# Bi-Integrable Couplings of a New Soliton Hierarchy Associated withSO(4)

Authors: Yan Cao; Liangyun Chen; Baiying He
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

Based on the six-dimensional real special orthogonal Lie algebraSO(4), a new Lax integrable hierarchy is obtained by constructing an isospectral problem. Furthermore, we construct bi-integrable couplings for this hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Hamiltonian structures of the obtained bi-integrable couplings are constructed by the variational identity.


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## Body

## 1. Introduction

As is well known, integrable equations are a remarkable class of nonlinear equations. To enrich multicomponent integrable equations, it has been an important task to explore more integrable properties for multi-integrable couplings. For example, one can find work on the integrable couplings [1, 2]. It is always interesting to explore any new procedure for generating integrable couplings for different soliton hierarchies, even from existing nonsemisimple Lie algebras.The trace identity provides a simple and effective way to search for Hamiltonian structures of the hierarchies, which is proposed by Tu [3, 4]. However the trace identity is based on the semisimple Lie algebras. It is no longer applicable, when the trace identity comes to the nonsemisimple Lie algebras. So the variational identity is proposed, which can be used to obtain the Hamiltonian structures in the case of nonsemisimple Lie algebras [5–7].Searching for integrable couplings of systems will become more and more meaningful in the area of research in mathematical physics. In [8], Ma et al. propose a new way to generate integrable couplings through a few classes of matrix Lie algebras consisting of block matrices. Recently, bi-integrable couplings and tri-integrable couplings for the KDV hierarchy and the AKNS hierarchy have been studied considerably [9, 10]. Bi-integrable couplings of a new soliton hierarchy associated with SO(3) have been constructed [11].An integrable coupling of a given system(1)ut=Ku=Kx,t,u,ux,ut,uxx,uxt,utt,…is a triangular integrable system of the following form [12]:(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.A bi-integrable coupling of given integrable system (1) is an enlarged triangular integrable system of the following form [8]: (3)ut=Ku,u1,t=S1u,u1,u2,t=S2u,u1,u2.Integrable couplings correspond to nonsemisimple Lie algebrasg-, and such Lie algebras can be written as semidirect sums [13]: (4)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}.In this paper, we will construct bi-integrable couplings associated withSO(4) for a new hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Our work is essentially motivated by [11, 14, 15].


## 2. A New Soliton Hierarchy

Firstly, we will use the six-dimensional real special orthogonal Lie algebraSO(4) [16] to construct a new soliton hierarchy and consider corresponding Hamiltonian structure by means of the trace identity [3, 4]. This Lie algebra G has the following basis:(5)e1=000-1000000001000,e2=000000-1001000000,e3=0-100100000000000,e4=00000000000-10010,e5=00100000-10000000,e6=0000000-100000100.Set [a,b]=ab-ba,∀a,b∈G. Then the circular commutator relations are (6)e1,e2=0,e1,e3=-e6,e1,e4=-e5,e1,e5=e4,e1,e6=e3,e2,e3=-e5,e2,e4=-e6,e2,e5=e3,e2,e6=e4,e3,e4=0,e3,e5=-e2,e3,e6=-e1,e4,e5=-e1,e4,e6=-e2,e5,e6=0.The corresponding loop algebra G~ is given by (7)G~=e1n,e2n,e3n,e4n,e5n,e6n,ein=eiλn,i=1,…,6,along with the commutators [ei(m),ej(n)]=[ei,ej]λm+n,m,n∈Z,1≤i,j≤6.Consider an isospectral problem(8)φx=Uφ,φt=Vφ,λt=0,where U,V are defined as (9)U=Uu,λ=0-u2u4-u1u20-λ-u5-u4λ0-u3u1u5u30,V=Vu,λ=0-cf-ac0-b-g-fb0-dagd0,(10)a=∑i≥0aiλ-i,b=∑i≥0biλ-i,c=∑i≥0ciλ-i,d=∑i≥0diλ-i,f=∑i≥0fiλ-i,g=∑i≥0giλ-i.The stationary zero curvature representationVx=[U,V] gives (11)ax=-u2g-u3f+u4d+u5c,bx=-u2f-u3g+u4c+u5d,cx=λf+u1g-u4b-u5a,dx=λg+u1f-u4a-u5b,fx=-λc-u1d+u2b+u3a,gx=-λd-u1c+u2a+u3b.Take the initial values(12)a0=α,b0=β,c0=d0=f0=g0=0.From (11), we have (13)a1=b1=0,c1=αu3+βu2,d1=αu2+βu3,f1=αu5+βu4,g1=αu4+βu5,a2=-βu2u3+u4u5-α2u22+u32+u42+u52,c2=-αu5x-βu4x-u1αu2+βu3,b2=-αu2u3+u4u5-β2u22+u32+u42+u52,d2=-αu4x-βu5x-u1αu3+βu2,f2=αu3x+βu2x-u1αu4+βu5,g2=αu2x+βu3x-u1αu5+βu4.Then the zero curvature equationUtm-Vx[m]+[U,V[m]]=0 leads to the following Lax integrable hierarchy: (14)utm=u1u2u3u4u5tm=-amxcmx-u1gm+u4bmdmx-u1fm+u5bmfmx+u1dm-u2bmgmx+u1cm-u3bm=J1-2am-2cm-2dm-2fm-2gm=J1P1,m,where(15)J1=12∂00000-∂+u4∂-1u4-u4∂-1u5u4∂-1u2E0-u5∂-1u4-∂+u5∂-1u5u5∂-1u2+u1F0u2∂-1u4-u1-u2∂-1u5-∂+u2∂-1u2G0-u1-u3∂-1u4u3∂-1u5-u3∂-1u2H,(16)E=u4∂-1u3+u1,F=u5∂-1u3,G=-u2∂-1u3,H=-∂+u3∂-1u3,where ∂=d/dx and ∂-1=∫(d/dx)dx.From recurrence relations (11), we have (17)P1,m=0l12l13l14l15u3u2∂-1u4l23l24-u2∂-1u3u2l32u3∂-1u5-u3∂-1u2l35u5l42u4∂-1u5-u4∂-1u2l45u4u5∂-1u4l53l54-u5∂-1u3P1,m-1=L1P1,m-1,where L1 is a recurrence operator and (18)l12=∂-1-u3∂-u4u1,l13=∂-1-u2∂-u5u1,l14=∂-1u2u1-u5∂,l15=∂-1u3u1-u4∂,l23=-u1+u2∂-1u5,l24=-∂-u2∂-1u2,l32=-u1+u3∂-1u4,l35=-∂-u3∂-1u3,l42=∂+u4∂-1u4,l45=-u1-u4∂-1u3,l53=∂+u5∂-1u5,l54=-u1-u5∂-1u2.In what follows, we will construct the Hamiltonian structure for soliton hierarchy (14) by the trace identity [3, 4]: (19)δδu∫tr∂U∂λVdx=λ-γ∂∂λλγtr∂U∂uV,γ=λ2ddλln⁡trV2.In thisSO(4) case, we can easily obtain (20)trV∂U∂λ=-2b,trV∂U∂u1=-2a,trV∂U∂u2=-2c,trV∂U∂u3=-2d,trV∂U∂u4=-2f,trV∂U∂u5=-2g.Substituting the above equations into trace identity (19) yields (21)δδu∫-2bdx=λ-γ∂∂λλγ-2a-2c-2d-2f-2g.Balancing coefficients of each power ofλ in the above equality gives rise to (22)δδu∫-2bm+1dx=γ-m-2am-2cm-2dm-2fm-2gm.Checking the case withm=1 gives γ=0. Thus, we get the Hamiltonian structure for soliton hierarchy (14): (23)utm=K1,mu=-amxcmx-u1gm+u4bmdmx-u1fm+u5bmfmx+u1dm-u2bmgmx+u1cm-u3bm=J1δH1,mδu,H1,m=2∫bm+1mdx,m≥1.It is said that hierarchy (14) has the Hamiltonian structure, and it is easy to verity that J1L1=L1∗J1. Therefore, hierarchy (14) is Liouville integrable.


