With the help of the known Lie algebra, a type of new 8-dimensional matrix Lie algebra is constructed in the paper. By using the 8-dimensional matrix Lie algebra, the nonlinear integrable couplings of the Levi hierarchy and the Wadati-Konno-Ichikawa (WKI) hierarchy are worked out, which are different from the linear integrable couplings. Based on the variational identity, the Hamiltonian structures of the above hierarchies are derived.

1. Introduction

The notion of integrable couplings was introduced when the study of Virasoro symmetric algebras [1, 2]. To find as many new integrable systems and their integrable couplings as possible and to elucidate in depth their algebraic and geometric properties are of both theoretical and practical value. During the past few years, some interesting integrable couplings and associated properties of some known interesting integrable hierarchies, such as the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and the Kaup-Newell (KN) hierarchy, were obtained [3–13]. Here it is necessary to point out that the above mentioned integrable couplings are linear for the supplementary variable, so they are called linear integrable couplings.

Recently, Professor Ma proposed the notion of nonlinear integrable couplings and gave the general scheme to construct nonlinear integrable couplings of hierarchies [14]. Based on the general scheme of constructing nonlinear integrable couplings, Professor Zhang introduced some new explicit Lie algebras and obtained the nonlinear integrable couplings of the Giachetti-Johnson (GJ) hierarchy, the Yang hierarchy, and the classical Boussinesq-Burgers (CBB) hierarchy [15, 16].

The aim of the paper is to seek the nonlinear integrable couplings of the Levi hierarchy and the WKI hierarchy as well as their Hamiltonian structures. The plan of the paper is as follows. In Section 2, with the help of the Lie algebra G={(g1g20g1+g2)∣g1,g2∈sl(2)}, an 8-dimensional matrix Lie algebra is presented. It is different from the Lie algebras given in [14–16]. By employing the 8-dimensional matrix Lie algebra, the nonlinear integrable couplings of the Levi hierarchy and the WKI hierarchy are derived in Section 3. Furthermore, the corresponding Hamiltonian structures are worked out by virtue of the variational identity in Section 4. Finally, some conclusions are obtained in Section 5.

2. 8-Dimensional Matrix Lie Algebra

The Lie algebra is presented as H=span{h1,h2,h3,h4} with the basis as follows:
(1)h1=(1000),h2=(0001),h3=(0100),h4=(0010),
equipped with the commutators
(2)[h1,h2]=0,[h1,h3]=h3,[h1,h4]=-h4,[h2,h3]=-h3,[h2,h4]=h4,[h3,h4]=h1-h2.
By virtue of the Lie algebra H, we construct an 8-dimensional matrix Lie algebra
(3)G=span{g1,g2,g3,g4,g5,g6,g7,g8}
with the basis as follows:
(4)g1=(h100h1),g2=(h200h2),g3=(h300h3),g4=(h400h4),g5=(0h10h1),g6=(0h20h2),g7=(0h30h3),g8=(0h40h4),
which have the commutative relations
(5)[g1,g2]=0,[g1,g3]=g3,[g1,g4]=-g4,[g2,g3]=-g3,[g2,g4]=g4,[g3,g4]=g1-g2,[g1,g5]=0,[g1,g6]=0,[g1,g7]=g7,[g1,g8]=-g8,[g2,g5]=0,[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]=0,[g5,g6]=0,[g5,g7]=g7,[g5,g8]=-g8,[g6,g7]=-g7,[g6,g8]=g8,[g7,g8]=g5-g6.
Denoting G1=span{g1,g2,g3,g4} and G2=span{g5,g6,g7,g8}, then we have
(6)G=G1⊕G2,G1≅H,[G1,G2]⊂G2.
Here we need to emphasize that the subalgebras G1 and G2 are both nonsemisimple, which is very important for deriving nonlinear integrable couplings of hierarchies. By using the Lie algebra G, we can construct a few kinds of loop algebras G~=G⊗λNn+j, N and j stand for natural numbers. Among these loop algebras, the simplest one is
(7)G~=span{gi(n)}i=18,gi(n)=giλn,hhhhhhhhhhhhhji=1,2,3,4,5,6,7,8,
along with the commutators [gi(m),gj(n)]=[gi,gj]λm+n, deg(gi(n))=n, m,n∈Z, 1≤i, and j≤8.

