=V2u,u1,u2,λ=0c′′a′′-c′′0-b′′-a′′b′′0=∑i≥00ci′′ai′′-ci′′0-bi′′-ai′′bi′′0λ-i.The enlarged stationary zero curvature equationV¯1x=[U¯1,V¯1] is equivalent to(18)Vx=U,V,V1x=U,V1+U1,V+αU1,V1,V2x=U,V2+U2,V+U1,V1+αU1,V2+αU2,V1.The above equation system equivalently leads to(19)ax=pc-qb,bx=-λc+qa,cx=λb-pa,ax′=pc′-qb′-q′b+p′c+αp′c′-q′b′,bx′=-λc′+qa′+q′a+αq′a′,cx′=λb′-pa′-p′a-αp′a′,ax′′=-qb′′+pc′′-q′′b+p′′c-q′b′+p′c′+α-q′b′′+p′c′′+α-q′′b′+p′′c′,bx′′=-λc′′+qa′′+q′′a+q′a′+αq′a′′+αq′′a′,cx′′=λb′′-pa′′-p′′a-p′a′-αp′a′′-αp′′a′.Now, we define the enlarged Lax matricesV¯1[m]=(λmV¯1)+=M2(V[m],V1[m],V2[m]),  m≥0, where V[m] is defined as V[m]=(λmV)+, and Vi[m]=(λmVi)+,  i=1,2.Solving the enlarged zero curvature equationsU¯1tm-V¯1x[m]+[U¯1,V¯1[m]]=0, m≥0, we get bi-integrable couplings of the soliton hierarchy in [18](20)u¯tm=pqp′q′p′′q′′tm=-cm+1bm+1-cm+1′bm+1′-cm+1′′bm+1′′=0120000-1200000000120000-1200000000120000-120-2bm+1-2cm+1-2bm+1′-2cm+1′-2bm+1′′-2cm+1′′=J1P1,m+1.


### 2.2. Hamiltonian Structures

In this section, for purpose of generating the Hamiltonian structure of hierarchy (20), we will use the corresponding variational identity [20]:(21)δδu¯∫V¯,U¯λdx=λ-γ∂∂λλγV¯,U¯u,where {·,·} is a required bilinear form, which is symmetric, nondegenerate, and invariant under the Lie bracket.∀
a
=
(
a
1
,
a
2
,
a
3
,
a
1
′
,
a
2
′
,
a
3
′
,
a
1
′
′
,
a
2
′
′
,
a
3
′
′
)
∈
R
9, b=(b1,b2,b3,b1′,b2′,b3′,b1′′,b2′′,b3′′)∈R9; we define the Lie bracket [·,·] on R9 as follows: (22)a,b=aTR¯1b,R¯1b=RbR1bR2b0Rb+αR1bR1b+αR2b00Rb+αR1b,where(23)Rb=0-b3b2b30-b1-b2b10,R1b=0-b3′b2′b3′0-b1′-b2′b1′0,R2b=0-b3′′b2′′b3′′0-b1′′-b2′′b1′′0.Following the properties of the matrixF1: