# A New Super Extension of Dirac Hierarchy

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

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## Abstract

We derive a new super extension of the Dirac hierarchy associated with a3×3
matrix super spectral problem with the help of the zero-curvature equation. Super trace identity is used to furnish the super Hamiltonian structures for the resulting nonlinear super integrable hierarchy.


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## Body

## 1. Introduction

In 1984, Kupershmidt [1] proposed a fermionic extension of the KdV equation with a Lax pair and a local super bi-Hamiltonian structure. Then, super integrable systems have received much attention with the development of integrable systems. Many experts and scholars do research on the topic. So far, many classical integrable hierarchies have been extended to the super ones by adding fermion fields, such as the super AKNS hierarchy [2, 3], the super Kaup-Newell hierarchy [4], the super Dirac hierarchy [2, 5], and the super Kadomtsev-Petviashvili (KP) hierarchy, and so on [6–10].The Dirac hierarchy is based on the spectral problem(1)(ϕ1ϕ2)x=(rλ+s-λ+s-r)(ϕ1ϕ2),
and a super extension Dirac hierarchy can be constructed by the matrix super spectral [2]
(2)(ϕ1ϕ2ϕ3)x=(rλ+sα-λ+s-rββ-α0)(ϕ1ϕ2ϕ3),
where ϕ3, α, and β are fermion fields. It reduces to the spectral Dirac system (1) as α=β=0.In this paper, we consider a new3×3 matrix super spectral problem which generates a generalized super Dirac hierarchy with four Fermi variables. As we will show, this spectral problem takes the spectral Dirac system (1) and the super Dirac system (2) as special cases.The paper is organized as follows. In the Section2, we will construct a generalized super Dirac hierarchy related to the 3×3 matrix super spectral problem and consider some special reductions. In Section 3, we prove the localness of the whole super soliton hierarchy. In Section 4, we present the super Hamiltonian structures for the generalized super Dirac hierarchy with the help of the super trace identity. The last section contains concluding remarks.


