Starting from a matrix Lie superalgebra, two-component super AKNS system is constructed. By making use of monononlinearization technique of Lax pairs, we find that the obtained two-component super AKNS system is a finite-dimensional integrable super Hamiltonian system. And its Lax representation and r-matrix are also given in this paper.

1. Introduction

The inverse scattering method provides us with a powerful tool to generate multicomponent soliton equations. In [1–4], they have constructed many multicomponent soliton equations, which are much more important for physicists and mathematicians than one-component ones, owing to the fact that they possess rich structure and have more extensive prospect.

Monononlinearization of Lax pair is a method to obtain finite-dimensional integrable Hamiltonian system, which was firstly proposed by Cao in [5]. The main idea of monononlinearization includes the following three aspects. Firstly, they find a symmetry constraint between potential and eigenfunctions. Secondly, substituting the symmetry constraint into the spectral problem, they obtain constrained finite-dimensional system. Lastly, they show that obtained constrained system is Hamiltonian system and completely integrable in the Liouville sense. Many finite-dimensional integrable Hamiltonian systems are constructed in [6–10]. This method was generalized by Ma and Strampp in [11]. The main difference of binary-nonlinearization and monononlinearization lies in the following two aspects. One is that binary-nonlinearization needs to introduce adjoint spectral problem, and, thus, both spectral problem and adjoint spectral problem constitute even-dimensional system. The other is that symmetry constraint proposed in the procedure of binary-nonlinearization is not only associated with eigenfunctions but also associated with adjoint eigenfunctions. According to the Liouville integrable theorem which states that a 2n-dimensional Hamiltonian system over some region Ω∈R2n with n independent integrals of motion in involution may be integrated by quadratures, we cannot apply monononlinearization to odd-dimensional spectral problem. However, we can apply both monononlinearization and binary-nonlinearization to the even-dimensional spectral problem. Many finite-dimensional integrable Hamiltonian systems are constructed in [12–14]. Owing to one-component super integrable system which is associated with a 3×3 spectral matrix, we just consider binary-nonlinearization in our previous papers [15–17]. From the above analysis, we propose the following questions. Does even-dimensional spectral problem of super integrable system exist? If it exists, can we apply monononlinearization to the super integrable system? All of these questions will be answered in this paper.

The paper is organized as follows. In Section 2, we derive two-component super AKNS system and write this new system as the super Hamiltonian form. In Section 3, we find a symmetry constraint which is only associated with eigenfunctions, and, after substituting the symmetry constraint into the N copies spectral problem, we obtained 6N-dimensional constrained system. Furthermore, we rewrite the 6N-dimensional system as the super Hamiltonian form. Lax representation and r-matrix of the constrained system are given in Section 4. In the last section, some conclusions and discussions are given.

2. Two-Component Super AKNS System

Let us start with the following linear space G={e1,e2,e3,e4,e5}:
(1)e1=(E000-E0000),e2=(0E0000000),e3=(000E00000),e4=(00E0000-E0),e5=(00000EE00),
where E=diag(1,1) is a 2×2 unit matrix, 0 is a 2×2 zero matrix, G0={e1,e2,e3} is even, and G1={e4,e5} is odd. After a direct calculation, we obtain that
(2)[e1,e2}=2e2,[e1,e3}=-2e3,[e2,e3}=e1,[e1,e4}=e4,[e1,e5}=-e5,[e2,e5}=e4,[e3,e4}=e5,[e4,e4}=-2e2,[e4,e5}=e1,[e5,e5}=2e3,
and the others are zeros, where
(3)[a,b}=ab-(-1)p(a)p(b)ba
is the super Lie bracket and p(f) denotes the parity of the arbitrary element f.

It is easy to prove that the linear space G is matrix Lie superalgebra. The corresponding loop superalgebra G~ is presented as
(4)ei(n)=eiλn,[e1(m),e2(n)}=2e2(m+n),[e1(m),e3(n)}=-2e3(m+n),[e2(m),e3(n)}=e1(m+n),[e1(m),e4(n)}=e4(m+n),[e1(m),e5(n)}=-e5(m+n),[e2(m),e4(n)}=[e3(m),e5(n)}=0,[e2(m),e5(n)}=e4(m+n),[e3(m),e4(n)}=e5(m+n),[e4(m),e4(n)}=-2e2(m+n),[e4(m),e5(n)}=e1(m+n),[e5(m),e5(n)}=2e3(m+n),degei(n)=n,(i=1,2,3,4,5).