F1(R¯1(b))T=-R¯1(b)F1 and F1=F1T, we get (24)F1=η1η22η3η2αη2+2η32αη32η32αη30⊗r0000r0000r0,where η1,η2,η3,r0 are arbitrary constants. We are easy to have (25)det⁡F1=-64α2η1-αη2+2η33η36r09≠0.In order to get the Hamiltonian structure of the Lax integrable system, we define a bilinear form{a,b} on R9 of the following form:(26)a,b=aTF1b.Now we can compute that(27)V¯1,U¯1,λ=aη1+r0a′η2+2r0a′′η3,V¯1,U¯1,p=r0bη1+r0b′η2+2r0b′′η3,V¯1,U¯1,q=r0cη1+r0c′η2+2r0c′′η3,V¯1,U¯1,p′=r0b+αr0b′η2+2r0b′+2αr0b′′η3,V¯1,U¯1,q′=r0c+αr0c′η2+2r0c′+2αr0c′′η3,V¯1,U¯1,p′′=2r0b+2αr0b′η3,V¯1,U¯1,q′′=2r0c+2αr0c′η3and furthermore, we use the following formular [20]:(28)γ=-λ2ddλln⁡V¯,V¯,to obtain that γ=0. Applying the corresponding variational identity, we obtain the following Hamiltonian structure for the hierarchy of bi-integrable coupling (20): (29)u¯tm=K¯1,mu¯=J¯1δH¯1,mδu¯;m≥0,where the Hamiltonian operator is (30)J¯1=η1η22η3η2αη2+2η32αη32η32αη30-1⊗012-120,and the Hamiltonian functions read (31)H¯1,m=-∫am+2η1+r0am+2′η2+2r0am+2′′η3m+1dx,m≥0.Based on (19), a direct computation yields a recursion relation: (32)P1,m+1=L¯1P1,m,where (33)L¯1=M2TL1,L11,L21=L100L11L1+αL110L21L11+αL21L1+αL11,with L1, L11, and L21 being defined by(34)L1=l11l12l21l22,L11=l11′l12′l21′l22′,(35)L21=l11′′l12′′l21′′l22′′,(36)l11=-p∂-1q,l12=∂+p∂-1p,l21=-∂-q∂-1q,l22=q∂-1p,l11′=-p∂-1q′-αp′∂-1q′-p′∂-1q,l12′=p∂-1p′+αp′∂-1p′+p′∂-1p,l21′=-q∂-1q′-αq′∂-1q′-q′∂-1q,l22′=q∂-1p′+αq′∂-1p′+q′∂-1p,l11′′=-p∂-1q′′-αp′∂-1q′′-p′∂-1q′-αp′′∂-1q′-p′′∂-1q,l12′′=p∂-1p′′+αp′∂-1p′′+p′∂-1p′+αp′′∂-1p′+p′′∂-1p,l21′′=-q∂-1q′′-αq′∂-1q′′-q′∂-1q′-αq′′∂-1q′-q′′∂-1q,l22′′=q∂-1p′′+αq′∂-1p′′+q′∂-1p′+αq′′∂-1p′+q′′∂-1p,where ∂=d/dx and ∂-1=∫‍(d/dx)dx.


