# The Bi-Integrable Couplings of Two-Component Casimir-Qiao-Liu Type Hierarchy and Their Hamiltonian Structures

Authors: Juhui Zhang; Yuqin Yao
Journal: Advances in Mathematical Physics
 (2016)

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## Abstract

A new type of two-component Casimir-Qiao-Liu type hierarchy (2-CQLTH) is produced from a new spectral problem and their bi-Hamiltonian structures are constructed. Particularly, a new completely integrable two-component Casimir-Qiao-Liu type equation (2-CQLTE) is presented. Furthermore, based on the semidirect sums of matrix Lie algebras consisting of3×3 block matrix Lie algebra, the bi-integrable couplings of the 2-CQLTH are constructed and their bi-Hamiltonian structures are furnished.


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## Body

## 1. Introduction

Solitons and integrable systems play an important role in nonlinear wave and dynamics systems. It has been significant in soliton theory to find more new integrable systems. In 1996, Olver and Rosenau obtained a Casimir equation [1]:(1)ρt=D±D3ρ-2,which is an integrable case of the general class of equations ρt=D1±D2ρk. In 2009, Qiao and Liu proposed a third-order integrable peakon equation [2, 3]:(2)qt=12q2xxx-12q2xwhich possesses Lax representation and bi-Hamiltonian structures. Here we can find that (2) can be obtained from the Casimir equation (1) by setting ρ=23q. We will use Casimir-Qiao-Liu equation (CQLE) to denote (2) in this paper. The CQLE (2) represents a third-order approximation of long wavelength, small amplitude waves of inviscid and incompressible fluids. Moreover, it may be reduced from the two-dimensional Euler equation by an approximation procedure, and its solutions may be useful to construct new solvable potentials in Newtonian dynamics and to model electrophysiological phenomena in neuroscience. It has attracted many scholars’ attention in recent years [4–6].In recent years, the construction of soliton hierarchies and integrable couplings have become important research fields in soliton theory [7–10]. The soliton hierarchies are generated from the zero curvature equations [11, 12] which are based on semisimple Lie algebras, while the integrable couplings are generated from the zero curvature equations based on semidirect sums of Lie algebras [13–15].The trace identity proposed by Tu is a useful tool for constructing the Hamiltonian structures for both continuous and discrete integrable systems [11, 12]. Many integrable Hamiltonian systems of infinite dimensions with various physics and mathematical backgrounds have been obtained [16–18]. But when it comes to the integrable couplings, since they are based on semidirect sums of Lie algebras, the trace identity cannot be used properly. In order to solve this problem, Ma proposes the variational identity [19] while Guo and Zhang propose the quadratic-form identity [20]. By using them, the Hamiltonian structures of many integrable couplings systems have been furnished [21–23].Integrable couplings correspond to nonsemisimple Lie algebrasg¯, and such Lie algebras can be written as semidirect sums [24]:(3)g¯=g⊕gc,g-semisimple,gc-solvable.The notion of semidirect sums g¯=g⊕gc means that the two Lie subalgebras g and gc satisfy(4)g,gc⊆gc,where g,gc=A,B∣A∈g,B∈gc, with ·,· denoting the Lie bracket of g¯. We also require the closure property between g and gc under the matrix multiplication:(5)ggc,gcg⊆gc,where g1g2=A,B∣A∈g1,B∈g2.Now we make the following assumptions:(6)U¯=U+Uc,V¯=V+Vc,where U,V∈g,Uc,Vc∈gc and U¯,V¯ satisfying (3), (4), and (5). In this condition, we can construct the enlarged spectral problem as(7)ϕ¯x=U¯ϕ¯=U¯u¯,λϕ¯,ϕ¯t=V¯ϕ¯=V¯u¯,u¯x,…,∂m0u¯∂xm0;λϕ¯.From the enlarged zero curvature equations(8)U¯-V¯x+U¯,V¯=0,we have(9)U-Vx+U,V=0,Uc,t-Vc,x+U,Vc+Uc,V+Uc,Vc=0.The first equation of (9) is the original soliton hierarchy while the second equation is the integrable couplings.The bi-Hamiltonian structures of (9)(10)u¯tm=J¯δHm+1δu¯,m≥0,can be obtained by using the following variational identity:(11)δδu∫V,∂U∂λdx=λ-γ∂∂λλγV,∂U∂u,γ=-λ2ddλlnV,V,where ·,· is a bilinear form [19, 20, 25].In this paper, starting from a new eigenvalue problem, the two-component Casimir-Qiao-Liu type hierarchy (2-CQLTH) is worked out. It is proved that the 2-CQLTH has Lax pairs and bi-Hamiltonian structures, so it is completely integrable. Particularly, the two-component Casimir-Qiao-Liu type equation (2-CQLTE) is given. Then, by constructing the enlarged spectral problem, we obtain the the bi-integrable couplings of the 2-CQLTH. Similarly, we have proved the integrability of the bi-integrable couplings by constructing their Lax pairs and bi-Hamiltonian structures.This paper is organized as follows. In Section2, a new 2-CQLTH is given. In Section 3, the bi-Hamiltonian structures of the 2-CQLTH are constructed. In Section 4, we obtain the bi-integrable couplings of the 2-CQLTH. In Section 5, we generate the bi-Hamiltonian structures of the integrable couplings. The conclusion is given in Section 6.