In this section, by virtue of the Lie algebra H, we construct an 8-dimensional matrix Lie algebra G and corresponding loop algebra G~; in what follows we will generate the nonlinear integrable couplings of hierarchies by using the loop algebra G~.

3. Nonlinear Integrable Couplings of Hierarchies

In this section, based on the loop algebra G~, we construct two isospectral problems to generate the nonlinear integrable couplings of the Levi hierarchy and the WKI hierarchy, respectively.

3.1. Nonlinear Integrable Couplings of Levi Hierarchy

Take the following isospectral problem:(8)ϕx=Uϕ,λt=0,U=12(-λ+u1-u2)(g1(0)-g2(0))+u1g3(0)+u2g4(0)+12(u3-u4)(g5(0)-g6(0))+u3g7(0)+u4g8(0).
Set V=v1(g1(0)-g2(0))+v2g3(0)+v3g4(0)+v4(g5(0)-g6(0))+v5g7(0)+v6g8(0), where vi=∑m≥0vimλ-m, i=1,2,3,4,5,6. Solving the stationary zero curvature equation Vx=[U,V] gives rise to the recursion relation as follows:
(9)v1mx=u1v3m-u2v2m,v2mx=-v2m+1+(u1-u2)v2m-2u1v1m,v3mx=v3m+1-(u1-u2)v3m+2u2v1m,v4mx=u1v6m-u4v2m-u2v5m+u3v3m+u3v6m-u4v5m,v5mx=(u1-u2)v5m-2u3v1m-v5m+1-2u1v4m+(u3-u4)(v2m+v5m)-2u3v4m,v6mx=2u4v1m-(u1-u2)v6m+v6m+1+2u2v4m-(u3-u4)(v3m+v6m)+2u4v4m,v10=-α2≠0,v20=v30=v40=v50=v60=0,v11=0,v21=αu1,v31=αu2,v41=0,v51=αu3,v61=αu4,v12=αu1u2,v22=αu1(u1-u2)-αu1x,v32=αu2(u1-u2)+αu2x,v42=α(u1u4+u2u3+u3u4),v52=αu3(u1-u2)-αu3x+α(u1+u3)(u3-u4),v62=αu4(u1-u2)+αu4x+α(u2+u4)(u3-u4).
Denoting V+(n)=∑m=0n(v1m,v2m,v3m,v4m,v5m,v6m)Tλn-m and V-(n)=λnV-V+(n), it is easy to compute
(10)-V+x(n)+[U,V+(n)]=v2n+1g3(0)-v3n+1g4(0)+v5n+1g7(0)-v6n+1g8(0).
Take
(11)V(n)=V+(n)+Δn,Δn=12(v2n-v3n-2v1n)(g1(0)-g2(0))+12(v5n-v6n-2v4n)(g5(0)-g6(0)).
Thus, the zero curvature equation
(12)Ut-Vx(n)+[U,V(n)]=0
leads to the following integrable system:
(13)Ut=(u1u2u3u4)t=(-v2n+1+(v2n-v3n-2v1n)u1v3n+1-(v2n-v3n-2v1n)u2-v5n+1+(v2n-v3n-2v1n)u3+(v5n-v6n-2v4n)(u1+u3)v6n+1-(v2n-v3n-2v1n)u4-(v5n-v6n-2v4n)(u2+u4))=(v2nx-v1nxv3nx+v1nxv5nx-v4nxv6nx+v4nx)=(000∂00∂00∂0-∂∂0-∂0)(v1n+v3n+v4n+v6n-v1n+v2n-v4n+v5nv1n+v3n-v1n+v2n)=J1Pn,
where J1 is a Hamiltonian operator and Pn+1=L1Pn, the recurrence operator L1 is given from (9) by
(14)L1=(L11L12L13L14L21L22L23L2400L33L3400L43L44),
where
(15)L11=∂-(u2+u4)+∂-1(u1+u3)∂,L12=(u2+u4)+∂-1(u2+u4)∂,L13=-u4+∂-1u3∂,L14=u4+∂-1u4∂,L21=-(u1+u3)-∂-1(u1+u3)∂,L22=-∂+(u1+u3)-∂-1(u2+u4)∂,L23=-u3-∂-1u3∂,L24=u3-∂-1u4∂,L33=∂-u2+∂-1u1∂,L34=u2+∂-1u2∂,L43=-u1-∂-1u1∂,(16)L44=u1-∂-1u2∂-∂.
Therefore, the system (13) can be written as
(17)(u1u2u3u4)t=J1L1n-1(v11+v31+v41+v61-v11+v21-v41+v51v11+v31-v11+v21).
When u3=u4=0, the system (13) reduces to the Levi hierarchy; therefore, in terms of the definition of integrable coupling, we conclude that the system (13) is an integrable coupling of the Levi hierarchy. Especially taking n=2, we have the following reduced equations:
(18)u1t=α(u12-2u1u2)x-αu1xx,u2t=α(-u22+2u1u2)x+αu2xx,u3t=2α(u1u3-u2u3-u1u4-u3u4)x-αu3xx+α(u32)x,u4t=2α(u1u4-u2u4+u3u4+u2u3)x+αu4xx-α(u42)x.
Obviously, (18) are nonlinear equations in u3 and u4, so we call (13) the nonlinear integrable coupling of the Levi hierarchy.