## 2.1. Bi-Integrable Couplings Associated withSO(3)

So as to generate bi-integrable couplings, we introduce a kind of block matrices:(6)M2A,A1,A2=AA1A20A+αA1A1+αA200A+αA1,where α is an arbitrary nonzero constant and A, A1, and A2 are square matrices of the same order. In the following, we define the corresponding non-semi-simple Lie algebra g¯ by a semidirect sum: (7)g¯λ=g⊎gc,with(8)g=M2A,0,0∣A∈SO3~,gc=M20,A1,A2∣A1,A2∈SO3~,where the loop algebra SO(3)~ is defined by (9)SO3~=Aλ∈SO3∣entries  ofAλ-Laurent  series  inλ.Obviously, we have the matrix commutator relation: (10)M2A,A1,A2,M2B,B1,B2=M2C,C1,C2,with C, C1, and C2 being defined by (11)C=A,B,C1=A,B1+A1,B+αA1,B1,C2=A,B2+A2,B+A1,B1+αA1,B2+αA2,B1.Let us consider the Lie algebraSO(3). It has a basis (12)e1=001000-100,e2=00000-1010,e3=010-100000,with which the structure equations of SO(3) are [e1,e2]=e3, [e2,e3]=e1,  [e3,e1]=e2.The soliton hierarchy introduced in [18] has a spectral problem (13)ϕx=Uϕ=Uu,λϕ,u=p,qT,with the spectral matrix U being chosen as(14)U=Uu,λ=λe1+pe2+qe3=0qλ-q0-p-λp0.Based on this special non-semi-simple Lie algebrag¯(λ), we begin with the corresponding enlarged spectral matrix U¯1=M2(U,U1,U2) and let supplementary spectral matrices be(15)U1=U1u1,λ=0q′0-q′0-p′0p′0,u1=p′q′,U2=U2u2,λ=0q′′0-q′′0-p′′0p′′0,u2=p′′q′′.For purpose of solving the enlarged stationary zero curvature equationV¯1x=[U¯1,V¯1], we take V¯1=M2(V,V1,V2), where V is defined as in [18](16)V=Vu,λ=0ca-c0-b-ab0=∑i≥00ciai-ci0-bi-aibi0λ-i,and the supplementary spectral matrices V1 and V2 read(17)V1=V1u,u1,λ=0c′a′-c′0-b′-a′b′0=∑i≥00ci′ai′-ci′0-bi′-ai′bi′0λ-i,V2=V2u,u1,u2,λ=0c′′a′′-c′′0-b′′-a′′b′′0=∑i≥00ci′′ai′′-ci′′0-bi′′-ai′′bi′′0λ-i.The enlarged stationary zero curvature equationV¯1x=[U¯1,V¯1] is equivalent to(18)Vx=U,V,V1x=U,V1+U1,V+αU1,V1,V2x=U,V2+U2,V+U1,V1+αU1,V2+αU2,V1.The above equation system equivalently leads to(19)ax=pc-qb,bx=-λc+qa,cx=λb-pa,ax′=pc′-qb′-q′b+p′c+αp′c′-q′b′,bx′=-λc′+qa′+q′a+αq′a′,cx′=λb′-pa′-p′a-αp′a′,ax′′=-qb′′+pc′′-q′′b+p′′c-q′b′+p′c′+α-q′b′′+p′c′′+α-q′′b′+p′′c′,bx′′=-λc′′+qa′′+q′′a+q′a′+αq′a′′+αq′′a′,cx′′=λb′′-pa′′-p′′a-p′a′-αp′a′′-αp′′a′.Now, we define the enlarged Lax matricesV¯1[m]=(λmV¯1)+=M2(V[m],V1[m],V2[m]),  m≥0, where V[m] is defined as V[m]=(λmV)+, and Vi[m]=(λmVi)+,  i=1,2.Solving the enlarged zero curvature equationsU¯1tm-V¯1x[m]+[U¯1,V¯1[m]]=0, m≥0, we get bi-integrable couplings of the soliton hierarchy in [18](20)u¯tm=pqp′q′p′′q′′tm=-cm+1bm+1-cm+1′bm+1′-cm+1′′bm+1′′=0120000-1200000000120000-1200000000120000-120-2bm+1-2cm+1-2bm+1′-2cm+1′-2bm+1′′-2cm+1′′=J1P1,m+1.