## 3. Bi-Integrable Couplings and Hamiltonian Structures

### 3.1. Bi-Integrable Couplings Associated withSO(4)

We take the class of3×3 block matrices in the form of(24)M2A1,A2,A3=A1A2A30A1αA200A1,where α is an arbitrary nonzero constant and A1, A2, and A3 are square matrices of the same order. In the following, we define the corresponding nonsemisimple Lie algebra g- by a semidirect sum (25)g-λ=g⊎gc,with(26)g=M2A1,0,0∣A∈SO4~,gc=M20,A2,A3∣A2,A3∈SO4~,where loop algebra SO(4)~ is defined by (27)SO4~=Aλ∈SO4∣entries  ofAλ—Laurent  series  inλ.The corresponding matrix product reads(28)M2A1,A2,A3,M2B1,B2,B3=M2C1,C2,C3,with C1, C2, and C3 being defined by (29)C1=A1,B1,C2=A1,B2+A2,B1,C3=A1,B3+A3,B1+αA2,B2.Based on this special nonsemisimple Lie algebrag-(λ), we choose the corresponding enlarged spectral matrix U-1=M2(U,U1,U2), where U is defined as in (9) U1 and U2 are defined as follows: (30)U1=U1u1,λ=0-u2′u4′-u1′u2′00-u5′-u4′00-u3′u1′u5′u3′0,u1=u1′u2′u3′u4′u5′,U2=U2u2,λ=0-u2′′u4′′-u1′′u2′′00-u5′′-u4′′00-u3′′u1′′u5′′u3′′0,u2=u1′′u2′′u3′′u4′′u5′′.In order to solve the enlarged stationary zero curvature equationsV-1x=[U-1,V-1], we take V-1=M2(V,V1,V2), where V is defined by (9): (31)V=Vu,λ=0-cf-ac0-b-g-fb0-dagd0=∑i≥00-cifi-aici0-bi-gi-fibi0-diaigidi0λ-i.V1 and V2 are defined similarly:(32)V1=V1u,u1,λ=0-c′f′-a′c′0-b′-g′-f′b′0-d′a′g′d′0=∑i≥00-ci′fi′-ai′ci′0-bi′-gi′-fi′bi′0-di′ai′gi′di′0λ-i,V2=V2u,u1,u2,λ=0-c′′f′′-a′′c′′0-b′′-g′′-f′′b′′0-d′′a′′g′′d′′0=∑i≥00-ci′′fi′′-ai′′ci′′0-bi′′-gi′′-fi′′bi′′0-di′′ai′′gi′′di′′0λ-i.It now follows from the enlarged stationary zero curvature equationsV-1x=[U-1,V-1] that (33)Vx=U,V,V1x=U,V1+U1,V,V2x=U,V2+U2,V+αU1,V1.The above equation system equivalently leads to (11) and the following equations:(34)ax′=u5c′-u3f′-u2g′+u4d′+u5′c-u3′f-u2′g+u4′d,bx′=u4c′-u3g′-u2f′+u5d′+u4′c-u3′g-u2′f+u5′d,cx′=λf′-u5a′-u4b′+u1g′-u5′a-u4′b+u1′g,dx′=λg′-u4a′-u5b′+u1f′-u4′a-u5′b+u1′f,fx′=-λc′+u2b′+u3a′-u1d′+u2′b+u3′a-u1′d,gx′=-λd′+u2a′+u3b′-u1c′+u2′a+u3′b-u1′c,ax′′=u5c′′-u3f′′-u2g′′+u4d′′+u5′′c-u3′′f-u2′′g+u4′′d+αu5′c′-u3′f′-u2′g′+u4′d′,bx′′=u4c′′-u3g′