3.2. Nonlinear Integrable Couplings of WKI Hierarchy

Consider an isospectral problem
(19)ϕx=Uϕ,λt=0,U=i[g2(1)-g1(1)]+u1g3(1)+u2g4(1)+u3g7(1)+u4g8(1).
Set V=∑m≥0V~, where
(20)V~=[λv1mg1(-m)-λv1mg2(-m)+(v2mx+iλu1v1m)FF×g3(-m)+(v3mx+iλu2v1m)g4(-m)FF+λv4mg5(-m)-λv4mg6(-m)FF+(v5mx+iλu3v1m+iλu1v4m+iλu3v4m)g7(-m)FF+(v6mx+iλu4v1m+iλu2v4m+iλu4v4m)g8(-m)].
Because every term in U includes λ, V is different from the common form and includes potentials u1, u2, u3, and u4 and v2mx, v3mx, v5mx, v6mx, and so on. Then the zero curvature equation Vx=[U,V] yields
(21)v1mx=u1v3mx-u2v2mx,i(u1v1m+1)x+v2mxx=-2iv2m+1x,i(u2v1m+1)x+v3mxx=2iv3m+1x,v4mx=u1v6mx-u2v5mx+u3v3mx-u4v2mx+u3v6mx-u4v5mx,i(u3v1m+1+u1v4m+1+u3v4m+1)x+v5mxx=-2iv5m+1x,i(u4v1m+1+u2v4m+1+u4v4m+1)x+v6mxx=2iv6m+1x,v10=α1,v20=α2,v30=α3,v40=α4,v50=α5,v60=α6,v11=2p,v21=-u1p,v31=u2p,v41=-2p-2p′,v51=u1p+u1+u3p′,v61=-u2p-u2+u4p′,p=1-u1u2,p′=1-(u1+u3)(u2+u4).
Denoting V+(n)=∑m=0nV~=λnV-V-(n), then we have -V+x(n)+[U,V+(n)]=V-x(n)-[U,V-(n)]. A direct calculation reads
(22)-V+x(n)+[U,V+(n)]=-λv2n-1xxg3(0)-λv3n-1xxg4(0)-λv5n-1xxg7(0)-λv6n-1xxg8(0).
Therefore, the zero curvature equation
(23)Ut-Vx(n)+[U,V(n)]=0
admits
(24)Ut=(u1u2u3u4)t=(v2n-1xxv3n-1xxv5n-1xxv6n-1xx)=(000∂200-∂200∂20-∂2-∂20∂20)×(-v3n-1-v6n-1v2n-1+v5n-1-v3n-1v2n-1)=J2Qn-1.
Here J2 is a Hamiltonian operator and Qn=L2Qn-1, the recurrence operator L2 is given from (21) by
(25)L2=(L11L12L13L14L21L22L23L2400L33L3400L43L44),
where
(26)L11=-i2∂-i4u2+u4p′∂-1u1+u3p′∂2,L12=i4u2+u4p′∂-1u2+u4p′∂2,L13=-i4[u2+u4p′∂-11p′(-u1∂2+q∂qp∂-1u1p∂2)ffffffff+u2+u4p∂-1u1p∂2],L14=i4[u2+u4p′∂-11p′(-u2∂2+q∂qp∂-1u2p∂2)fffffff+u2+u4p∂-1u2p∂2],L21=-i4u1+u3p′∂-1u1+u3p′∂2,L22=i2∂+i4u1+u3p′∂-1u2+u4p′∂2,L23=-i4[u1+u3p′∂-11p′(-u1∂2+q∂qp∂-1u1p∂2)ffffffff+u1+u3p∂-1u1p∂2],L24=i4[u1+u3p′∂-11p′∂21p′∂-1fffffff×1p′(-u2∂2+q∂qp∂-1u2p∂2)fffffff+u1+u3p∂-1u2p∂2],L33=-i2∂-i4u2p∂-1u1p∂2,L34=i4u2p∂-1u2p∂2,L43=-i4u1p∂-1u1p∂2,L44=i2∂+i4u1p∂-1u2p∂2.
Here p, p′ are given in (21) and q=u1u4+u2u3+u3u4. Hence, the system (24) can be written as
(27)(u1u2u3u4)t=J2L2n-2(v31+v61v21+v51v31v21).
When u3=u4=0, the system (24) is just the WKI hierarchy. Taking n=2, the system (24) reduces the following equations:
(28)u1t=(-u11-u1u2)xx,u2t=(u21-u1u2)xx,u3t=(u11-u1u2+u1+u31-(u1+u3)(u2+u4))xx,u4t=(-u21-u1u2-u2+u41-(u1+u3)(u2+u4))xx.
It is easy to find that (28) are nonlinear equations in u3 and u4, so we call (24) the nonlinear integrable coupling of the WKI hierarchy.