## 2.2. Hamiltonian Structures

In this section, for purpose of generating the Hamiltonian structure of hierarchy (20), we will use the corresponding variational identity [20]:(21)δδu¯∫V¯,U¯λdx=λ-γ∂∂λλγV¯,U¯u,where {·,·} is a required bilinear form, which is symmetric, nondegenerate, and invariant under the Lie bracket.∀
a
=
(
a
1
,
a
2
,
a
3
,
a
1
′
,
a
2
′
,
a
3
′
,
a
1
′
′
,
a
2
′
′
,
a
3
′
′
)
∈
R
9, b=(b1,b2,b3,b1′,b2′,b3′,b1′′,b2′′,b3′′)∈R9; we define the Lie bracket [·,·] on R9 as follows: (22)a,b=aTR¯1b,R¯1b=RbR1bR2b0Rb+αR1bR1b+αR2b00Rb+αR1b,where(23)Rb=0-b3b2b30-b1-b2b10,R1b=0-b3′b2′b3′0-b1′-b2′b1′0,R2b=0-b3′′b2′′b3′′0-b1′′-b2′′b1′′0.Following the properties of the matrixF1: F1(R¯1(b))T=-R¯1(b)F1 and F1=F1T, we get (24)F1=η1η22η3η2αη2+2η32αη32η32αη30⊗r0000r0000r0,where η1,η2,η3,r0 are arbitrary constants. We are easy to have (25)det⁡F1=-64α2η1-αη2+2η33η36r09≠0.In order to get the Hamiltonian structure of the Lax integrable system, we define a bilinear form{a,b} on R9 of the following form:(26)a,b=aTF1b.Now we can compute that(27)V¯1,U¯1,λ=aη1+r0a′η2+2r0a′′η3,V¯1,U¯1,p=r0bη1+r0b′η2+2r0b′′η3,V¯1,U¯1,q=r0cη1+r0c′η2+2r0c′′η3,V¯1,U¯1,p′=r0b+αr0b′η2+2r0b′+2αr0b′′η3,V¯1,U¯1,q′=r0c+αr0c′η2+2r0c′+2αr0c′′η3,V¯1,U¯1,p′′=2r0b+2αr0b′η3,V¯1,U¯1,q′′=2r0c+2αr0c′η3and furthermore, we use the following formular [20]:(28)γ=-λ2ddλln⁡V¯,V¯,to obtain that γ=0. Applying the corresponding variational identity, we obtain the following Hamiltonian structure for the hierarchy of bi-integrable coupling (20): (29)u¯tm=K¯1,mu¯=J¯1δH¯1,mδu¯;m≥0,where the Hamiltonian operator is (30)J¯1=η1η22η3η2αη2+2η32αη32η32αη30-1⊗012-120,and the Hamiltonian functions read (31)H¯1,m=-∫am+2η1+r0am+2′η2+2r0am+2′′η3m+1dx,m≥0.Based on (19), a direct computation yields a recursion relation: (32)P1,m+1=L¯1P1,m,where (33)L¯1=M2TL1,L11,L21=L100L11L1+αL110L21L11+αL21L1+αL11,with L1, L11, and L21 being defined by(34)L1=l11l12l21l22,L11=l11′l12′l21′l22′,(35)L21=l11′′l12′′l21′′l22′′,(36)l11=-p∂-1q,l12=∂+p∂-1p,l21=-∂-q∂-1q,l22=q∂-1p,l11′=-p∂-1q′-αp′∂-1q′-p′∂-1q,l12′=p∂-1p′+αp′∂-1p′+p′∂-1p,l21′=-q∂-1q′-αq′∂-1q′-q′∂-1q,l22′=q∂-1p′+αq′∂-1p′+q′∂-1p,l11′′=-p∂-1q′′-αp′∂-1q′′-p′∂-1q′-αp′′∂-1q′-p′′∂-1q,l12′′=p∂-1p′′+αp′∂-1p′′+p′∂-1p′+αp′′∂-1p′+p′′∂-1p,l21′′=-q∂-1q′′-αq′∂-1q′′-q′∂-1q′-αq′′∂-1q′-q′′∂-1q,l22′′=q∂-1p′′+αq′∂-1p′′+q′∂-1p′+αq′′∂-1p′+q′′∂-1p,where ∂=d/dx and ∂-1=∫‍(d/dx)dx.


