# Super-Hamiltonian Structures and Conservation Laws of a New Six-Component Super-Ablowitz-Kaup-Newell-Segur Hierarchy

Authors: Fucai You; Jiao Zhang; Yan Zhao
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

A six-component super-Ablowitz-Kaup-Newell-Segur (-AKNS) hierarchy is proposed by the zero curvature equation associated with Lie superalgebras. Supertrace identity is used to furnish the super-Hamiltonian structures for the resulting nonlinear superintegrable  hierarchy. Furthermore, we derive the infinite conservation laws of the first two nonlinear super-AKNS equations in the hierarchy by utilizing spectral parameter expansions. PACS: 02.30.Ik; 02.30.Jr; 02.20.Sv.


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## Body

## 1. Introduction

It is well known that many physically important integrable partial differential equations belong to the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy [1–3], such as the KdV equation, the mKdV equation, the nonlinear Schrödinger equation, the Sin-Gordon equation, and the mixed KdV-mKdV equation. The AKNS hierarchy is based on the Zakharov and Shabat [4] spectral problem
(1)(ϕ1ϕ2)x=(-λqrλ)(ϕ1ϕ2).
It possesses Lax representation, Hamiltonian structures, and infinitely many conserved quantities and can be solved by the method of inverse scattering, Hirota method, Darboux transform, and others.The superintegrable systems had already aroused strong interest in theoretical physics [5, 6], where the fermion fields are added and equally treated with the boson fields. Many classical integrable equations have been extended to the super ones by adding fermion fields, such as the super-AKNS [6–10], the super-KdV [5], the super-Dirac [9, 11, 12], and the super-Kadomtsev Petviashvili (KP) [13–15]. The super-AKNS hierarchy was first proposed in [6] based on the superalgebra sl(2,R). Extension of this work to other and higher dimensional superalgebras is given in [16]. The super-AKNS matrix superspectral problem is
(2)(ψ1ψ2ψ3)x=(-λqαrλβ-βα0)(ψ1ψ2ψ3),
where ψ3, α, and β are fermion fields. It reduces to the spectral AKNS’s system as α=β=0.In this paper, we consider a new3×3 matrix superspectral problem which generates a six-component super-AKNS hierarchy. As we will show this spectral problem takes the spectral AKNS’s system and super-AKNS’s system as special cases.The paper is organized as follows. In Section2, we will construct a six-component super-AKNS hierarchy related to the 3×3 matrix superspectral problem. In Section 3, we present the super-Hamiltonian structures for the six-component super-AKNS hierarchy with the help of the supertrace identity. In Section 4, we consider some special reductions of the superintegrable hierarchy. In Section 5, we derive the infinite conservation laws for the associated hierarchy. The last section contains concluding remarks.