In what follows, we will construct multicomponent super integrable equations via the matrix Lie superalgebra G. Firstly, let us consider the following superisospectral problem:
(5)ϕx=U(u,λ)ϕ,
where
(6)U(u,λ)=-e1(1)+qe2(0)+re3(0)+αe4(0)+βe5(0),
which can be written as the following matrix form:
(7)ϕx=(-λEqαrλEββ-α0)ϕ,
where q=diag(q1,q2), r=diag(r1,r2), α=diag(α1,α2), β=diag(β1,β2), u=(q,r,α,β)T is a potential, λ is a spectral parameter which satisfies λtn=0, and ϕ=(ϕ1,…,ϕ6)T is an eigenfunction.

Taking
(8)V=(abρc-aδδ-ρ0),
where a=diag(a1,a2), b=diag(b1,b2), c=diag(c1,c2), ρ=diag(ρ1,ρ2), and δ=diag(δ1,δ2), the adjoint representation equation
(9)Vx=[U,V]
leads to
(10)ak,x=qkck-rkbk+αkδk+βkρk,k=1,2,bk,x=-2λbk-2qkak-2αkρk,k=1,2,ck,x=2λck+2rkak+2βkδk,k=1,2,ρk,x=-λρk+qkδk-αkak-βkbk,k=1,2,δk,x=λδk+rkρk-αkck+βkak,k=1,2.
On setting ak=∑m≥0ak(m)λ-m, bk=∑m≥0bk(m)λ-m, ck=∑m≥0ck(m)λ-m, ρk=∑m≥0ρk(m)λ-m, and δk=∑m≥0δk(m)λ-m(k=1,2), (10) engenders equivalently
(11)bk(0)=ck(0)=ρk(0)=δk(0)=0,k=1,2,ak,x(m)=qkck(m)-rkbk(m)+αkδk(m)+βkρk(m),k=1,2,m≥0,bk,x(m)=-2bk(m+1)-2qkak(m)-2αkρk(m),k=1,2,m≥0,ck,x(m)=2ck(m+1)+2rkak(m)+2βkδk(m),k=1,2,m≥0,ρk,x(m)=-ρk(m+1)+qkδk(m)-αkak(m)-βkbk(m),hhhhhhhhhhhhlhhhhhhhk=1,2,m≥0,δk,x(m)=δk(m+1)+rkρk(m)-αkck(m)+βkak(m),k=1,2,m≥0,
which results in a recursion relation to determine ak(m),bk(m),ck(m),ρk(m),δk(m)(k=1,2):
(12)(c1(m+1)c2(m+1)b1(m+1)b2(m+1)2δ1(m+1)2δ2(m+1)-2ρ1(m+1)-2ρ2(m+1))=L(c1(m)c2(m)b1(m)b2(m)2δ1(m)2δ2(m)-2ρ1(m)-2ρ2(m)),
where
(13)L=(L11L12L13L14L21L22L23L24L31L32L33L34L41L42L43L44),
with
(14)L11=diag(12∂-r1∂-1q1,12∂-r2∂-1q2),L12=diag(r1∂-1r1,r2∂-1r2),L13=diag(-12r1∂-1α1-12β1,-12r2∂-1α2-12β2),L14=diag(12r1∂-1β1,12r2∂-1β2),L21=diag(-q1∂-1q1,-q2∂-1q2),L22=diag(-12∂+q1∂-1r1,-12∂+q2∂-1r2),L23=diag(-12q1∂-1α1,-12q2∂-1α2),L24=diag(12α1+12q1∂-1β1,12α2+12q2∂-1β2),L31=diag(2α1-2β1∂-1q1,2α2-2β2∂-1q2),L32=diag(2β1∂-1r1,2β2∂-1r2),L33=diag(∂-β1∂-1α1,∂-β2∂-1α2),L34=diag(r1+β1∂-1β1,r2+β2∂-1β2),L41=diag(2α1∂-1q1,2α2∂-1q2),L42=diag(2β1-2α1∂-1r1,2β2-2α2∂-1r2),L43=diag(-q1+α1∂-1α1,-q2+α2∂-1α2),L44=diag(-∂-α1∂-1β1,-∂-α2∂-1β2).
It is easy to obtain that ak,x(0)=0(k=1,2). Therefore, we choose ak(0)=-1(k=1,2) and select constants of integral to be zero, which lead to the first few terms being worked out as follows:
(15)ak(1)=0,bk(1)=qk,ck(1)=rk,ρk(1)=αk,δk(1)=βk,ak(2)=12qkrk+αkβk,bk(2)=-12qk,x,ck(2)=12rk,x,ρk(2)=-αk,x,δk(2)=βk,x,
where k=1,2.