## 3. Tri-Integrable Couplings and Hamiltonian Structures

### 3.1. Tri-Integrable Couplings Associated withSO(3)

So as to generate bi-integrable couplings, we introduce a kind of block matrices:(37)M3A,A1,A2,A3=AA1A2A30A+βA1βA2A1+βA300A+βA1+μA2νA2000A+βA1,where β, μ, and ν are arbitrary nonzero constants and A, A1, A2, and A3 are square matrices of the same order. In the following, we define the corresponding non-semi-simple Lie algebra g¯(λ) by a semidirect sum: (38)g¯λ=g⊎gc,with(39)g=M3A,0,0,0∣A∈SO3~,gc=M30,A1,A2,A3∣A1,A2,A3∈SO3~,where the loop algebra SO(3)~ is defined by (40)SO3~=Aλ∈SO3∣entries  ofAλ-Laurent  series  inλ.Obviously, we have the matrix commutator relation: (41)M3A,A1,A2,A3,M3B,B1,B2,B3=M3C,C1,C2,C3,with C,  C1,  C2, and C3 being defined by (42)C=A,B,C1=A,B1+A1,B+βA1,B1,C2=A,B2+A2,B+μA2,B2+βA1,B2+βA2,B1,C3=A,B3+A3,B+βA3,B1+βA1,B3+A1,B1+νA2,B2.We introduce the following enlarged spectral matrix to construct tri-integrable couplings forSO(3) hierarchy: (43)U¯2=U¯2u¯,λ=M3U,U1,U2,U3∈g¯λ,with U=U(u,λ) being defined as in (14), where U1 and U2 are defined by (15), and also the supplementary spectral matrix U3 reads (44)U3=U3u3,λ=0q′′′0-q′′′0-p′′′0p′′′0,u3=p′′′q′′′.As usual, we take a solution of the following form:(45)V¯2=V¯2u¯,λ=M3V,V1,V2,V3=VV1V2V30V+βV1βV2V1+βV300V+βV1+μV2νV2000V+βV1,where V, V1, and V2 are defined by (16) and (17); also V3 reads(46)V3=V3u,u1,u2,u3,λ=0c′′′a′′′-c′′′0-b′′′-a′′′b′′′0,a′′′=∑i≥0ai′′′λ-i,b′′′=∑i≥0bi′′′λ-i,c′′′=∑i≥0ci′′′λ-i,f′′′=∑i≥0fi′′′λ-i,g′′′=∑i≥0gi′′′λ-i.It now follows from the enlarged stationary zero curvature equationV¯2x=[U¯2,V¯2] that (47)Vx=U,V,V1x=U,V1+U1,V+βU1,V1,V2x=U,V2+U2,V+μU2,V2+βU1,V2+βU2,V1,V3x=U,V3+U3,V+βU3,V1+U1,V1+βU1,V3+νU2,V2.The above equation system is equivalent to(48)ax=pc-qb,bx=-λc+qa,cx=λb-pa,ax′=-qb′+pc′-q′b+p′c+β-q′b′+p′c′,bx′=-λc′+qa′+q′a+βq′a′,cx′=λb′-pa′-p′a-βp′a′,ax′′=-qb′′+pc′′-q′′b+p′′c+μ-q′′b′′+p′′c′′+β-q′b′′+p′c′′+β-q′′b′+p′′c′,bx′′=-λc′′+qa′′+q′′a+μq′′a′′+βq′a′′+βq′′a′,cx′′=λb′′-pa′′-p′′a-μp′′a′′-βp′a′′-βp′′a′,ax′′′=-qb′′′+pc′′′-q′′′b+p′′′c+β-q′′′b′+p′′′c′+β-q′b′′′+p′c′′′-q′b′+p′c′+ν-q′′b′′+p′′c′′,bx′′′=-λc′′′+qa′′′+q′′′a+βq′′′a′+βq′a′′′+q′a′+νq′′a′′,cx′′′=λb′′′-pa′′′-p′′′a-βp′′′a′-βp′a′′′-p′a′-νq′′a′′.Now, we define the enlarged Lax matricesV¯2[m]=(λmV¯2)+=M3(V[m],V1[m],V2[m],V3[m]), m≥0, where V[m] is defined as V[m]=(λmV)+, and Vi[m]=(λmVi)+, i=1,2,3.Solving the enlarged zero curvature equationsU¯2tm-V¯2x[m]+[U¯2,V¯2[m]]=0, m≥0, we get tri-integrable couplings of the soliton hierarchy in [18](49)u¯tm=pqp′q′p′′q′′p′′′q′′′tm=-cm+1bm+1-cm+1′bm+1′-cm+1′′bm+1′′-cm+1′′′bm+1′′′=012000000-12000000000012000000-12000000000012000000-12000000000012000000-120-2bm+1-2cm+1-2bm+1′-2cm+1′-2bm+1′′-2cm+1′′-2bm+1′′′-2cm+1′′′=J2P2,m+1.