## 2. The Two-Component Casimir-Qiao-Liu Type Hierarchy

Consider a spatial matrix isospectral problem:(12)ϕx=Uϕ=Uu,λϕ,Uu,λ=-p12λq-12λqp,u=pq,ϕ=ϕ1ϕ2,where p, q are potential functions and λ is the spectral parameter.Assuming thatW has the form(13)W=abc-a, then the stationary zero curvature equation(14)Wx=U,Wgives rise to(15)ax=12bqλ+12cqλ,bx=-aqλ-2bp,cx=-aqλ+2cp.Further, let a,b,c possess the Laurent expansions: (16)a=∑i≥0aiλ-2i,b=∑i≥0biλ-2i-1,c=∑i≥0ciλ-2i-1,i≥0, where the initial values are as follows: (17)a0=0,b0=14,c0=-14. In this case, system (15) becomes(18)ai,x=q2bi+ci,bi,x=-2pbi-qai+1,ci,x=2pci-qai+1,i≥0.Imposing the following conditions on constants of integration: (19)aiu=0=biu=0=ciu=0=0,i≥1, the sequence of ai,bi,ci∣i≥1 can be uniquely determined and the first two sets are as follows: (20)a1=-p2q,b1=-12qpqx+12pq2,c1=-12qpqx-12pq2,a2=-p3q3+pxx2q3-3pxqx2q4-pqxx2q4+3pqx22q5,b2=a2,xq-2∫pqa2,xdx,c2=a2,xq+2∫pqa2,xdx.Now we introduce(21)Vm=λλ2m+1W++Δm,m≥0,where(22)Δm=-am+1λqpam+1-λqpam+1am+1.By considering(23)ϕtm=Vmϕ,the compatibility of (12) and (23) gives the zero curvature equation:(24)Utm-Vxm+U,Vm=0.From the zero curvature equation (24), we can obtain the 2-CQLTH:(25)utm=Km=am+1,x2qpam+1x=J-4am+1cm+1-bm+1,where (26)J=-14∂0012∂qp∂-1qp∂.Remark 1.
The Lax pair for the 2-CQLTH (see (25)) is given by (12) and (23). This implies that the 2-CQLTH (see (25)) is integrable in Lax sense.Particularly, when we takem=1, we obtain(27)pt=-3p3q3+2pqxxq4-15pqx22q5pxp-qxq-3pqxq4pxxp-qxxq+p2q3pxxxp-qxxxq,qt=-4p2q2+pxxpq2-3pxqxpq3+3qxxq3-12qx2q4pxp-qxq-6qxq3pxxp-qxxq+1q2pxxxp-qxxxq.Remark 2.
Takingp=1/2, the second equation of (27) can be reduced to the CQLE (2). Thus, (27) is called 2-CQLTE.