## 2. A Six-Component Super-AKNS Hierarchy

In this section, we will derive a new six-component super-AKNS hierarchy associated with the3×3 matrix spectral problem. Let G be a commutative superalgebra over R and G~ a matrix loop superalgebra over G with the nondegenerate Killing form. We take a matrix superspectral problem
(3)ϕx=Uϕ=U(u,λ)ϕ,ϕ=(ϕ1ϕ2ϕ3),hhhhhhhhhhhhU=(-λqα1rλβ1β2α20),
where ϕx and ϕt denote the partial derivatives with respect to x and t, u=(q,r,α1,α2,β1,β2)T∈G6 is a potential consisting of commuting and anticommuting variables, q,r,λ,ϕ1,ϕ2 are the commuting variables, which can be indicated by the degree p as p(q)=p(r)=p(λ)=p(ϕ1)=p(ϕ2)=0, and α1,α2,β1,β2,ϕ3 are the anticommuting variables, which can be indicated by the degree p as p(α1)=p(α2)=p(β1)=p(β2)=p(ϕ3)=1. Here λ is assumed to be a constant spectral parameter (i.e., λt=0).Let us find the following temporal evolution equation associated with (3):
(4)ϕt=Vϕ,V=(A+G2BρC-A+G2δετG),
where A,B,C, and G are commuting fields and ρ,δ,ε, and τ are anticommuting fields. From the stationary zero curvature equation
(5)Vx=[U,V],
it gives rise to
(6)Ax=2qC-2rB+α1ε-β1τ-α2δ+β2ρ,Bx=-2λB-qA+α1τ+α2ρ,Cx=2λC+rA+β1ε+β2δ,Gx=α1ε+β1τ+α2δ+β2ρ,ρx=-λρ+qδ-12α1A+12α1G-β1B,δx=λδ+rρ-α1C+12β1A+12β1G,τx=-λτ-qε-12α2A-12α2G+β2B,εx=ε-rτ+α2C+12β2A-12β2G.
We put A,B,C,G,ρ,δ,ε, and τ to be polynomial of λ:
(7)A=∑m≥0∞‍Amλ-m,B=∑m≥0∞‍Bmλ-m,C=∑m≥0∞‍Cmλ-m,G=∑m≥0∞‍Gmλ-m,ρ=∑m≥0∞‍ρmλ-m,ε=∑m≥0∞‍εmλ-m,δ=∑m≥0∞‍δmλ-m,τ=∑m≥0∞‍τmλ-m,
and substituting (7) into (6) and equating the coefficients of λ, we obtain
(8)Am,x=2qCm-2rBm+α1εm-β1τm-α2δm+β2ρm,Bm,x=-2Bm+1-qAm+α1τm+α2ρm,Cm,x=2Cm+1+rAm+β1εm+β2δm,Gm,x=α1εm+β1τm+α2δm+β2ρm,ρm,x=-ρm+1+qδm-12α1Am+12α1Gm-β1Bm,δm,x=δm+1+rρm-α1Cm+12β1Am+12β1Gm,τm,x=-τm+1-qεm-12α2Am-12α2Gm+β2Bm,εm,x=εm+1-rτm+α2Cm+12β2Am-12β2Gm.
Upon choosing the initial data
(9)A0=-2,B0=C0=ρ0=δ0=ε0=τ0=0,G0=-g0=constant,
then the recursion relations in (8) uniquely define a series of sets of differential polynomial functions in u with respect to x. The first two sets are as follows:
(10)B1=q,C1=r,A1=0,G1=0,ρ1=-12α1(g0-2),τ1=12α2(g0+2),δ1=12β1(g0+2),ε1=-12β2(g0-2),B2=-12(qx-g0α1α2),C2=12(rx+g0β1β2),ρ2=-α1,x+12g0(α1,x+qβ1),τ2=-α2,x-12g0(α2,x-qβ2),δ2=β1,x+12g0(β1,x+rα1),A2=qr+α1β2-α2β1-12g0(α1β2+α2β1),ε2=β2,x-12g0(β2,x-rα2),G2=α1β2+α2β1-12g0(α1β2-α2β1).
From the recursion relations in (8), we can obtain the hereditary recursion operator L which satisfies that
(11)(Cm+1,Bm+1,-εm+1,-τm+1,δm+1,ρm+1)T=L(Cm,Bm,-εm,-τm,δm,ρm)T,
where
(12)L=(L11L12L13L21L22L23L31L32L33),L11=(-12∂+r∂-1q-r∂-1rq∂-1q-12α112∂-q∂-1r),L12=(-r2∂-1α1r2∂-1β1-∂-1α1∂-1β1),L13=(-r2∂-1α2r2∂-1β2-∂-1α2∂-1β2-12α2),L21=(-α2-β2∂-1qβ2∂-1r-α2∂-1qα2∂-1r+β2),L22=(-∂-∂-1α1-12β2∂-1α1-r-β2∂-1β1q+α2∂-1α1∂),L23=(β2∂-1α200-α2∂-1β2),L31=(-α1+β1∂-1q-β1∂-1rα1∂-1q-α1∂-1r+β1),L32=(-β1∂-1α100α1∂-1β1),L33=(-∂r+β1∂-1β2-q-α1∂-1α2∂+∂-1β2-12α1∂-1β2).
Taking
(13)V(n)=(λnV)+=∑j=0n‍(Aj+Gj2BjρjCj-Aj+Gj2δjεjτjGj)λn-j,
here (λnV)+ denotes the polynomial part of λnV.The compatibility conditions (i.e., zero curvature equation)(14)Ut-Vx(n)+[U,V(n)]=0
of the matrix superspectral problems
(15)ϕx=Uϕ,ϕtn=V(n)ϕ,n≥0,
determine a new six-component super-AKNS integrable soliton hierarchy
(16)utn=(qrα1β1α2β2)tn=Kn(u)=(-2Bn+12Cn+1-ρn+1δn+1-τn+1εn+1),n≥0.