4. Hamiltonian Structures

In this section, we will seek the Hamiltonian structures of the nonlinear integrable couplings of the Levi hierarchy (13) and the WKI hierarchy (24) by virtue of the variational identity. First, we construct a linear map G→R8, g=∑i=18vigi(0)→v=(v1,v2,v3,v4,v5,v6,v7,v8)T, g∈G,v∈R8. We can conclude that the linear map is an isomorphism from G to R8. Let a,b∈R8; matrix R(b) is determined by [9]
(29)[a,b]T=aTR(b),
where a=(a1,…,a8)T and b=(b1,…,b8)T. From (29), we have(30)R(b)=(00b3-b400b7-b800-b3b400-b7b8b4-b4b2-b10b8-b8b6-b50-b3b30b1-b2-b7b70b5-b6000000b3+b7-b4-b8000000-b3-b7b4+b80000b4+b8-b4-b8b2-b1-b5+b600000-b3-b7b3+b70b1-b2+b5-b6).

Solving the matrix equation for the constant matrix F, R(b)F=-(R(b)F)T, FT=F,
(31)F=(1000100001000100000100010010001010000000010000000001000000100000).
Then in terms of F, define a linear functional in the R8(32){a,b}=aTFb=(a1+a5)b1+(a2+a6)b2+(a4+a8)b3+(a3+a7)b4+a1b5+a2b6+a4b7+a3b8.
It is easy to find that {a,b} satisfies the variational identity
(33)δδu∫{V,Uλ}dx=λ-γ∂∂λλγ{V,Uu}.
Rewrite the Lax pair of nonlinear integrable coupling of the Levi hierarchy as follows:
(34)U=(12(u1-u2-λ),12(λ-u1+u2),u1,u2,hhh12(u3-u4),12(u4-u3),u3,u4)T,V=(v1,-v1,v2,v3,v4,-v4,v5,v6)T.
By using (32), we have
(35){V,Uλ}=-v1-v4,{V,Uu1}=v1+v3+v4+v6,{V,Uu2}=-v1+v2-v4+v5,{V,Uu3}=v1+v3,{V,Uu4}=-v1+v2.
According the variational identity (33), we have
(36)δδu∫(-v1-v4)dx=λ-γ∂∂λλγ(v1+v3+v4+v6,-v1+v2jjjjjgg-v4+v5,v1+v3,-v1+v2)T.
Comparing the coefficients of λ-n-2 yields
(37)δδu∫(-v1n+1-v4n+1)dx=(γ-n)×(v1n+v3n+v4n+v6n-v1n+v2n-v4n+v5nv1n+v3n-v1n+v2n).
Taking n=1 gives rise to γ=0. Therefore,
(38)Pn=δHnδu,Hn=∫an+1+dn+1ndx,ggggggggggn≥1.
Hence, the nonlinear integrable coupling of the Levi hierarchy has the following Hamiltonian structure:
(39)Ut=J1δHnδu,n≥1.
Similar to (34), in order to deduce to the Hamiltonian structure of the nonlinear integrable coupling of the WKI hierarchy, we rewrite the Lax pair as follows:
(40)U=(-iλ,iλ,λu1,λu2,0,0,λu3,λu4)TV=(λv1,-λv1,v2x+iλu1v1,v3x+iλu2v1,λv4,hhhh-λv4,v5x+iλu3(v1+v4)hhhh+iλu1v4,v6x+iλu4(v1+v4)+iλu2v4)T.
Repeat the above procedure; we have
(41){V,Uλ}=-2i(v1+v4)+2iu1(v3+v6)-2iu2(v2+v5)+2iu3v3-2iu4v2,{V,Uu1}=2i(v3+v6),{V,Uu2}=-2i(v2+v5),{V,Uu3}=2iv3,{V,Uu4}=-2iv2,δδu∫2i[-(v1n-1+v4n-1)+u1(v3n-1+v6n-1)ggggggg-u2(v2n-1+v5n-1)+u3v3n-1-u4v2n-1]dxgggj=2i(2+γ-n)(-v3n-1-v6n-1v2n-1+v5n-1-v3n-1v2n-1).
Taking n=2 in above equation gives γ=-1. Therefore, Qn-1=(δH~n-1/δu), n≥2, where
(42)H~n-1=∫((v1n-1-v4n-1)-u1(v3n-1-v6n-1)hhh.+u2(v2n-1-v5n-1)-u3v3n-1hhh.+u4v2n-1)hhh.×(n-1)-1dx.
Hence, the nonlinear integrable coupling of the WKI hierarchy has the following Hamiltonian structure:
(43)Ut=J2δH~n-1δu,n≥2.

5. Conclusions

In this paper, we presented a set of new 8-dimensional matrix Lie algebra by virtue of the Lie algebra given in [14–16]. With the help of the Lie algebra, we obtain the nonlinear integrable couplings of the Levi hierarchy and the WKI hierarchy. Their Hamiltonian structures are also worked out by the variational identity. The Lie algebra constructed in this paper can be used to generate the nonlinear integrable couplings of other hierarchies. We will study these problems in the future.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the Strategic Pioneering Program of Chinese Academy of Sciences (no. XDA 10020104), the Global Change and Air-Sea Interaction (no. GASI-03-01-01-02), the Nature Science Foundation of Shandong Province of China (no. ZR2013AQ017), the Science and Technology Plan Project of Qingdao (no. 14-2-4-77-jch), the Open Fund of the Key Laboratory of Ocean Circulation and Waves, the Chinese Academy of Science (no. KLOCAW1401), the Open Fund of the Key Laboratory of Data Analysis and Application, and the State Oceanic Administration (no. LDAA-2013-04).

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