′-u2f′′+u5d′′+u4′′c-u3′′g-u2′′f+u5′′d+αu4′c′-u3′g′-u2′f′+u5′d′,cx′′=λf′′-u5a′′-u4b′′+u1g′′-u5′′a-u4′′b+u1′′g-αu5′a′+u4′b′-u1′g′,dx′′=λg′′-u4a′′-u5b′′+u1f′′-u4′′a-u5′′b+u1′′f-αu4′a′+u5′b′-u1′f′,fx′′=-λc′′+u2b′′+u3a′′-u1d′′+u2′′b+u3′′a-u1′′d+αu2′b′+u3′a′-u1′d′,gx′′=-λd′′+u2a′′+u3b′′-u1c′′+u2′′a+u3′′b-u1′′c+αu2′a′+u3′b′-u1′c′.Now, let us 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)+,m≥0, 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 for soliton hierarchy (14): (35)u-tm=utmu1tmu2tm=J100J2J10J3αJ2J1P1mP2mP3m=J100J2J10J3αJ2J1P-1,m,with utm, P1m, and J1 being defined by (14) and (15), with u1tm, u2tm, P2m, P3m, J2, and J3 being defined as follows:(36)u1tm=u1′u2′u3′u4′u5′tm=-amx′cmx′-u1gm′+u4bm′-u1′gm+u4′bmdmx′-u1fm′+u5bm′-u1′fm+u5′bmfmx′+u1dm′-u2bm′+u1′dm-u2′bmgmx′+u1cm′-u3bm′+u1′cm-u3′bm,u2tm=u1′′u2′′u3′′u4′′u5′′tm=-amx′′cmx′′+u4bm′′-u1gm′′+u4′′bm-u1′′gm+αu4′bm′-u1′gm′dmx′′+u5bm′′-u1fm′′+u5′′bm-u1′′fm+αu5′bm′-u1′fm′fmx′′-u2bm′′+u1dm′′-u2′′bm+u1′′dm+αu1′dm′-u2′bm′gmx′′-u3bm′′+u1cm′′-u3′′bm+u1′′cm+αu1′cm′-u3′bm′,P2m=-2am′-2cm′-2dm′-2fm′-2gm′,P3m=-2am′′-2cm′′-2dm′′-2fm′′-2gm′′,J2=12∂00000-u4∂-1u4′+u4′∂-1u4-u4′∂-1u5+u4∂-1u5′B1B20-u5∂-1u4′+u5′∂-1u4-u5∂-1u5′+u5′∂-1u5B3B40u2∂-1u4′+u2′∂-1u4u2∂-1u5′+u2′∂-1u5-u1′B5B60u3∂-1u4′+u3′∂-1u4u3∂-1u5′+u3′∂-1u5B7B8,where (37)B1=u4∂-1u2′+u4′∂-1u2,B2=u4∂-1u3′+u4′∂-1u3+u1′,B3=u5∂-1u2′+u5′∂-1u2+u1′,B4=u5∂-1u3′+u5′∂-1u3,B5=-u2∂-1u2′+u2′∂-1u2,B6=-u2∂-1u3′+u2′∂-1u3,B7=-u3∂-1u2′+u3′∂-1u2,B8=-u3∂-1u3′+u3′∂-1u3,(38)J3=12∂00000-u4∂-1u4′′+u4′′∂-1u4+αu4′∂-1u4′C1C2C30-u5∂-1u4′′+u5′′∂-1u4+αu5′∂-1u4′C4C5C60u2∂-1u4′′+u2′′∂-1u4+αu2′∂-1u4′C7C8C90u3∂-1u4′′+u3′′∂-1u4+αu3′∂-1u4′-u1′′C10C11C12,where(39)C1=-u4∂-1u5′′+u4′′∂-1u5+αu4′∂-1u5′,C2=u4∂-1u2′′+u4′′∂-1u2+αu4′∂-1u2′,C3=u4∂-1u3′′+u4′′∂-1u3+αu4′∂-1u3′+u1′′,C4=-u5∂-1u5′′+u5′′∂-1u5+αu5′∂-1u5′,C5=u5∂-1u2′′+u5′′∂-1u2+αu5′∂-1u2′+u1′′,C6=u5∂-1u3′′+u5′′∂-1u3+αu5′∂-1u3′,C7=u2∂-1u5′′+u2′′∂-1u5+αu2′∂-1u5′-u1′′,C8=-u2∂-1u2′′+u2′′∂-1u2+αu2′∂-1u2′,C9=-u2∂-1u3′′+u2′′∂-1u3+αu2′∂-1u3′,C10=u3∂-1u5′′+u3′′∂-1u5+αu3′∂-1u5′,C11=-u3∂-1u2′′+u3′′∂-1u2+αu3′∂-1u2′,C12=-u3∂-1u3′′+u3′′∂-1u3+αu3′∂-1u3′.