Secondly, let us associate (5) with the following auxiliary spectral problem:
(16)ϕtn=V(n)ϕ,
where
(17)V(n)=∑m=0n(a1(m)0b1(m)0ρ1(m)00a2(m)0b2(m)0ρ2(m)c1(m)0-a1(m)0δ1(m)00c2(m)0-a2(m)0δ2(m)δ1(m)0-ρ1(m)0000δ2(m)0-ρ2(m)00)λn-m.
The compatibility condition of (5) and (16) leads to the famous zero curvature equations
(18)Utn-Vx(n)+[U,V(n)]=0,n≥1,
which lead to isospectral super integrable equations
(19)(q1q2r1r2α1α2β1β2)tn=(-2b1(n+1)-2b2(n+1)2c1(n+1)2c2(n+1)-ρ1(n+1)-ρ2(n+1)δ1(n+1)δ2(n+1)).
Under the special reduction of q2=r2=α2=δ2=0, (19) is equivalent to the super AKNS equations [15, 18, 19], and thus (19) is called a two-component super AKNS equation.

Lastly, super Hamiltonian structures of the two-component super AKNS equations (19) may be established by applying a powerful tool, that is, the so-called supertrace identity [20, 21]
(20)δδu∫str(V∂U∂λ)dx=λ-γ∂∂λλγstr(∂U∂uV).
As is usual, we need the following equalities which are easy to calculate:
(21)str(V∂U∂λ)=-2(a1+a2),str(∂U∂qkV)=ck,str(∂U∂rkV)=bk,str(∂U∂αkV)=2δk,str(∂U∂βkV)=-2ρk,
where k=1,2. Substituting the above equality (21) into the supertrace identity (20) and comparing the coefficients of λ-m-2 on two sides, we arrive at
(22)(δδq1δδq2δδr1δδr2δδα1δδα2δδβ1δδβ2)∫-2(a1(m+2)+a2(m+2))dx=(γ-m-1)(c1(m+1)c2(m+1)b1(m+1)b2(m+1)2δ1(m+1)2δ2(m+1)-2ρ1(m+1)-2ρ2(m+1)),
which leads to the constant γ=0 with m=0. Thus, we have
(23)(c1(m+1)c2(m+1)b1(m+1)b2(m+1)2δ1(m+1)2δ2(m+1)-2ρ1(m+1)-2ρ2(m+1))=(δδq1δδq2δδr1δδr2δδα1δδα2δδβ1δδβ2)H¯m,H¯m=∫2m+2(a1(m+2)+a2(m+2))dx.
Now, it follows from (23) that the two-component super AKNS system (19) has the following super bi-Hamiltonian structure:
(24)utn=(q1q2r1r2α1α2β1β2)tn=J(c1(n+1)c2(n+1)b1(n+1)b2(n+1)2δ1(n+1)2δ2(n+1)-2ρ1(n+1)-2ρ2(n+1))=JδH¯nδu=MδH¯n-1δu,
where the super Hamiltonian pair (J,M=JL) reads as
(25)J=(0-2E002E00000012E0012E0),M=JL=(-2L21-2L22-2L23-2L242L112L122L132L1412L4112L4212L4312L4412L3112L3212L3312L34),
where Lmn(1≤m,n≤4) are given by (13).