### 3.2. Hamiltonian Structures

In this section, for the purpose of generating the Hamiltonian structure of the hierarchy (49), we will use the corresponding variational identity [20]: (50)δδu¯∫V¯,U¯λdx=λ-γ∂∂λλγV¯,U¯u,where {·,·} is a required bilinear form, which is symmetric, nondegenerate, and invariant under the Lie bracket.∀
a
=
(
a
1
,
a
2
,
a
3
,
a
1
′
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a
2
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a
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12, b=(b1,b2,b3,b1′,b2′,b3′,b1′′,b2′′,b3′′,b1′′′,b2′′′,b3′′′)∈R12; we define the Lie bracket [·,·] on R12 as follows:(51)a,b=aTR¯2b,where (52)R¯2b=RbR1bR2bR3b0Rb+βR1bβR2bR1b+βR3b00Rb+βR1b+μR2bνR2b000Rb+βR1b,with R(b),R1(b), and R2(b) being defined by (23), and (53)R3b=0-b3′′′b2′′′b3′′′0-b1′′′-b2′′′b1′′′0.Following the properties of the matrixF2, F2(R¯2(b))T=R¯2(b)F2 and F2=F2T, we have (54)F2=η1η2η3η4η2βη2+η4βη3βη4η3βη3μη3+νη40η4βη400⊗r0000r0000r0,where ⊗ is the Kronecker product and η1,η2,η3,η4,r0 are arbitrary constants. It is easy to have(55)det⁡F2=η46β2η1-βη2+η43μη3+νη43r012≠0.In order to get the Hamiltonian structure of the Lax integrable system, we define a bilinear form{a,b} on R12 of the following form:(56)a,b=aTF2b.Now, we further compute that(57)V¯2,U¯2,λ=-ar0η1-a′r0η2-a′′r0η3-a′′′r0η4,V¯2,U¯2,p=r0bη1+r0b′η2+r0b′′η3+r0b′′′η4,V¯2,U¯2,q=-r0cη1-r0c′η2-r0c′′η3-r0c′′′η4,V¯2,U¯2,p′=r0b+r0b′βη2+r0b′′βη3+r0b′+r0b′′′βη4,V¯2,U¯2,q′=-r0c+βr0c′η2+r0c′′βη3+r0c′+βr0c′′′η4,V¯2,U¯2,p′′=r0b+βr0b′+μr0b′′η3+νr0b′′η4,V¯2,U¯2,q′′=-r0c-βr0c′-μr0c′′η3-νr0c′′η4,V¯2,U¯2,p′′′=r0b+βr0b′η4,V¯2,U¯2,q′′′=-r0c-βr0c′η4.We use formula (28) and find that γ=0. Applying the corresponding variational identity, we obtain the following Hamiltonian structure for the hierarchy of tri-integrable couplings (49): (58)u¯tm=K¯2,mu¯=J¯2δH¯2,mδu¯,m≥0,where the Hamiltonian operator is (59)J¯2=η1η2η3η4η2βη2+η4βη3βη4η3βη3μη3+νη40η4βη400-1⊗012-120,and the Hamiltonian functions read (60)H¯2,m=∫r0am+2η1+r0am+2′η2+r0am+2′′η3+r0am+2′′′η4m+1dx,m≥0.Based on (48), a direct computation shows a recursion relation: (61)P2,m+1=L¯2P2,m,where the recursion operator L¯2 is given by (62)L¯2=M3TL1,L11,L31,L41=L1000L11L1+βL1100L31βL31L1+βL11+μL310L41L1+βL41νL31L1+βL11,with L1 and L11 being given as in (34), and (63)L31=x11x12x21x22,L41=y11y12y21y22,x11=-p+μp′′+βp′∂-1q′′-βp′′∂-1q′-p′′∂-1q,x12=p+μp′′+βp′∂-1p′′+βp′′∂-1p′+p′′∂-1p,x21=-q+μq′′+βq′∂-1q′′-βq′′∂-1q′-q′′∂-1q,x22=q+μq′′+βq′∂-1p′′+βq′′∂-1p′+q′′∂-1p,y11=-p+βp′∂-1q′′′-νp′′∂-1q′′-βp′′′+p′∂-1q′-p′′′∂-1q,y12=p+βp′∂-1p′′′+νp′′∂-1p′′+βp′′′+p′∂-1p′+p′′′∂-1p,y21=-q+βq′∂-1q′′′-νq′′∂-1q′′-βq′′′+q′∂-1q′-q′′′∂-1q,y22=q+βq′∂-1p′′′+νq′′∂-1p′′+βq′′′+q′∂-1p′+q′′′∂-1p.