### 3.2. Hamiltonian Structures

In this section, in order to generate the Hamiltonian structures of hierarchy (35), we use the corresponding variational identity [17] (40)δδu-∫V-,U-λdx=λ-γ∂∂λλγV-,U-u,where {·,·} is a required bilinear form, which is symmetric, nondegenerate, and invariant under the loop algebra g-(λ).First, we need to define a mapping [8] (41)σ:g-λ⟶R18,A⟼a1,a2,…,a18T,where(42)A=MA1,A2,A3∈g-λ,Ai=0-a6i-3a6i-1-a6i-5a6i-30-a6i-4-a6i-a6i-1a6i-40-a6i-2a6i-5a6ia6i-20,i=1,2,3.This mapping σ induces a Lie algebraic structure on R18. The Lie bracket [·,·] on R18 can be computed as follows: (43)a,b=aTR-1b,∀a=a1,…,a18T,b=b1,…,b18T∈R18,where (44)R-1b=M2R1,R2,R3,Ri=00b6ib6i-1-b6i-2-b6i-300b6i-1b6i-b6i-3-b6i-2-b6i-b6i-100b6i-4b6i-5-b6i-1-b6i00b6i-5b6i-4b6i-2b6i-3-b6i-4-b6i-500b6i-3b6i-2-b6i-5-b6i-400,i=1,2,3.Mapping (41) is a Lie algebra isomorphism between the two Lie algebras.Then we define a bilinear form onR18 as follows: (45)a,b=aTF1b,∀a=a1,…,a18T,b=b1,…,b18T∈R18.Following the properties of the matrix F1: F1(R-1(b))T=-R-1(b)F1 and F1=F1T, we have (46)F1=η1η2η3η2αη30η300⊗r1r20000r2r1000000r1r20000r2r1000000r1r20000r2r1,where η1,η2,η3,r1, and r2 are arbitrary constants. We choose r1≠±r2; then the nondegenerate condition requires (47)detF1=α6η318r12-r229≠0.So, the bilinear form on the semidirect sumg-(λ) is given by(48)A,B=a1,…,a18F1b1,…,b18=r1a1b1+r2a2b1+r2a1b2+r1a2b2+r1a3b3+r2a4b3+r2a3b4+r1a4b4+r1a5b5+r2a6b5+r2a5b6+r1a6b6η1+r1a7b1+r2a8b1+r2a7b2+r1a8b2+r1a9b3+r2a10b3+r2a9b4+r1a10b4+r1a11b5+r2a12b5+r2a11b6+r1a12b6+r1a1b7+r2a2b7+r2a1b8+r1a2b8+r1a3b9+r2a4b9+r2a3b10+r1a4b10+r1a5b11+r2a6b11+r2a5b12+r1a6b12η2+r1a13b1+r2a14b1+r2a13b2+r1a14b2+r1a15b3+r2a16b3+r2a15b4+r1a16b4+r1a17b5+r2a18b5+r1a18b6+r2a17b6+αr1a7b7+αr2a8b7+αr2a7b8+αr1a8b8+αr1a9b9+αr2a10b9+αr2a9b10+αr1a10b10+αr1a11b11+αr2a12b11+αr2a11b12+αr1a12b12+r1a1b13+r2a2b13+r2a1b14+r1a2b14+r1a3b15+r2a4b15+r2a3b16+r1a4b16+r1a5b17+r2a6b17+r2a5b18+r1a6b18η3.As mentioned above, now we can compute that(49)V-1,U-1,λ=r2a+r1bη1+r2a′+r1b′η2+r2a′′+r1b′′η3,V-1,U-1,u1=r1a+r2bη1+r1a′+r2b′η2+r1a′′+r2b′′η3,V-1,U-1,u2=r1c+r2dη1+r1c′+r2d′η2+r1c′′+r2d′′η3,V-1,U-1,u3=r2c+r1dη1+r2c′+r1d′η2+r2c′′+r1d′′η3,V-1,U-1,u4=r1f+r2gη1+r1f′+r2g′η2+r1f′′+r2g′′η3,V-1,U-1,u5=r2f+r1gη1+r2f′+r1g′η2+r2f′′+r1g′′η3,V-1,U-1,u1′=r1a+r2bη2+αr1a′+r2b′η3,V-1,U-1,u1′′=r2b+r1aη3,V-1,U-1,u2′=r1c+r2dη2+αr1c′+r2d′η3,V-1,U-1,u2′′=r1c+r2dη3,V-1,U-1,u3′=r2c+r1dη2+αr2c′+r1d′η3,V-1,U-1,u3′′=r2c+r1dη3,V-1,U-1,u4′=r1f+r2gη2+αr1f′+r2g′η3,V-1,U-1,u4′′=r1f+r2gη3,V-1,U-1,u5′=r2f+r1gη2+αr2f′+r1g′η3,V-1,U-1,u5′′=r2f+r1gη3.and