The first nonlinear equations in two-component AKNS system (24) are given by
(26)q1,t2=-12q1,xx+q12r1+2q1α1β1-2α1α1,x,q2,t2=-12q2,xx+q22r2+2q2α2β2-2α2α2,x,r1,t2=12r1,xx-q1r12-2r1α1β1-2β1β1,x,r2,t2=12r2,xx-q2r22-2r2α2β2-2β2β2,x,α1,t2=-α1,xx-q1β1,x+12q1r1α1-12q1,xβ1,α2,t2=-α2,xx-q2β2,x+12q2r2α2-12q2,xβ2,β1,t2=β1,xx+r1α1,x+12r1,xα1-12q1r1β1,β2,t2=β2,xx+r2α2,x+12r2,xα2-12q2r2β2,
and the corresponding temporal part of the Lax system is
(27)ϕt2=V(2)ϕ,
where(28)V(2)=(-λ2+v10q1λ-12q1,x0α1λ-α1,x00-λ2+v20q2λ-12q2,x0α2λ-α2,xr1λ+12r1,x0λ2-v10β1λ+β1,x00r2λ+12r2,x0λ2-v20β2λ+β2,xβ1λ+β1,x0-α1λ+α1,x0000β2λ+β2,x0-α2λ+α2,x00),with vk=(1/2)qkrk+αkβk(k=1,2).

3. Finite-Dimensional Super Hamiltonian System

In this section, we will apply monononlinearization to the two-component super AKNS system (24). It is easy to find that, for the super spectral problem (1), variational derivative of λ with respect to the potential u reads (up to a constant factor)
(29)δλδu=(δλδq1δλδq2δλδr1δλδr2δλδα1δλδα2δλδβ1δλδβ2)=(ϕ32ϕ42-ϕ12-ϕ222ϕ3ϕ52ϕ4ϕ6-2ϕ1ϕ5-2ϕ2ϕ6).
When zero boundary conditions (lim|x|→+∞ϕj=0, 1≤j≤6) are imposed, we can verify a simple characteristic property of the above variational derivative:
(30)Lδλδu=λδλδu,
where L is given by (13).

To carry out nonlinearization of N copies of systems (7) and (27) with N distinct parameters λj(1≤j≤N), we consider the following Bargmann symmetry constraint:
(31)q1=-〈Φ1,Φ1〉,q2=-〈Φ2,Φ2〉,r1=〈Φ3,Φ3〉,r2=〈Φ4,Φ4〉,α1=〈Φ1,Φ5〉,α2=〈Φ2,Φ6〉,β1=〈Φ3,Φ5〉,β2=〈Φ4,Φ6〉,
where Φj=(ϕj1,…,ϕjN)T(1≤j≤6) and 〈·,·〉 refers to the standard inner product of the Euclidian space ℝN. Now, substituting the constraint (31) into N copies of system (7), we obtain the following constrained vector system:
(32)Φ1,x=-ΛΦ1-〈Φ1,Φ1〉Φ3+〈Φ1,Φ5〉Φ5=∂H1∂Φ3,Φ2,x=-ΛΦ2-〈Φ2,Φ2〉Φ4+〈Φ2,Φ6〉Φ6=∂H1∂Φ4,Φ3,x=〈Φ3,Φ3〉Φ1+ΛΦ3+〈Φ3,Φ5〉Φ5=-∂H1∂Φ1,Φ4,x=〈Φ4,Φ4〉Φ2+ΛΦ4+〈Φ4,Φ6〉Φ6=-∂H1∂Φ2,Φ5,x=〈Φ3,Φ5〉Φ1-〈Φ1,Φ5〉Φ3=∂H1∂Φ5,Φ6,x=〈Φ4,Φ6〉Φ2-〈Φ2,Φ6〉Φ4=∂H1∂Φ6,
where
(33)H1=-〈ΛΦ1,Φ3〉-〈ΛΦ2,Φ4〉-12〈Φ1,Φ1〉〈Φ3,Φ3〉-12〈Φ2,Φ2〉〈Φ4,Φ4〉+〈Φ1,Φ5〉〈Φ3,Φ5〉+〈Φ2,Φ6〉〈Φ4,Φ6〉,
with Λ=diag(λ1,…,λN) and the Poisson bracket is defined by
(34){f,g}=∑j=1N(∂f∂ϕ1j∂g∂ϕ3j+∂f∂ϕ2j∂g∂ϕ4j-∂f∂ϕ3j∂g∂ϕ1j-∂f∂ϕ4j∂g∂ϕ2j+∂f∂ϕ5j∂g∂ϕ5j+∂f∂ϕ6j∂g∂ϕ6j).