## 3. Bi-Hamiltonian Structures and Liouville Integrability

By direct computation, we can get(28)∂U∂λ=012q-12q0,∂U∂p=-1001,∂U∂q=012λ-12λ0, so we have (29)trW∂U∂λ=12qc-12qb,trW∂U∂p=-2a,trW∂U∂q=12cλ-12bλ. Now, the trace identity [11](30)δδu∫trW∂U∂λdx=λ-γ∂∂λλγtrW∂U∂u,γ=λ2ddλlntrW2gives(31)δδu∫12qc-12qbdx=λ-γ∂∂λλγ-2a12cλ-12bλ.Comparing coefficients of all powers of λ in the equality (31), we have(32)δδu∫qcm-qbmγ-2mdx=-4amcm-bm,m≥0.Checking a particular case with m=1 in (32), we have γ=1.Thus, we obtain(33)δδuHm=-4amcm-bm,m≥0,where the Hamiltonian functionals are defined by (34)Hm=∫qcm-qbm1-2mdx. It now follows that the soliton hierarchy (25) has the following Hamiltonian structures:(35)utm=Km=JδHmδu,m≥0.From the recursion relations (18), we can get (36)-4am+1cm+1-bm+1=L-4amcm-bm,m≥0, where (37)L=-1q∂1q∂+4pq∂-1pq∂00-∂-1pq∂1q∂1p∂+4∂-1pq∂pq. It is easy to verify that JL=L∗J. Actually, we know that the hierarchy (25) is bi-Hamiltonian:(38)utm=Km=JδHmδu=MδHm-1δu,m≥1,where the second Hamiltonian operator is(39)M=JL=-∂pq∂-1pq∂+14∂1q∂1q∂002∂-12∂1p∂1p∂and J and M constitute the Hamiltonian pairs. Thus, the soliton hierarchy (25) is Liouville integrable.Remark 3.
Based onSO(3) [26, 27], a 3×3 matrix isospectral problem is constructed:(40)ϕx=Uϕ=Uu,λϕ,u=pq,ϕ=ϕ1ϕ2, with the spectral matrix as (41)U=0-qλ2-pqλ20-qλ2pqλ20. By a standard procedure similar to Sections 2 and 3, another 2-QLTH can be worked out:(42)utm=-am+1,x-2qpam+1x=JδHmδu=MδHm-1δu,m≥1,with the Hamiltonian operators (43)J=-∂1q∂1q∂-∂pq∂-1pq∂002∂1p∂1p∂+2∂,M=M100M2,M1=2∂1q∂1q∂1q∂1q∂+2∂1q∂1q∂pq∂-1pq∂+2∂pq∂-1pq∂1q∂1q∂+2∂pq∂-1pq∂pq∂-1pq∂,M2=-4∂1p∂1q∂1q∂1p∂-4∂1p∂1q∂pq-4pq∂1q∂1p∂-4pq∂pq, and the Hamiltonian functionals (44)Hm=∫-qbm-qcm1-2mdx,m≥0. Particularly, when we take m=1, a new 2-CQLTE is presented as(45)pt=3p34q3-2pqxxq4+15pqx22q5pxp-qxq-3pqxq4pxxp-qxxq+p2q3pxxxp-qxxxq,qt=p2q2-pxxpq2-3qxxq3+3pxqxpq3+12qx2q4pxp-qxq-6qxq3pxxp-qxxq+1q2pxxxp-qxxxq.Equation (45) can be transformed into (27) in complex field, but they are different equations in real field.