## 3.1. Tri-Integrable Couplings Associated withSO(3)

So as to generate bi-integrable couplings, we introduce a kind of block matrices:(37)M3A,A1,A2,A3=AA1A2A30A+βA1βA2A1+βA300A+βA1+μA2νA2000A+βA1,where β, μ, and ν are arbitrary nonzero constants and A, A1, A2, and A3 are square matrices of the same order. In the following, we define the corresponding non-semi-simple Lie algebra g¯(λ) by a semidirect sum: (38)g¯λ=g⊎gc,with(39)g=M3A,0,0,0∣A∈SO3~,gc=M30,A1,A2,A3∣A1,A2,A3∈SO3~,where the loop algebra SO(3)~ is defined by (40)SO3~=Aλ∈SO3∣entries  ofAλ-Laurent  series  inλ.Obviously, we have the matrix commutator relation: (41)M3A,A1,A2,A3,M3B,B1,B2,B3=M3C,C1,C2,C3,with C,  C1,  C2, and C3 being defined by (42)C=A,B,C1=A,B1+A1,B+βA1,B1,C2=A,B2+A2,B+μA2,B2+βA1,B2+βA2,B1,C3=A,B3+A3,B+βA3,B1+βA1,B3+A1,B1+νA2,B2.We introduce the following enlarged spectral matrix to construct tri-integrable couplings forSO(3) hierarchy: (43)U¯2=U¯2u¯,λ=M3U,U1,U2,U3∈g¯λ,with U=U(u,λ) being defined as in (14), where U1 and U2 are defined by (15), and also the supplementary spectral matrix U3 reads (44)U3=U3u3,λ=0q′′′0-q′′′0-p′′′0p′′′0,u3=p′′′q′′′.As usual, we take a solution of the following form:(45)V¯2=V¯2u¯,λ=M3V,V1,V2,V3=VV1V2V30V+βV1βV2V1+βV300V+βV1+μV2νV2000V+βV1,where V, V1, and V2 are defined by (16) and (17); also V3 reads(46)V3=V3u,u1,u2,u3,λ=0c′′′a′′′-c′′′0-b′′′-a′′′b′′′0,a′′′=∑i≥0ai′′′λ-i,b′′′=∑i≥0bi′′′λ-i,c′′′=∑i≥0ci′′′λ-i,f′′′=∑i≥0fi′′′λ-i,g′′′=∑i≥0gi′′′λ-i.It now follows from the enlarged stationary zero curvature equationV¯2x=[U¯2,V¯2] that (47)Vx=U,V,V1x=U,V1+U1,V+βU1,V1,V2x=U,V2+U2,V+μU2,V2+βU1,V2+βU2,V1,V3x=U,V3+U3,V+βU3,V1+U1,V1+βU1,V3+νU2,V2.The above equation system is equivalent to(48)ax=pc-qb,bx=-λc+qa,cx=λb-pa,ax′=-qb′+pc′-q′b+p′c+β-q′b′+p′c′,bx′=-λc′+qa′+q′a+βq′a′,cx′=λb′-pa′-p′a-βp′a′,ax′′=-qb′′+pc′′-q′′b+p′′c+μ-q′′b′′+p′′c′′+β-q′b′′+p′c′′+β-q′′b′+p′′c′,bx′′=-λc′′+qa′′+q′′a+μq′′a′′+βq′a′′+βq′′a′,cx′′=λb′′-pa′′-p′′a-μp′′a′′-βp′a′′-βp′′a′,ax′′′=-qb′′′+pc′′′-q′′′b+p′′′c+β-q′′′b′+p′′′c′+β-q′b′′′+p′c′′′-q′b′+p′c′+ν-q′′b′′+p′′c′′,bx′′′=-λc′′′+qa′′′+q′′′a+βq′′′a′+βq′a′′′+q′a′+νq′′a′′,cx′′′=λb′′′-pa′′′-p′′′a-βp′′′a′-βp′a′′′-p′a′-νq′′a′′.Now, we define the enlarged Lax matricesV¯2[m]=(λmV¯2)+=M3(V[m],V1[m],V2[m],V3[m]), m≥0, where V[m] is defined as V[m]=(λmV)+, and Vi[m]=(λmVi)+, i=1,2,3.Solving the enlarged zero curvature equationsU¯2tm-V¯2x[m]+[U¯2,V¯2[m]]=0, m≥0, we get tri-integrable couplings of the soliton hierarchy in [18](49)u¯tm=pqp′q′p′′q′′p′′′q′′′tm=-cm+1bm+1-cm+1′bm+1′-cm+1′′bm+1′′-cm+1′′′bm+1′′′=012000000-12000000000012000000-12000000000012000000-12000000000012000000-120-2bm+1-2cm+1-2bm+1′-2cm+1′-2bm+1′′-2cm+1′′-2bm+1′′′-2cm+1′′′=J2P2,m+1.