## 2.  Super Extension Dirac Hierarchy

In this section, we will derive a generalized super Dirac hierarchy. To this end, we take a matrix super spectral problem(3)ϕx=Uϕ=U(u,λ)ϕ,ϕ=(ϕ1ϕ2ϕ3),U=(rλ+sα1-λ+s-rβ1β2α20),
where r, s,  λ,  ϕ1, and ϕ2 are the commuting variables, which can be indicated by the degree (mod 2) p as p(r)=p(s)=p(λ)=p(ϕ1)=p(ϕ2)=0 and α1, α2, β1, β2, and 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.We first solve the stationary zero-curvature equation(4)Vx=[U,V],V=(Vij)3×3,
where
(5)p(V11)=p(V12)=p(V21)=p(V22)=p(V33)=0,p(V13)=p(V23)=p(V31)=p(V32)=1,
and each entry Vij=Vij(A,B,C,G,ρ,δ,τ,ε) with the function of A,B,C,G,ρ,δ,τ,ε with p(A)=p(B)=p(C)=p(G)=0, p(ρ)=p(δ)=p(τ)=p(ε)=1:
(6)V11=12(C+G),V12=A+B,V13=ρ,V21=A-B,V22=12(-C+G),V23=δ,V31=ε,V32=τ,V33=G.
Substituting (6) into (4) yields
(7)Ax=-λC+2rB+12α1τ+12α2ρ+12β1ε+12β2δ,Bx=2rA-sC+12α1τ+12α2ρ-12β1ε-12β2δ,Cx=4λA-4sB+α1ε+β2ρ-β1τ-α2δ,Gx=α1ε+β2ρ+β1τ+α2δ,ρx=λδ+rρ+sδ+12α1G-12α1C-β1A-β1B,δx=-λρ+sρ-rδ+12β1G-α1A+α1B+12β1C,τx=-λε+β2A+β2B-12α2C-12α2G-sε+rτ,εx=λτ+12β2C-12β2G+α2A-α2B-rε-sτ.
We put A,B,C,G,ρ,δ,ε, and τ to be polynomials of λ:
(8)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.
Substituting (8) into (7) and equating the coefficients of λ, we obtain
(9)Am,x=-Cm+1+2rBm+12α1τm+12α2ρm+12β1εm+12β2δm,Bm,x=2rAm-sCm+12α1τm+12α2ρm-12β1εm-12β2δm,Cm,x=4Am+1-4sBm+α1εm+β2ρm-β1τm-α2δm,Gm,x=α1εm+β2ρm+β1τm+α2δm,ρm,x=δm+1+rρm+sδm+12α1Gm-12α1Cm-β1Am-β1Bm,δm,x=-ρm+1+sρm-rδm+12β1Gm-α1Am+α1Bm+12β1Cm,τm,x=-εm+1+β2Am+β2Bm-12α2Cm-12α2Gm-sεm+rτm,εm,x=τm+1+12β2Cm-12β2Gm+α2Am-α2Bm-rεm-sτm.
Upon choosing the initial data
(10)A0=C0=ρ0=τ0=δ0=ε0=0,B0=1,G0=-g0=constant,
then the recursion relations in (9) uniquely define a series of sets of differential polynomial functions in u with respect to x. The first two sets are as follows:
(11)A1=s,C1=2r,δ1=β1+12g0α1,ρ1=α1-12g0β1,τ1=α2-12g0β2,ε1=β2+12g0α2,B1=0,G1=0,A2=12rx-14g0(α1α2+β1β2),C2=-sx-12g0(α1β2+α2β1),δ2=α1,x+12g0(rβ1-sα1-β1,x),ρ2=-β1,x-12g0(β1s+α1r+α1,x),τ2=β2,x+12g0(rα2-sβ2+α2,x),ε2=-α2,x-12g0(sα2+rβ2-β2,x),B2=12(r2+s2)+12α1β2+12β1α2+14g0(α1α2-β1β2),G2=α2α1+β1β2+12g0(α1β2-α2β1).
From the recursion relations in (9), we can obtain the hereditary recursion operator L which satisfies that(12)(Cm+1,2Am+1,-εm+1,-τm+1,δm+1,ρm+1)T=L(Cm,2Am,-εm,-τm,δm,ρm)T,
where