## 4. Bi-Integrable Couplings of the 2-CQLTH

Supposing that the triangular block matricesM have the following form [28, 29]: (46)MA1,A2,A3=A1A2A30A1δA200A1, where δ is a constant, and defining g¯=MA1,A2,A3, g=MA1,0,0, and gc=M0,A2,A3, we know that g¯,g,gc satisfy (3), (4), and (5).Considering the enlarged spectral matrix(47)U¯=UU1U20UδU100U,W¯=WW1W20WδW100W, where (48)U=-pqλ2-qλ2p,U1=-r1r2λ2-r2λ2r1,U2=-s1s2λ2-s2λ2s1,W=abc-a,W1=def-d,W2=ghk-g, and p,q,r1,r2,s1,s2 are dependent variables, if we substitute U¯,W¯ into the corresponding enlarged stationary zero curvature equation, we can get(49)2ax-bqλ-cqλ=0,bx+2bp+aqλ=0,cx-2cp+aqλ=0,2dx-eqλ-fqλ-bλr2-cλr2=0,ex+2ep+dqλ+2br1+aλr2=0,fx-2fp+dqλ-2cr1+aλr2=0,2gx-hqλ-kqλ-eδλr2-fδλr2-bλs2-cλs2=0,hx+2hp+gqλ+2eδr1+dδλr2+2bs1+aλs2=0,kx-2kp+gqλ-2fδr1+dδλr2-2cs1+aλs2=0.When a,b,c,d,e,f,g,h,k are assumed to be(50)a=∑i≥0aiλ-2i,b=∑i≥0biλ-2i-1,c=∑i≥0ciλ-2i-1,i≥0,d=∑i≥0diλ-2i,e=∑i≥0eiλ-2i-1,f=∑i≥0fiλ-2i-1,i≥0,g=∑i≥0giλ-2i,h=∑i≥0hiλ-2i-1,k=∑i≥0kiλ-2i-1,i≥0, and the initial values are taken as(51)a0=0,b0=α,c0=-α,d0=0,e0=β,f0=-β,g0=0,h0=ι,k0=-ι, we can write (49) in the following forms:(52)-12biq-ciq2+ai,x=0,2bip+ai+1q+bi,x=0,-2cip+ai+1qλ+ci,x=0,-12eiq-fiq2+di,x-12bir2-12cir2=0,2eip+di+1qλ+ei,x+2br1+ai+1r2=0,-2fip+di+1q+fi,x-2cir1+ai+1r2=0,-12hiq-kiq2+gi,x-12eiδr2-12fiδr2-12bis2-12cis2=0,2hip+gi+1q+hi,x+2eiδr1+di+1δr2+2bis1+ai+1s2=0,-2kip+gi+1q+ki,x-2fiδr1+di+1δλr2-2cis1+ai+1λs2=0.Furthermore, by imposing(53)aiu=0=biu=0=ciu=0=0,i≥1,diu=0=eiu=0=fiu=0=0,i≥1,giu=0=hiu=0=kiu=0=0,i≥1, the sequence of ai,bi,ci,di,ei,fi,gi,hi,ki∣i≥1 has the unique expressions (54)a0=0,b0=α,c0=-α,a1=-2αpq,b1=2αpq2-2α1qpqx,c1=-2αpq2-2α1qpqx,d0=0,e0=β,f0=-β,d1=-2αpq-2βr1q+2βpr2q2,e1=2αpq2-2α1qpqx-2β1qr1qx+4βpr1q2x+2β1qpr2q2x+2βr2q2pqx-4βp2r2q3x,f1=-2αpq2-2α1qpqx-2β1qr1qx-4βpr1q2x+2β1qpr2q2x+2βr2q2pqx+4βp2r2q3x,g0=0,h0=ι,k0=-ι,g1=-2ιpq-2βδr1q-2αs1q+2αδpr2q2+2βδr1r2q2-2βδpr2r2q3+2αps2q2,