furthermore we use the formula [17] (50)γ=-λ2ddλln⁡V-,V-,to obtain that γ=0. Applying corresponding variational identity (40), we obtain the following Hamiltonian structures for the hierarchy of bi-integrable couplings (35): (51)u-tm=K-1,mu-=J-1δH-1,mδu-;m≥1,with the Hamiltonian operator (52)J-1=η1η2η3η2αη30η300-1⊗J1,where J1 is defined by (15) and the Hamiltonian functionals (53)H-1,m=-∫r2am+1+r1bm+1η1+r1am+1′+r2bm+1′η2+r2am+1′′+r1bm+1′′η3mdx,m≥1.Based on (11) and (34), a direct computation yields a recursion relation (54)P1,m=L-1P1,m-1,where (55)L-1=M2TL1,L11,L21=L100L11L10L21αL11L1,with L1 being defined by (17) L11 and L21 are defined as follows:(56)L11=0∂-1-u4u1′-u4′u1-u3′∂l13′l14′l15′u3′u2∂-1u4′+u2′∂-1u4l23′l24′l25′u2′u3∂-1u4′+u3′∂-1u4-u1′l33′l34′l35′u5′u4∂-1u4′+u4′∂-1u4l43′l44′l45′u4′u5∂-1u4′+u5′∂-1u4l53′l54′l55′,(57)l13′=∂-1-u5u1′-u5′u1-u2′∂,l14′=∂-1u2u1′+u2′u1-u5′∂,l15′=∂-1u3u1′+u3′u1-u4′∂,l23′=u2∂-1u5′+u2′∂-1u5-u1′,l24′=-u2∂-1u2′-u2′∂-1u2,l25′=-u2∂-1u3′-u2′∂-1u3,l33′=u3∂-1u5′+u3′∂-1u5,l34′=-u3∂-1u2′-u3′∂-1u2,l35′=-u3∂-1u3′-u3′∂-1u3,l43′=u4∂-1u5′+u4′∂-1u5,l44′=-u4∂-1u2′-u4′∂-1u2,l45′=-u4∂-1u3′-u4′∂-1u3-u1′,l53′=u5∂-1u5′+u5′∂-1u5,l54′=-u5∂-1u2′-u5′∂-1u2-u1′,l55′=-u5∂-1u3′-u5′∂-1u3,(58)L21=0∂-1-u4u1′′-u3′′∂-u4′′u1-αu4′u1′l13′′l14′′l15′′u3′′u2∂-1u4′′+u2′′∂-1u4+αu2′∂-1u4′l23′′l24′′l25′′u2′′u3∂-1u4′′+u3′′∂-1u4+αu3′∂-1u4′-u1′′l33′′l34′′l35′′u5′′u4′′∂-1u4+u4∂-1u4′′+αu4′∂-1u4′l43′′l44′′l45′′u4′′u5∂-1u4′′+u5′′∂-1u4+αu5′∂-1u4′l53′′l54′′l55′′,(59)l13′′=∂-1-u5u1′′-u5′′u1-u2′′∂-αu5′u1′,l14′′=∂-1u2u1′′+u2′′u1-u5′′∂+αu2′u1′,l15′′=∂-1u3u1′′+u3′′u1-u4′′∂+αu3′u1′,l23′′=u2∂-1u5′′+u2′′∂-1u5+αu2′∂-1u5′-u1′′,l24′′=-u2∂-1u2′′-u2′′∂-1u2-αu2′∂-1u2′,l25′′=-u2∂-1u3′′-u2′′∂-1u3-αu2′∂-1u3′,l33′′=u3∂-1u5′′+u3′′∂-1u5+αu3′∂-1u5′,l34′′=-u3∂-1u2′′-u3′′∂-1u2-αu3′∂-1u2′,l35′′=-u3∂-1u3′′-u3′′∂-1u3-αu3′∂-1u3′,l43′′=u4∂-1u5′′+u4′′∂-1u5+αu4′∂-1u5′,l44′′=-u4∂-1u2′′-u4′′∂-1u2-αu4′∂-1u2′,l45′′=-u4∂-1u3′′-u4′′∂-1u3-αu4′∂-1u3′-u1′′,l53′′=u5∂-1u5′′+u5′′∂-1u5+αu5′∂-1u5′,l54′′=-u5∂-1u2′′-u5′′∂-1u2-αu5′∂-1u2′-u1′′,l55′′=-u5∂-1u3′′-u5′′∂-1u3-αu5′∂-1u3′.