## 3. The Super-Hamiltonian Structures

In this section, we will establish the super-Hamiltonian structure of the six-component super-AKNS hierarchy by supertrace identity [9, 17]
(17)δδu∫‍Str(V∂U∂λ)dx=λ-γ∂∂λλγStr(V∂U∂u),
where the constant γ is determined by
(18)γ=-λ2ddλln⁡|Str(VV)|.
Through direct calculations, we have
(19)Str(V∂U∂λ)=-A,Str(V∂U∂q)=C,Str(V∂U∂r)=B,Str(V∂U∂α1)=-ε,Str(V∂U∂β1)=-τ,Str(V∂U∂α2)=δ,Str(V∂U∂β2)=ρ.
Substituting the above results into the supertrace identity (17) yields that
(20)δδu∫‍-Adx=λ-γ∂∂λλγ(C,B,-ε,-τ,δ,ρ)T.
Comparing the coefficients of λ-m-2 on both sides of (20) gives rise to
(21)δδu∫‍-Am+2dx=(γ-m-1)(Cm+1Bm+1-εm+1-τm+1δm+1ρm+1),m≥0.
By employing the computing formula (18) on the constant γ, we obtain γ=0. Thus we have
(22)δHmδu=(Cm+1Bm+1-εm+1-τm+1δm+1ρm+1),Hm=∫‍2m+1Am+1dx,m≥0.
It then follows that the superintegrable hierarchy (16) possesses the following super-Hamiltonian form:
(23)utn=Kn(u)=JδHnδu,n≥0,
where the super-Hamiltonian operator J is given by
(24)J=(0-2000020000000000-100001000010000-1000).We note that the recursion operatorL is an integrodifferential operator, but the generalized superintegrable system (23) is pure differential equations according to [12].