k1=-2ι1qpqx-2βδ1qr1qx+2βδ1qr1r2q2x+2βδr2q2r1qx-2βδ1qpr2r2q3x-2βδr2q2pr2q2x-2βδr2r2q3pqx-2α1qs1qx+2αδr2q2pqx+2αδ1qpr2q2x+2αs2q2pqx+2α1qps2q2x-2ιpq2-4βδpr1q2+4βδpr1r2q3+4βδpr1r2q3-6βδpr2q22-2βδr1q2+4αδp2r2q3-4αps1q2+4αp2s2q3,h1=-2ι1qpqx-2βδ1qr1qx+2βδ1qr1r2q2x+2βδr2q2r1qx-2βδ1qpr2r2q3x-2βδr2q2pr2q2x-2βδr2r2q3pqx-2α1qs1qx+2αδr2q2pqx+2αδ1qpr2q2x+2αs2q2pqx+2α1qps2q2x+2ιpq2+4βδpr1q2-4βδpr1r2q3-4βδpr1r2q3+6βδpr2q22+2βδr1q2-4αδp2r2q3+4αps1q2-4αp2s2q3.In order to get the integrable couplings systems from the enlarged zero curvature equations(55)U¯tm-Vxm+U,Vm=0,we define the Lax matrices and modification terms as(56)Vm=λλ2m+1W++Δm,m≥0,where (57)Δm=Δ1mΔ2mΔ3m0Δ1mδΔ2m00Δ1m,Δ1m=-am+1λqpam+1-λqpam+1am+1,Δ2m=-dm+1-2a1+mr1pλqpdm+1+λr2pa1+m-λqpdm+1-λr2pa1+mdm+1+2a1+mr1p,Δ1m=-gm+1-2δr1pdm+1-2s1pa1+mλqpgm+1+δλr2pdm+1+λs2pa1+m-λqpgm+1-δλr2pdm+1-λs2pa1+mgm+1+2δr1pdm+1+2s1pa1+m, and P+ denotes the polynomial part of P.In this condition, the integrable couplings system can be obtained as(58)utm=ptmqtmr1tmr2tms1tms2tm=∂am+12∂qpam+1∂dm+1+2∂a1+mr1p2∂qpdm+1+2∂r2pa1+m∂gm+1+2δ∂r1pdm+1+2∂s1pa1+m2∂qpgm+1+2δ∂r2pdm+1+2∂s2pa1+m,which is equal to(59)utm=J¯-4am+1η1-4dm+1η2-4gm+1η3-bm+1η1+cm+1η1-em+1η2+fm+1η2-hm+1η3+km+1η3-4am+1η2-4dm+1δη3-bm+1η2+cm+1η2-em+1δη3+fm+1δη3-4am+1η3-bm+1η3+cm+1η3,where(60)J¯=00J101δJ1J2J1J2J3,J1=j1100j14,J2=j21j22j23j24,J3=j31j32j33j34,j11=-14η3∂,j14=12η3∂qp∂-1qp∂,j21=η24δη3η3∂,j22=12η3∂r1p∂-1qp∂,j23=12η3∂qp∂-1r1p∂,j24=12η3∂qp∂-1r2p∂+12η3∂r2p∂-1qp∂-η22δη3η3∂qp∂-1qp∂,j31=12η3δ∂r1p∂-1r1p∂+η14η3η3∂-η2η24δη3η3η3∂,j32=12η3δ∂r1p∂-1r2p∂+12η3∂s1p∂-1qp∂-η22η3η3∂r1p∂-1qp∂,j33=12η3δ∂r2p∂-1r1p∂+12η3∂qp∂-1s1p∂-η22η3η3∂qp∂-1r1p∂,j34=12η3∂qp∂-1s2p∂+12η3∂s2p∂-1qp∂+12η3δ∂r2p∂-1r2p∂-η12η3η3∂qp∂-1qp∂-η22η3η3∂qp∂-1r2p∂-η22η3η3∂r2p∂-1qp∂+η2η22δη3η3η3∂qp∂-1qp∂.Remark 4.
U ¯ and V[m] are the Lax pairs of the integrable couplings of the 2-CQLTH; this implies that the integrable couplings are integrable in Lax sense.