## 3.1. Bi-Integrable Couplings Associated withSO(4)

We take the class of3×3 block matrices in the form of(24)M2A1,A2,A3=A1A2A30A1αA200A1,where α is an arbitrary nonzero constant and A1, A2, and A3 are square matrices of the same order. In the following, we define the corresponding nonsemisimple Lie algebra g- by a semidirect sum (25)g-λ=g⊎gc,with(26)g=M2A1,0,0∣A∈SO4~,gc=M20,A2,A3∣A2,A3∈SO4~,where loop algebra SO(4)~ is defined by (27)SO4~=Aλ∈SO4∣entries  ofAλ—Laurent  series  inλ.The corresponding matrix product reads(28)M2A1,A2,A3,M2B1,B2,B3=M2C1,C2,C3,with C1, C2, and C3 being defined by (29)C1=A1,B1,C2=A1,B2+A2,B1,C3=A1,B3+A3,B1+αA2,B2.Based on this special nonsemisimple Lie algebrag-(λ), we choose the corresponding enlarged spectral matrix U-1=M2(U,U1,U2), where U is defined as in (9) U1 and U2 are defined as follows: (30)U1=U1u1,λ=0-u2′u4′-u1′u2′00-u5′-u4′00-u3′u1′u5′u3′0,u1=u1′u2′u3′u4′u5′,U2=U2u2,λ=0-u2′′u4′′-u1′′u2′′00-u5′′-u4′′00-u3′′u1′′u5′′u3′′0,u2=u1′′u2′′u3′′u4′′u5′′.In order to solve the enlarged stationary zero curvature equationsV-1x=[U-1,V-1], we take V-1=M2(V,V1,V2), where V is defined by (9): (31)V=Vu,λ=0-cf-ac0-b-g-fb0-dagd0=∑i≥00-cifi-aici0-bi-gi-fibi0-diaigidi0λ-i.V1 and V2 are defined similarly:(32)V1=V1u,u1,λ=0-c′f′-a′c′0-b′-g′-f′b′0-d′a′g′d′0=∑i≥00-ci′fi′-ai′ci′0-bi′-gi′-fi′bi′0-di′ai′gi′di′0λ-i,V2=V2u,u1,u2,λ=0-c′′f′′-a′′c′′0-b′′-g′′-f′′b′′0-d′′a′′g′′d′′0=∑i≥00-ci′′fi′′-ai′′ci′′0-bi′′-gi′′-fi′′bi′′0-di′′ai′′gi′′di′′0λ-i.It now follows from the enlarged stationary zero curvature equationsV-1x=[U-1,V-1] that (33)Vx=U,V,V1x=U,V1+U1,V,V2x=U,V2+U2,V+αU1,V1.The above equation system equivalently leads to (11) and the following equations:(34)ax′=u5c′-u3f′-u2g′+u4d′+u5′c-u3′f-u2′g+u4′d,bx′=u4c′-u3g′-u2f′+u5d′+u4′c-u3′g-u2′f+u5′d,cx′=λf′-u5a′-u4b′+u1g′-u5′a-u4′b+u1′g,dx′=λg′-u4a′-u5b′+u1f′-u4′a-u5′b+u1′f,fx′=-λc′+u2b′+u3a′-u1d′+u2′b+u3′a-u1′d,gx′=-λd′+u2a′+u3b′-u1c′+u2′a+u3′b-u1′c,ax′′=u5c′′-u3f′′-u2g′′+u4d′′+u5′′c-u3′′f-u2′′g+u4′′d+αu5′c′-u3′f′-u2′g′+u4′d′,bx′′=u4c′′-u3g′′-u2f′′+u5d′′+u4′′c-u3′′g-u2′′f+u5′′d+αu4′c′-u3′g′-u2′f′+u5′d′,cx′′=λf′′-u5a′′-u4b′′+u1g′′-u5′′a-u4′′b+u1′′g-αu5′a′+u4′b′-u1′g′,dx′′=λg′′-u4a′′-u5b′′+u1f′′-u4′′a-u5′′b+u1′′f-αu4′a′+u5′b′-u1′f′,fx′′=-λc′′+u2b′′+u3a′′-u1d′′+u2′′b+u3′′a-u1′′d+αu2′b′+u3′a′-u1′d′,gx′′=-λd′′+u2a′′+u3b′′-u1c′′+u2′′a+u3′′b-u1′′c+αu2′a′+u3′b′-u1′c′.Now, let us 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)+,m≥0, 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 for soliton hierarchy (14): (35)u-tm=utmu1tmu2tm=J100J2J10J3αJ2J1P1mP2mP3m=J100J2J10J3αJ2J1P-1,m,with utm, P1m, and J1 being defined by (14) and (15), with u1tm, u2tm, P2m, P3m, J2, and J3 being defined as follows:(36)u1tm=u1′u2′u3′u4′u5′tm=-amx′cmx′-u1gm′+u4bm′-u1′gm+u4′bmdmx′-u1fm′+u5bm′-u1′fm+u5′bmfmx′+u1dm′-u2bm′+u1′dm-u2′bmgmx′+u1cm′-u3bm′+u1′cm-u3′bm,u2tm=u1′′u2′′u3′′u4′′u5′′tm=-amx′′cmx′′+u4bm′′-u1gm′′+u4′′bm-u1′′gm+αu4′bm′-u1′gm′dmx′′+u5bm′′-u1fm′′+u5′′bm-u1′′fm+αu5′bm′-u1′fm′fmx′′-u2bm′′+u1dm′′-u2′′bm+u1′′dm+αu1′dm′-u2′bm′gmx′′-u3bm′′+u1cm′′-u3′′bm+u1′′cm+αu1′cm′-u3′bm′,P2m=-2am′-2cm′-2dm′-2fm′-2gm′,P3m=-2am′′-2cm′′-2dm′′-2fm′′-2gm′′,J2=12∂00000-u4∂-1u4′+u4′∂-1u4-u4′∂-1u5+u4∂-1u5′B1B20-u5∂-1u4′+u5′∂-1u4-u5∂-1u5′+u5′∂-1u5B3B40u2∂-1u4′+u2′∂-1u4u2∂-1u5′+u2′∂-1u5-u1′B5B60u3∂-1u4′+u3′∂-1u4u3∂-1u5′+u3′∂-1u5B7B8,where (37)B1=u4∂-1u2′+u4′∂-1u2,B2=u4∂-1u3′+u4′∂-1u3+u1′,B3=u5∂-1u2′+u5′∂-1u2+u1′,B4=u5∂-1u3′+u5′∂-1u3,B5=-u2∂-1u2′+u2′∂-1u2,B6=-u2∂-1u3′+u2′∂-1u3,B7=-u3∂-1u2′+u3′∂-1u2,B8=-u3∂-1u3′+u3′∂-1u3,(38)J3=12∂00000-u4∂-1u4′′+u4′′∂-1u4+αu4′∂-1u4′C1C2C30-u5∂-1u4′′+u5′′∂-1u4+αu5′∂-1u4′C4C5C60u2∂-1u4′′+u2′′∂-1u4+αu2′∂-1u4′C7C8C90u3∂-1u4′′+u3′′∂-1u4+αu3′∂-1u4′-u1′′C10C11C12,where(39)C1=-u4∂-1u5′′+u4′′∂-1u5+αu4′∂-1u5′,C2=u4∂-1u2′′+u4′′∂-1u2+αu4′∂-1u2′,C3=u4∂-1u3′′+u4′′∂-1u3+αu4′∂-1u3′+u1′′,C4=-u5∂-1u5′′+u5′′∂-1u5+αu5′∂-1u5′,C5=u5∂-1u2′′+u5′′∂-1u2+αu5′∂-1u2′+u1′′,C6=u5∂-1u3′′+u5′′∂-1u3+αu5′∂-1u3′,C7=u2∂-1u5′′+u2′′∂-1u5+αu2′∂-1u5′-u1′′,C8=-u2∂-1u2′′+u2′′∂-1u2+αu2′∂-1u2′,C9=-u2∂-1u3′′+u2′′∂-1u3+αu2′∂-1u3′,C10=u3∂-1u5′′+u3′′∂-1u5+αu3′∂-1u5′,C11=-u3∂-1u2′′+u3′′∂-1u2+αu3′∂-1u2′,C12=-u3∂-1u3′′+u3′′∂-1u3+αu3′∂-1u3′.