Analogously, making use of (31) and (32), the t2-part (27) of the two-component AKNS equations (24) is constrained as the following 6N-dimensional system:
(35)ϕ1j,t2=(-λj2+12q~1r~1+α~1β~1)ϕ1j+(q~1λj-12q~1,x)ϕ3j+(α~1λj-α~1,x)ϕ5j,1≤j≤N,ϕ2j,t2=(-λj2+12q~2r~2+α~2β~2)ϕ2j+(q~2λj-12q~2,x)ϕ4j+(α~2λj-α~2,x)ϕ6j,1≤j≤N,ϕ3j,t2=(r~1λj+12r~1,x)ϕ1j+(λj2-12q~1r~1-α~1β~1)ϕ3j+(β~1λj+β~1,x)ϕ5j,1≤j≤N,ϕ4j,t2=(r~2λj+12r~2,x)ϕ2j+(λj2-12q~2r~2-α~2β~2)ϕ4j+(β~2λj+β~2,x)ϕ6j,1≤j≤N,ϕ5j,t2=(β~1λj+β~1,x)ϕ1j+(-α~1λj+α~1,x)ϕ3j,mmnmmmmmmmmmnm1≤j≤N,ϕ6j,t2=(β~2λj+β~2,x)ϕ2j+(-α~2λj+α~2,x)ϕ4j,mnmmmmmmmmmmmm1≤j≤N,
where q~k, r~k, α~k, β~k denote the constrained qk, rk, αk, βk and q~k,x, r~k,x, α~k,x, β~k,x are given as follows:
(36)q~1,x=2〈ΛΦ1,Φ1〉+2〈Φ1,Φ1〉〈Φ1,Φ3〉,q~2,x=2〈ΛΦ2,Φ2〉+2〈Φ2,Φ2〉〈Φ2,Φ4〉,r~1,x=2〈ΛΦ3,Φ3〉+2〈Φ3,Φ3〉〈Φ1,Φ3〉,r~2,x=2〈ΛΦ4,Φ4〉+2〈Φ4,Φ4〉〈Φ2,Φ4〉,α~1,x=-〈ΛΦ1,Φ5〉-〈Φ1,Φ3〉〈Φ1,Φ5〉,α~2,x=-〈ΛΦ2,Φ6〉-〈Φ2,Φ4〉〈Φ2,Φ6〉,β~1,x=〈ΛΦ3,Φ5〉+〈Φ1,Φ3〉〈Φ3,Φ5〉,β~2,x=〈ΛΦ4,Φ6〉+〈Φ2,Φ4〉〈Φ4,Φ6〉.
After a direct calculation, constrained system (35) can be written as the vector form
(37)Φ1,t2=∂H2∂Φ3,Φ2,t2=∂H2∂Φ4,Φ3,t2=-∂H2∂Φ1,Φ4,t2=-∂H2∂Φ2,Φ5,t2=∂H2∂Φ5,Φ6,t2=∂H2∂Φ6,
where
(38)H2=-〈Λ2Φ1,Φ3〉-〈Λ2Φ2,Φ4〉-12〈Φ1,Φ1〉〈ΛΦ3,Φ3〉-12〈Φ2,Φ2〉〈ΛΦ4,Φ4〉-12〈ΛΦ1,Φ1〉〈Φ3,Φ3〉-12〈ΛΦ2,Φ2〉〈Φ4,Φ4〉-12〈Φ1,Φ1〉〈Φ1,Φ3〉〈Φ3,Φ3〉-12〈Φ2,Φ2〉〈Φ2,Φ4〉〈Φ4,Φ4〉+〈Φ1,Φ5〉〈ΛΦ3,Φ5〉+〈Φ2,Φ6〉〈ΛΦ4,Φ6〉+〈ΛΦ1,Φ5〉〈Φ3,Φ5〉+〈ΛΦ2,Φ6〉〈Φ4,Φ6〉+〈Φ1,Φ3〉〈Φ1,Φ5〉〈Φ3,Φ5〉+〈Φ2,Φ4〉〈Φ2,Φ6〉〈Φ4,Φ6〉.