## 4. Reductions

We now consider the possible reductions of our six-component super-AKNS hierarchy.Assumingα1=α2=β1=β2=0, hierarchy (23) reduces to the classical AKNS hierarchy [1]. Taking α1=-α2,  β2=β1, we can have the super-AKNS hierarchy [6, 9, 10].Whenn=1 in (23), we obtain the first-order nonlinear superintegrable equations
(25)qt1=qx-g0α1α2,rt1=rx+g0β1β2,α1,t1=α1,x-12g0α1,x-12g0qβ1,α2,t1=α2,x+12g0α2,x-12g0qβ2,β1,t1=β1,x+12g0β1,x+12g0rα1,β2,t1=β2,x-12g0β2,x+12g0rα2.
Taking n=2 in (23), we can obtain the second-order nonlinear superintegrable equations
(26)qt2=-12qxx+q2r+(g0-1)(α1,xα2-qα1β2)-(g0+1)(α2,xα1+qα2β1),rt2=12rxx-qr2+(g0+1)(β1,xβ2+rα2β1)+(g0-1)(rα1β2-β2,xβ1),α1,t2=(12g0-1)α1,xx+12(g0-1)β1qx-qβ1,x+12(1-g0)qrα1-α1α2β1,α2,t2=(-12g0-1)α2,xx+12(g0+1)β2qx+qβ2,x+12(1+g0)qrα2-α1α2β2,β1,t2=(12g0+1)β1,xx+12(g0+1)α1rx+rα1,x-12(1+g0)qrβ1+α1β1β2,β2,t2=(1-12g0)β2,xx+12(g0-1)α2rx-rα2,x+12(g0-1)qrβ2+α2β1β2.
In particular, letting g0=1 in (26), we have
(27)qt2=-12qxx+q2r-2α2,xα1-2qα2β1,rt2=12rxx-qr2+2β1,xβ2+2rα2β1,α1t2=-12α1,xx-qβ1,x-α1α2β1,α2t2=-32α2,xx+β2qx+qβ2,x+qrα2-α1α2β2,β1t2=32β1,xx+α1rx+rα1,x-qrβ1+α1β1β2,β2t2=12β2,xx-rα2,x+α2β1β2,
and taking g0=-1, (26) becomes
(28)qt2=-12qxx+q2r-2α1,xα2+2qα1β2,rt2=12rxx-qr2+2β2,xβ1-2rα1β2,α1t2=-32α1,xx-β1qx-qβ1,x+2qrα1-α1α2β1,α2t2=-12α2,xx+qβ2,x-α1α2β2,β1t2=12β1,xx+rα1,x+α1β1β2,β2t2=32β2,xx-rxα2-rα2,x-qrβ2+α2β1β2.
If we choose α1=-α2=α, β2=β1=β, and g0=0, (26) can be reduced to the second-order super-AKNS equations [10]
(29)qt2=-12qxx+q2r+2qαβ-2ααx,rt2=12rxx-qr2-2rαβ-2ββx,αt2=-αxx-qβx+12qrα-12qxβ,βt2=βxx+rαx+12rxα-12qrβ,
which is just the coupled nonlinear Schrödinger equations, also called Manakov equations
(30)uT=-uXX+u2v,vT=vXX-uv2,
as α=β=0,  q=u,  r=v,  t2=T, and x=2X.