## 3.2. Hamiltonian Structures

In this section, for the purpose of generating the Hamiltonian structure of the hierarchy (49), we will use the corresponding variational identity [20]: (50)δδu¯∫V¯,U¯λdx=λ-γ∂∂λλγV¯,U¯u,where {·,·} is a required bilinear form, which is symmetric, nondegenerate, and invariant under the Lie bracket.∀
a
=
(
a
1
,
a
2
,
a
3
,
a
1
′
,
a
2
′
,
a
3
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a
1
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′
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a
2
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1
′
′
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a
2
′
′
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a
3
′
′
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∈
R
12, b=(b1,b2,b3,b1′,b2′,b3′,b1′′,b2′′,b3′′,b1′′′,b2′′′,b3′′′)∈R12; we define the Lie bracket [·,·] on R12 as follows:(51)a,b=aTR¯2b,where (52)R¯2b=RbR1bR2bR3b0Rb+βR1bβR2bR1b+βR3b00Rb+βR1b+μR2bνR2b000Rb+βR1b,with R(b),R1(b), and R2(b) being defined by (23), and (53)R3b=0-b3′′′b2′′′b3′′′0-b1′′′-b2′′′b1′′′0.Following the properties of the matrixF2, F2(R¯2(b))T=R¯2(b)F2 and F2=F2T, we have (54)F2=η1η2η3η4η2βη2+η4βη3βη4η3βη3μη3+νη40η4βη400⊗r0000r0000r0,where ⊗ is the Kronecker product and η1,η2,η3,η4,r0 are arbitrary constants. It is easy to have(55)det⁡F2=η46β2η1-βη2+η43μη3+νη43r012≠0.In order to get the Hamiltonian structure of the Lax integrable system, we define a bilinear form{a,b} on R12 of the following form:(56)a,b=aTF2b.Now, we further compute that(57)V¯2,U¯2,λ=-ar0η1-a′r0η2-a′′r0η3-a′′′r0η4,V¯2,U¯2,p=r0bη1+r0b′η2+r0b′′η3+r0b′′′η4,V¯2,U¯2,q=-r0cη1-r0c′η2-r0c′′η3-r0c′′′η4,V¯2,U¯2,p′=r0b+r0b′βη2+r0b′′βη3+r0b′+r0b′′′βη4,V¯2,U¯2,q′=-r0c+βr0c′η2+r0c′′βη3+r0c′+βr0c′′′η4,V¯2,U¯2,p′′=r0b+βr0b′+μr0b′′η3+νr0b′′η4,V¯2,U¯2,q′′=-r0c-βr0c′-μr0c′′η3-νr0c′′η4,V¯2,U¯2,p′′′=r0b+βr0b′η4,V¯2,U¯2,q′′′=-r0c-βr0c′η4.We use formula (28) and find that γ=0. Applying the corresponding variational identity, we obtain the following Hamiltonian structure for the hierarchy of tri-integrable couplings (49): (58)u¯tm=K¯2,mu¯=J¯2δH¯2,mδu¯,m≥0,where the Hamiltonian operator is (59)J¯2=η1η2η3η4η2βη2+η4βη3βη4η3βη3μη3+νη40η4βη400-1⊗012-120,and the Hamiltonian functions read (60)H¯2,m=∫r0am+2η1+r0am+2′η2+r0am+2′′η3+r0am+2′′′η4m+1dx,m≥0.Based on (48), a direct computation shows a recursion relation: (61)P2,m+1=L¯2P2,m,where the recursion operator L¯2 is given by (62)L¯2=M3TL1,L11,L31,L41=L1000L11L1+βL1100L31βL31L1+βL11+μL310L41L1+βL41νL31L1+βL11,with L1 and L11 being given as in (34), and (63)L31=x11x12x21x22,L41=y11y12y21y22,x11=-p+μp′′+βp′∂-1q′′-βp′′∂-1q′-p′′∂-1q,x12=p+μp′′+βp′∂-1p′′+βp′′∂-1p′+p′′∂-1p,x21=-q+μq′′+βq′∂-1q′′-βq′′∂-1q′-q′′∂-1q,x22=q+μq′′+βq′∂-1p′′+βq′′∂-1p′+q′′∂-1p,y11=-p+βp′∂-1q′′′-νp′′∂-1q′′-βp′′′+p′∂-1q′-p′′′∂-1q,y12=p+βp′∂-1p′′′+νp′′∂-1p′′+βp′′′+p′∂-1p′+p′′′∂-1p,y21=-q+βq′∂-1q′′′-νq′′∂-1q′′-βq′′′+q′∂-1q′-q′′′∂-1q,y22=q+βq′∂-1p′′′+νq′′∂-1p′′+βq′′′+q′∂-1p′+q′′′∂-1p.


## 4. Conclusion

In this paper, we take advantage of the non-semi-simple Lie algebras consisting of3×3, 4×4 block matrices and apply them to the construction of bi-integrable couplings and tri-integrable couplings associated with SO(3), based on the enlarged zero curvature equations. According to the associated variational identities, their Hamiltonian structures can be generated.We can think about other related issues, for example, how we can get integrable couplings and their Hamiltonian structures when irreducible representations ofSO(3) and SO(4) are used to form matrix loop algebras. In addition, we can also consider the relations between the hierarchy of tri-integrable couplings associated with SO(3) and the hierarchy of tri-integrable couplings associated with SO(4).



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*Source: 9743475-2017-06-04.xml*