To this end, we show that the constrained N copies of systems (7) and (27) are super Hamiltonian systems (32) and (37).

In what follows, we will show that super Hamiltonian systems (32) and (37) are both completely integrable in the Liouville sense. First of all, through a straightforward and tedious calculation, we arrive at the following proposition.

Proposition 1.

Super Hamiltonian systems (32) and (37) have the following Lax representations, respectively:
(39)Lx(λ)=[U~,L(λ)],(40)Lt2(λ)=[V~(2),L(λ)],
where U~ and V~(2) are, respectively, U and V(2) under symmetry constraint (31), and
(41)L(λ)=(A1(λ)0B1(λ)0ϱ1(λ)00A2(λ)0B2(λ)0ϱ2(λ)C1(λ)0-A1(λ)0ϖ1(λ)00C2(λ)0-A2(λ)0ϖ2(λ)ϖ1(λ)0-ϱ1(λ)0000ϖ2(λ)0-ϱ2(λ)00),
with
(42)A1(λ)=-1+∑j=1N1λ-λjϕ1jϕ3j,A2(λ)=-1+∑j=1N1λ-λjϕ2jϕ4j,B1(λ)=-∑j=1N1λ-λjϕ1j2,B2(λ)=-∑j=1N1λ-λjϕ2j2,C1(λ)=∑j=1N1λ-λjϕ3j2,C2(λ)=∑j=1N1λ-λjϕ4j2,ϱ1(λ)=∑j=1N1λ-λjϕ1jϕ5j,ϱ2(λ)=∑j=1N1λ-λjϕ2jϕ6j,ϖ1(λ)=∑j=1N1λ-λjϕ3jϕ5j,ϖ2(λ)=∑j=1N1λ-λjϕ4jϕ6j.

Under the Poisson bracket (34), we have
(43){Ak(λ),Bk(λ)}=2μ-λ(-Bk(λ)+Bk(μ))=-{Bk(λ),Ak(λ)},k=1,2,{Ak(λ),Ck(λ)}=2μ-λ(Ck(λ)-Ck(μ))=-{Ck(λ),Ak(λ)},k=1,2,{Ak(λ),ϱk(λ)}=1μ-λ(-ϱk(λ)+ϱk(μ))=-{ϱk(λ),Ak(λ)},k=1,2,{Ak(λ),ϖk(λ)}=1μ-λ(ϖk(λ)-ϖk(μ))=-{ϖk(λ),Ak(λ)},k=1,2,{Bk(λ),Ck(λ)}=4μ-λ(-Ak(λ)+Ak(μ))=-{Ck(λ),Bk(λ)},k=1,2,{Bk(λ),ϖk(λ)}=2μ-λ(-ϱk(λ)+ϱk(μ))=-{ϖk(λ),Bk(λ)},k=1,2,{Ck(λ),ϱk(λ)}=2μ-λ(-ϖk(λ)+ϖk(μ))=-{ϱk(λ),Ck(λ)},k=1,2,{ϱk(λ),ϱk(λ)}=1μ-λ(-Bk(λ)+Bk(μ)),k=1,2,{ϖk(λ),ϖk(λ)}=1μ-λ(Ck(λ)-Ck(μ)),k=1,2,{ϱk(λ),ϖk(λ)}=1μ-λ(Ak(λ)-Ak(μ))={ϖk(λ),ϱk(λ)},k=1,2,
and the others are zero.

These relations imply the following proposition immediately.

Proposition 2.