## 5. Infinite Conservation Laws

In what follows, we will derive infinite conservation laws of (25) and (26). From the spectral problem (3), we can introduce the variables
(31)M=ϕ2ϕ1,N=ϕ3ϕ1,
and then we obtain
(32)Mx=r+2λM+β1N-qM2-α1MN,Nx=β2+α2M+λN-qMN.
Next, we expand M and N as series of the spectral parameter λ,
(33)M=∑j=1∞‍Mjλ-j,N=∑j=1∞‍Njλ-j,
where Mj are even, p(Mj)=0, and Nj are odd, p(Nj)=1.By substituting (33) into (32) and comparing the coefficients of λ, we raise the recursion formulas for Mj and Nj,
(34)Mj+1=12(-rδn,0+Mj,x-β1Nj∑l=1j-1‍kkkkkik+q∑l=1j-1‍MjMj-l+α1∑l=1j-1‍MjNj-l),kkkkkkkkkkkikkkikkkkkkkkkk(j≥2),Nj+1=-β2δn,0+Nj,x-α2Mj+q∑l=1j-1‍MjNj-l,(j≥2).
We write below the first few terms of Mj and Nj:
(35)M1=-12r,N1=-β2,M2=-14rx+β1β2,N2=-β2,x+12rα2,M3=-18rxx+12β1,xβ2-β2,xβ1+14rα2β1+14rα1β2+18qr2,N3=-β2,xx+34α2rx+12rα2,x-α2β1β2+12qrβ2.
On the other hand, it is easy to see that
(36)∂∂t∂∂xlog⁡ϕ1=∂∂x∂∂tlog⁡ϕ1or∂∂tϕ1,xϕ1=∂∂xϕ1,tϕ1,
which implies
(37)∂∂t(-λ+qM+α1N)=∂∂x(12(A+G)+BM+ρN),
and then the form of the conservation law is
(38)μt=νx,
with the assumption that μ=-λ+qM+α1N, ν=(1/2)(A+G)+BM+ρN.With regard to (25), we have
(39)A=-2λ,B=q,G=-g0λ,ρ=(1-12g0)α1.
Expanding μ and ν as
(40)μ=-λ+∑j=1∞‍μjλ-j,ν=-(1+12g0)λ+∑j=1∞‍νjλ-j,
then we have
(41)μj=qMj+α1Nj,νj=qMj+(1-12g0)α1Nj,hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhj≥1,
with the coefficients μj and νj, which are called conserved densities and currents, respectively. Then the first two conserved densities and currents of (25) read
(42)μ1=-12qr-α1β2,μ2=-14qrx+qβ1β2-α1β2,x+12rα1α2,ν1=-12qr-(1-12g0)α1β2,ν2=-14qrx+qβ1β2-(1-12g0)α1β2,x+12(1-12g0)rα1α2.
So the first two conservation laws of (25) are revealed as
(43)(-12qr-α1β2)t=[-12qr-(1-12g0)α1β2]x,(-14qrx+qβ1β2-α1β2,x+12rα1α2)t=[-14qrx+qβ1β2-(1-12g0)α1β2,xlllllllkk+12(1-12g0)rα1α2]x.For (26), one infers
(44)A=-2λ2+qr+α1β2-α2β1-12g0α1β2-12g0α2β1,G=-g0λ2+α1β2+α2β1-12g0α1β2+12g0α2β1,ρ=(1-12g0)λα1+(12g0-1)α1,x+12g0qβ1,B=λq-12qx+12g0α1α2.
If we write μ and ν as
(45)μ=-λ+∑j=1∞‍μjλ-j,ν=-(1+12g0)λ2+∑j=1∞‍νjλ-j,
the first two are
(46)μ1=-12qr-α1β2,μ2=-14qrx+qβ1β2-α1β2,x+12rα1α2,ν1=-12g0(β2,xα1+rα1α2+α1,xβ2+qβ1β2)-14qrx+qβ1β2+14qxr+α1,xβ2+β2,xα1+12rα1α2,ν2=-12g0(α1,xβ2,x+12qrα2β1+12qrα1β2-qβ2,xβ1kkkkkkkk-12rα1,xα2+rxα1α2-12rα2,xα1kkkkkkk12-2α1α2β1β2+β2,xxα1)+α1,xβ2,x+14qrα2β1-12rα1,xα2-12rα2,xα1-α1α2β1β2+β2,xxα1+34rxα1α2+18q2r2-18qrxx+12qβ1,xβ2-qβ2,xβ1+18qxrx-12qxβ1β2+34qrα1β2.
The recursion relations for μj and νj(j≥1) are as follows:
(47)μj=qMj+α1Nj,νj=qMj+1+12(g0α1α2-qx)Mj+(1-12g0)α1Nj+1-(α1,x-12g0α1,x-12g0qβ2)Nj,
where Mj and Nj can be recursively calculated from (34). Then we display the first two conservation laws of (26) as
(48)μ1,t=ν1,x,μ2,t=ν2,x,
where μ1,μ2,ν1, and ν2 are defined in (46).


## 6. Concluding Remarks

In this paper, we proposed a six-component super-AKNS system from a3×3 matrix superspectral problem. We obtained the super-Hamiltonian structure and different reductions for the superintegrable equations. Infinitely many conservation laws were also considered. The obtained results supplement the existing theories on the superintegrable systems. The bosonization approach for supersymmetric systems is a powerful tool to generate exact solutions. The superintegrable system (23) may admit bosonization. This and other related issues may be considered in further publication.



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