## 3.2. Hamiltonian Structures

In this section, in order to generate the Hamiltonian structures of hierarchy (35), we use the corresponding variational identity [17] (40)δδu-∫V-,U-λdx=λ-γ∂∂λλγV-,U-u,where {·,·} is a required bilinear form, which is symmetric, nondegenerate, and invariant under the loop algebra g-(λ).First, we need to define a mapping [8] (41)σ:g-λ⟶R18,A⟼a1,a2,…,a18T,where(42)A=MA1,A2,A3∈g-λ,Ai=0-a6i-3a6i-1-a6i-5a6i-30-a6i-4-a6i-a6i-1a6i-40-a6i-2a6i-5a6ia6i-20,i=1,2,3.This mapping σ induces a Lie algebraic structure on R18. The Lie bracket [·,·] on R18 can be computed as follows: (43)a,b=aTR-1b,∀a=a1,…,a18T,b=b1,…,b18T∈R18,where (44)R-1b=M2R1,R2,R3,Ri=00b6ib6i-1-b6i-2-b6i-300b6i-1b6i-b6i-3-b6i-2-b6i-b6i-100b6i-4b6i-5-b6i-1-b6i00b6i-5b6i-4b6i-2b6i-3-b6i-4-b6i-500b6i-3b6i-2-b6i-5-b6i-400,i=1,2,3.Mapping (41) is a Lie algebra isomorphism between the two Lie algebras.Then we define a bilinear form onR18 as follows: (45)a,b=aTF1b,∀a=a1,…,a18T,b=b1,…,b18T∈R18.Following the properties of the matrix F1: F1(R-1(b))T=-R-1(b)F1 and F1=F1T, we have (46)F1=η1η2η3η2αη30η300⊗r1r20000r2r1000000r1r20000r2r1000000r1r20000r2r1,where η1,η2,η3,r1, and r2 are arbitrary constants. We choose r1≠±r2; then the nondegenerate condition requires (47)detF1=α6η318r12-r229≠0.So, the bilinear form on the semidirect sumg-(λ) is given by(48)A,B=a1,…,a18F1b1,…,b18=r1a1b1+r2a2b1+r2a1b2+r1a2b2+r1a3b3+r2a4b3+r2a3b4+r1a4b4+r1a5b5+r2a6b5+r2a5b6+r1a6b6η1+r1a7b1+r2a8b1+r2a7b2+r1a8b2+r1a9b3+r2a10b3+r2a9b4+r1a10b4+r1a11b5+r2a12b5+r2a11b6+r1a12b6+r1a1b7+r2a2b7+r2a1b8+r1a2b8+r1a3b9+r2a4b9+r2a3b10+r1a4b10+r1a5b11+r2a6b11+r2a5b12+r1a6b12η2+r1a13b1+r2a14b1+r2a13b2+r1a14b2+r1a15b3+r2a16b3+r2a15b4+r1a16b4+r1a17b5+r2a18b5+r1a18b6+r2a17b6+αr1a7b7+αr2a8b7+αr2a7b8+αr1a8b8+αr1a9b9+αr2a10b9+αr2a9b10+αr1a10b10+αr1a11b11+αr2a12b11+αr2a11b12+αr1a12b12+r1a1b13+r2a2b13+r2a1b14+r1a2b14+r1a3b15+r2a4b15+r2a3b16+r1a4b16+r1a5b17+r2a6b17+r2a5b18+r1a6b18η3.As mentioned above, now we can compute that(49)V-1,U-1,λ=r2a+r1bη1+r2a′+r1b′η2+r2a′′+r1b′′η3,V-1,U-1,u1=r1a+r2bη1+r1a′+r2b′η2+r1a′′+r2b′′η3,V-1,U-1,u2=r1c+r2dη1+r1c′+r2d′η2+r1c′′+r2d′′η3,V-1,U-1,u3=r2c+r1dη1+r2c′+r1d′η2+r2c′′+r1d′′η3,V-1,U-1,u4=r1f+r2gη1+r1f′+r2g′η2+r1f′′+r2g′′η3,V-1,U-1,u5=r2f+r1gη1+r2f′+r1g′η2+r2f′′+r1g′′η3,V-1,U-1,u1′=r1a+r2bη2+αr1a′+r2b′η3,V-1,U-1,u1′′=r2b+r1aη3,V-1,U-1,u2′=r1c+r2dη2+αr1c′+r2d′η3,V-1,U-1,u2′′=r1c+r2dη3,V-1,U-1,u3′=r2c+r1dη2+αr2c′+r1d′η3,V-1,U-1,u3′′=r2c+r1dη3,V-1,U-1,u4′=r1f+r2gη2+αr1f′+r2g′η3,V-1,U-1,u4′′=r1f+r2gη3,V-1,U-1,u5′=r2f+r1gη2+αr2f′+r1g′η3,V-1,U-1,u5′′=r2f+r1gη3.and furthermore we use the formula [17] (50)γ=-λ2ddλln⁡V-,V-,to obtain that γ=0. Applying corresponding variational identity (40), we obtain the following Hamiltonian structures for the hierarchy of bi-integrable couplings (35): (51)u-tm=K-1,mu-=J-1δH-1,mδu-;m≥1,with the Hamiltonian operator (52)J-1=η1η2η3η2αη30η300-1⊗J1,where J1 is defined by (15) and the Hamiltonian functionals (53)H-1,m=-∫r2am+1+r1bm+1η1+r1am+1′+r2bm+1′η2+r2am+1′′+r1bm+1′′η3mdx,m≥1.Based on (11) and (34), a direct computation yields a recursion relation (54)P1,m=L-1P1,m-1,where (55)L-1=M2TL1,L11,L21=L100L11L10L21αL11L1,with L1 being defined by (17) L11 and L21 are defined as follows:(56)L11=0∂-1-u4u1′-u4′u1-u3′∂l13′l14′l15′u3′u2∂-1u4′+u2′∂-1u4l23′l24′l25′u2′u3∂-1u4′+u3′∂-1u4-u1′l33′l34′l35′u5′u4∂-1u4′+u4′∂-1u4l43′l44′l45′u4′u5∂-1u4′+u5′∂-1u4l53′l54′l55′,(57)l13′=∂-1-u5u1′-u5′u1-u2′∂,l14′=∂-1u2u1′+u2′u1-u5′∂,l15′=∂-1u3u1′+u3′u1-u4′∂,l23′=u2∂-1u5′+u2′∂-1u5-u1′,l24′=-u2∂-1u2′-u2′∂-1u2,l25′=-u2∂-1u3′-u2′∂-1u3,l33′=u3∂-1u5′+u3′∂-1u5,l34′=-u3∂-1u2′-u3′∂-1u2,l35′=-u3∂-1u3′-u3′∂-1u3,l43′=u4∂-1u5′+u4′∂-1u5,l44′=-u4∂-1u2′-u4′∂-1u2,l45′=-u4∂-1u3′-u4′∂-1u3-u1′,l53′=u5∂-1u5′+u5′∂-1u5,l54′=-u5∂-1u2′-u5′∂-1u2-u1′,l55′=-u5∂-1u3′-u5′∂-1u3,(58)L21=0∂-1-u4u1′′-u3′′∂-u4′′u1-αu4′u1′l13′′l14′′l15′′u3′′u2∂-1u4′′+u2′′∂-1u4+αu2′∂-1u4′l23′′l24′′l25′′u2′′u3∂-1u4′′+u3′′∂-1u4+αu3′∂-1u4′-u1′′l33′′l34′′l35′′u5′′u4′′∂-1u4+u4∂-1u4′′+αu4′∂-1u4′l43′′l44′′l45′′u4′′u5∂-1u4′′+u5′′∂-1u4+αu5′∂-1u4′l53′′l54′′l55′′,(59)l13′′=∂-1-u5u1′′-u5′′u1-u2′′∂-αu5′u1′,l14′′=∂-1u2u1′′+u2′′u1-u5′′∂+αu2′u1′,l15′′=∂-1u3u1′′+u3′′u1-u4′′∂+αu3′u1′,l23′′=u2∂-1u5′′+u2′′∂-1u5+αu2′∂-1u5′-u1′′,l24′′=-u2∂-1u2′′-u2′′∂-1u2-αu2′∂-1u2′,l25′′=-u2∂-1u3′′-u2′′∂-1u3-αu2′∂-1u3′,l33′′=u3∂-1u5′′+u3′′∂-1u5+αu3′∂-1u5′,l34′′=-u3∂-1u2′′-u3′′∂-1u2-αu3′∂-1u2′,l35′′=-u3∂-1u3′′-u3′′∂-1u3-αu3′∂-1u3′,l43′′=u4∂-1u5′′+u4′′∂-1u5+αu4′∂-1u5′,l44′′=-u4∂-1u2′′-u4′′∂-1u2-αu4′∂-1u2′,l45′′=-u4∂-1u3′′-u4′′∂-1u3-αu4′∂-1u3′-u1′′,l53′′=u5∂-1u5′′+u5′′∂-1u5+αu5′∂-1u5′,l54′′=-u5∂-1u2′′-u5′′∂-1u2-αu5′∂-1u2′-u1′′,l55′′=-u5∂-1u3′′-u5′′∂-1u3-αu5′∂-1u3′.


## 4. Conclusion

In this paper, new Lax integrable hierarchy (14) is obtained by constructing an isospectral problem associated with SO(4). Furthermore, we have obtained a new class of bi-integrable couplings (35) for soliton hierarchy (14) based on nonsemisimple Lie algebra (25).There are many soliton hierarchies obtained by using matrix spectral problems based on certain real Lie algebras such asSO(3) and SO(4). Recently seeking for integrable couplings of systems has become more and more meaningful in the area of mathematics and physics. Certainly, there are many interesting aspects of integrable couplings we have not solved, for example, the hierarchies associated with SO(5) and SO(6), the relations between the hierarchy of tri-integrable couplings associated with SO(4), and the hierarchy of tri-integrable couplings associated with SO(3). These problems are very interesting.



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*Source: 857684-2015-12-31.xml*