The Lax matrix L(λ) satisfied the following r-matrix relation:
(44){L1(λ),L2(μ)}=1μ-λ[P,L1(λ)+L2(μ)],
where
(45)P=σ1⊗σ1+σ2⊗σ2+2(σ3⊗σ5+σ5⊗σ3+σ4⊗σ6+σ6⊗σ4)+σ7⊗σ9+σ9⊗σ7+σ8⊗σ10+σ10⊗σ8,
with
(46)σ1=E11-E33,σ2=E22-E44,σ3=E13,σ4=E24,σ5=E31,σ6=E42,σ7=E15-E53,σ8=E26-E64,σ9=E35+E51,σ10=E46+E62,Eij is the 6×6 matrix having 1 in the (i,j)th position and zeros elsewhere, and L1(λ)=L(λ)⊗I and L2(μ)=I⊗L(μ).

Therefore, according to the general theory of the r-matrix [22], we know that (1/2)StrL2(λ) is a generating function of conserved integrals of motions. Explicitly, we can expand (1/2)StrL2(λ) as follows:
(47)12StrL2(λ)=∑n≥0Fnλ-n,
where
(48)F1=-2〈Φ1,Φ3〉-2〈Φ2,Φ4〉,Fn=∑j=1n-1[〈Λj-1Φ1,Φ3〉〈Λn-j-1Φ1,Φ3〉-〈Λj-1Φ1,Φ1〉〈Λn-j-1Φ3,Φ3〉+2〈Λj-1Φ1,Φ5〉〈Λn-j-1Φ3,Φ5〉+〈Λj-1Φ2,Φ4〉〈Λn-j-1Φ2,Φ4〉-〈Λj-1Φ2,Φ2〉〈Λn-j-1Φ4,Φ4〉+2〈Λj-1Φ2,Φ6〉〈Λn-j-1Φ4,Φ6〉]-2〈Λn-1Φ1,Φ3〉-2〈Λn-1Φ2,Φ4〉,n≥2.

Therefore, from (44), we have
(49){Fm,Fn}=0,1≤m,n≤3N,
which means that {Fn}n=13N are in involution. Moreover, referring to proof of functional independence in [15], it is easy to find that {Fn}n=13N are functionally independent over some region of ℝ6N.

To sum up the above results, we have the following theorem.

Theorem 3.

The super Hamiltonian systems given by (32) and (37) are both completely integrable in the sense of Liouville. That is to say, Hamiltonian systems (32) and (37) constitute an integrable decomposition of two-component super AKNS equations (26).

5. Conclusions and Discussions

Starting from a matrix Lie superalgebra, we constructed a two-component super AKNS system (19). For this new system, we considered a Bargmann symmetry constraint (31). Then, substituting the constraint (31) into N copies of systems (7) and (27), we obtained the constrained super Hamiltonian systems (32) and (37), and we showed that systems (32) and (37) are completely integrable in the Liouville sense. Accordingly, Lax matrix L(λ) and r-matrix representation were, respectively, given in Propositions 1 and 2. The difference between [15] and this paper will be listed in the following.

In [15], we applied binary-nonlinearization to one-component AKNS system which was associated with a 3×3 spectral matrix, while, in this paper, we applied monononlinearization to two-component AKNS system which was associated with a 6×6 spectral matrix.

In [15], symmetry constraint was associated with both eigenfunctions and adjoint eigenfunctions, while, in this paper, constraint was just associated with eigenfunctions.

Construction of integrals of motion is also different. In [15], we made use of constrained stationary equation (N~2)x=[M~,N~2]. And, in this paper, we made use of general theory of r-matrix.

According to the above conclusions, some future work is listed as follows.

Is this method applied to the other multicomponent super integrable system?

If potentials q and r are chosen as (2) in [23], is nonlinearization of Lax pairs applied to this multicomponent integrable system?

Is nonlinearization (including monononlinearization and binary-nonlinearization) extended to supersymmetry integrable system, such as supersymmetric Kadomtsev-Petviashvili system [24, 25]?

Conflict of Interests

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

Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant nos. 11001069 and 61273077 and Zhejiang Provincial Natural Science Foundation of China under Grant no. LQ12A01002.

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