## 5. Bi-Hamiltonian Structures and Liouville Integrability of the Bi-Integrable Couplings

Because the associated matrix Lie algebras are nonsemisimple, we should use the variational identity (11) to furnish the Hamiltonian structures of the integrable couplings system (58) (or (59)). Before doing this, we should find out the nondegenerate, symmetric, and adinvariant bilinear form corresponding to the nonsemisimple matrix loop algebras first [22].By defining a mapping fromg¯ to R9 as(61)σ:g¯⟶R9,A⟼a1,…,a9T,where (62)A=MA1,A2,A3∈g¯,Ai=a3i-2a3i-1a3i-a3i-2,i=1,2,3, we have the following results according to [22, 28, 30].The mapping (61) is a Lie algebra isomorphism from g¯ to R9 and the Lie bracket ·,· can be calculated as (63)a,b=aTRb,a=a1,a2,…,a9,b=b1,b2,…,b9∈R9, where (64)Rb=MR1,R2,R3,R=02b3i-1-2b3ib3i-2b3i-20-b3i-102b3i-2,i=1,2,3.Assuming that the bilinear form onR9 possesses the following form: (65)a,b=aTFb, where F is a constant matrix, the symmetric property and adinvariance property of F(66)a,b=b,a,a,b,c=a,b,c give rise to(67)F=FT,FRbT=-RbF,b∈R9.So the form of the constant matrix F can be written as(68)F=η1η2η3η2δη30η300⊗200001010,where η1, η2, and η3 are arbitrary constants. Thus, the bilinear form on the semidirect sums g¯ of g and gc can be defined as (69)A,B=σA,σBR9=a1,a2,…,a9Fb1,b2,…,b9T, where A=σ-1a1,a2,…,a9T,  B=σ-1b1,b2,…,b9T∈g¯.In order to guarantee that the bilinear form is nondegenerate, we should require(70)detF=8δ3η39≠0; that is to say, δ≠0 and η3≠0.By calculation, we have(71)∂U¯∂λ=0q20r220s22-q20-r220-s220000q20δr2200-q20-δr22000000q20000-q20,∂U¯∂p=-10000001000000-10000001000000-10000001,∂U¯∂q=0λ20000-λ200000000λ20000-λ200000000λ20000-λ20,∂U¯∂r1=00-10000001000000-δ000000δ000000000000,∂U¯∂r2=000λ20000-λ200000000δλ20000-δλ20000000000000,∂U¯∂s1=0000-10000001000000000000000000000000,∂U¯∂s2=00000λ20000-λ20000000000000000000000000. Further, we have (72)W∂U∂λ=-12bqη1+12cqη1-12eqη2+12fqη2-12br2η2+12cr2η2-12hqη3+12kqη3-12eδr2η3+12fδr2η3-12bs2η3+12cs2η3,W∂U∂u=-2aη1-2dη2-2gη3-12bλη1+12cλη1-12eλη2+12fλη2-12hλη3+12kλη3-2aη2-2dδη3-12bλη2+12cλη2-12eδλη3+12fδλη3-2aη3-12bλη3+12cλη3. Substituting the above expressions into the variational identity (11), we can obtain(73)δδu¯∫-12bqη1+12cqη1-12eqη2+12fqη2-12br2η2+12cr2η2-12hqη3+12kqη3-12eδr2η3+12fδr2η3-12bs2η3+12cs2η3dx=λ-γ∂∂λλγ-2aη1-2dη2-2gη3-12bλη1+12cλη1-12eλη2+12fλη2-12hλη3+12kλη3-2aη2-2dδη3-12bλη2+12cλη2-12eδλη3+12fδλη3-2aη3-12bλη3+12cλη3.Comparing the coefficients of all powers of λ in (73), we know that(74)δδu¯∫zm1-2mdx=-4amη1-4dmη2-4gmη3-bmη1+cmη1-emη2+fmη2-hmη3+kmη3-4amη2-4dmδη3-bmη2+cmη2-emδη3+fmδη3-4amη3-bmη3+cmη3,m≥0,where (75)zm=-bmqη1+cmqη1-emqη2+fmqη2-bmr2η2+cmr2η2-hmqη3+kmqη3-emδr2η3+fmδr2η3-bms2η3+cms2η3. Setting m=1 in (74), we know that the constant γ=1, so we have(76)δHmδu=-4amη1-4dmη2-4gmη3-bmη1+cmη1-emη2+fmη2-hmη3+kmη3-4amη2-4dmδη3-bmη2+cmη2-emδη3+fmδη3-4amη3-bmη3+cmη3.Thus, the Hamiltonian functions can be expressed as (77)Hm=∫zm1-2mdx and the Hamiltonian structures of (58) as(78)utm=JδHm+1δu,m≥0.Furthermore, by using (52), we can get(79)utm=JδHm+1δu=MδHmδu,where (80)M¯=00J¯101δJ¯1J¯2J¯1J¯2J¯3,J¯1=j¯1100j¯14,J¯2=j¯21j¯22j¯23j¯24,J¯3=j¯31j¯32j¯33j¯34,j¯11=14η3∂1q∂1q∂-1η3∂pq∂-1pq∂,j¯14=21η3∂-12η3∂1p∂1p∂,j¯21=-14η3∂r2q2∂1q∂-14η3∂1q∂r2q2∂+1η3∂pq∂-1pr2q2∂+1η3∂pr2q2∂-1pq∂-1η3∂pq∂-1r1q∂-1η3∂r1q∂-1pq∂+η2δη3η3∂pq∂-1pq∂-η24δη3η3∂1q∂1q∂,j¯22=-12η3∂r1pq∂1p∂+21η3∂r1q,j¯23=-12η3∂1p∂r1pq∂+21η3r1q∂,j¯24=21η3∂r1p+21η3r1p∂-2η2δη3η3∂+η22δη3η3∂1p∂1p∂,j¯31=-14η34∂pq∂-1s1q∂+4∂s1q∂-1pq∂-δ∂1q∂r2r2q3∂-δ∂r2r2q3∂1q∂+∂1q∂s2q2∂+∂s2q2∂1q∂-4δ∂r1q∂-1pr2q2∂-4δ∂pr2q2∂-1r1q∂+4δ∂r1q∂-1r1q∂-δ∂r2q2∂r2q2∂+4δ∂pr2q2∂-1pr2q2∂+4δ∂pr2r2q3∂-1pq∂+4δ∂pq∂-1pr2r2q3∂-4∂ps2q2∂-1pq∂-4∂pq∂-1ps2q2∂-4δ∂pq∂-1r1r2q2∂-4δ∂r1r2q2∂-1pq∂+η1η3∂1q∂1q∂-η2η2δη3η3∂1q∂1q∂-4η1η3∂pq∂-1pq∂+4η2η2δη3η3∂pq∂-1pq∂-η2η3∂r2q2∂1q∂-η2η3∂1q∂r2q2∂-4η2η3∂pq∂-1r1q∂-4η2η3∂r1q∂-1pq∂+2δ∂r1pq∂r1pq∂+4η2η3∂pq∂-1pr2q2∂+4η2η3∂pr2q2∂-1pq∂,j¯32=12η3δ∂r1r2pq2∂1p∂-21η3δ∂r1r2q2+21η3δ∂r1r1pq-12η3∂s1pq∂1p∂+21η3∂s1q+12η2η3η3∂r1pq∂1p∂-2η2η3η3∂r1q,j¯33=12η3δ∂1p∂r1r2pq2∂-21η3δr1r2q2∂+21η3δr1r1pq∂-12η3∂1p∂s1pq∂+21η3s1q∂+12η2η3η3∂1p∂r1pq∂-2η2η3η3r1q∂,j¯34=-21η3∂s1p-21η3s1p∂+η12η3η3∂1p∂1p∂+2η2η3η3∂r1p+2η2η3η3r1p∂-η2η22δη3η3η3∂1p∂1p∂-2η2η2δη3η3η3∂+2η1η3η3∂.By a complicated computation, we know that(81)J¯=-J¯∗M¯=-M¯∗; namely, the hierarchy (58) possess bi-Hamiltonian structures(82)utm=JδHm+1δu=MδHmδu,m≥0,where J¯ and M¯ are the Hamiltonian pairs of the integrable couplings of the 2-CQLTH. So the integrable couplings (58) are Liouville integrable.


## 6. Conclusion

In this paper, we construct a new eigenvalue problem and give the two-component Casimir-Qiao-Liu type hierarchy (2-CQLTH). The 2-CQLTH has Lax representation and possesses bi-Hamiltonian structures, which implies that it is integrable in both Lax and Liouville sense. Furthermore, we obtain the integrable couplings of the integrable hierarchy as well as its Lax pairs and bi-Hamiltonian structures and prove that the integrable couplings are also integrable in both Lax and Liouville senses.



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*Source: 6347961-2016-08-28.xml*