(13)L=(L11L12L13L21L22L23L31L32L33),L11=(-2r∂-1s2r∂-1r-12∂-2s∂-1s+12∂2s∂-1r),L12=(r∂-1β1-12β1-r∂-1α1-12α1s∂-1β1+12α1-12β1-s∂-1α1),L13=(-r∂-1β2+12β2r∂-1α2+12α2-s∂-1β2+12α2s∂-1α2-12β2),L21=(β2∂-1s+12α2-β2∂-1r-12β2α2∂-1s+12β2-α2∂-1r+12α2),L22=(-12β2∂-1β1-12α2∂-1α1-s12β2∂-1α1-12α2∂-1β1+r-∂12β2∂-1α1-12α2∂-1β1+r+∂12β2∂-1β1+12α2∂-1α1+s),L23=(12β2∂-1β2+12α2∂-1α2-12β2∂-1α2+12α2∂-1β2-12β2∂-1α2+12α2∂-1β2-12β2∂-1β2-12α2∂-1α2),L31=(-β1∂-1s+12α1β1∂-1r+12β1-α1∂-1s+12β1α1∂-1r-12α1),L32=(12α1∂-1α1+12β1∂-1β112α1∂-1β1-12β1∂-1α112α1∂-1β1-12β1∂-1α1-12α1∂-1α1-12β1∂-1β1),L33=(-12α1∂-1α2-12β1∂-1β2-s-12α1∂-1β2+12β1∂-1α2+∂-r12β1∂-1α2-12α1∂-1β2-r-∂12β1∂-1β2+12α1∂-1α2+s).Letϕ satisfy the spectral problem (3) and an auxiliary problem
(14)ϕtn=V(n)ϕ,n≥0,
where
(15)V(n)=(λnV)+=∑j=0n‍(Cj+Gj2Aj+BjρjAj-Bj-Cj+Gj2δjεjτjGj)λn-j,
where (λnV)+ denotes the polynomial part of λnV.Then, the compatibility condition of (3) and (14) yields the zero-curvature equation
(16)Ut-Vx(n)+[U,V(n)]=0,
which is equivalent to a hierarchy of generalized super Dirac equations
(17)utn=(qrα1β1α2β2)tn=Kn(u)=(2An+1-Cn+1δn+1-ρn+1-εn+1τn+1),dddddddddddddddddddddddddn≥0.The first nontrivial member in the hierarchy (17) is as follows:
(18)rt1=rx-12g0(α1α2+β1β2),st1=-β2,x-12(α2,x+rα2-sβ2),α1,t1=α1,x-12g0(β1,x-rβ1+sα1),β1,t1=β1,x+12g0(α1,x+rα1+sβ1),α2,t1=α2,x-12g0(β2,x-rβ2-sα2),β2,t1=β2,x+12g0(α2,x+rα2-sβ2).
When n=2 and g0=0 in (17), we can obtain the second-order nonlinear super integrable equations
(19)rt2=-12sxx-12α2,xα1-12α1,xα2-12β2,xβ1-12β1,xβ2+sr2+s3-sα2β1+sα1β2,st2=α2,xx+12rxα2+rα2,x-12sxβ2-sβ2,x+β2α1α2-12α2r2-12α2s2,α1,t2=-β1,xx+12rxβ1+rβ1,x-12sxα1-sα1,x-α1β1β2+12β1r2+12β1s2,β1,t2=α1,xx+β1,xs+12sxβ1+α1,xr+12rxα1-12α1r2-12α1s2+α1α2β1,α2,t2=β2,xx-α2,xs-12α2sx-β2,xr-12β2rx-12β2r2-12β2s2+α2β1β2,β2,t2=-α2,xx-12rxα2-rα2,x+12sxβ2+sβ2,x+12α2r2+12α2s2-β2α1α2,
which are, respectively, reduced to the super Dirac equations
(20)rt2=-12sxx-ααx+ββx+r2s+s3+2sαβ,st2=12rxx-αβx+αxβ-rs2-r3-2rαβ,αt2=-βxx+rβx-sαx-12sxα+12rxβ+12r2β+12s2β,βt2=αxx+rαx+sβx+12rxα+12sxβ-12r2α-12s2α,
or the famous Dirac equations
(21)rt2=-12sxx+r2s+s3,st2=12rxx-rs2-r3,
when α1=-α2=α, β2=β1=β, or α1=α2=β1=β2=0.


## 3. Localness

We note that the recursion operatorL is an integrodifferential operator, but the generalized super Dirac system (17) are pure differential equations.We put(22)B¯=1+B¯1λ+B¯2λ2+⋯,G¯=-g0+G¯1λ+G¯2λ2+⋯,V¯=v-1λ+v-2λ2+v-3λ3+⋯,V¯T=(A-,C-,δ-,ρ-,ϵ-,τ-),v-jT=(A¯j,C¯j,δ-j,ρ-j,ϵ-j,τ-j),
then, B¯, C¯, and V¯ satisfy the following equations:
(23)-G¯x+α1ε¯+β2ρ¯+β1τ¯+α2δ¯=0,-B¯x+2rA¯-sC¯+12α1τ¯+12α2ρ¯-12β1ε¯-12β2δ¯=0,-C¯x-4λA¯-4sB¯+α1ε¯+β2ρ¯-β1τ¯-α2δ¯=0,-A¯x-λC¯+2rB¯+12α1τ¯+12α2ρ¯+12β1ε¯+12β2δ¯=0,-ρ¯x+λδ¯+rρ¯+sδ¯+12α1G¯-12α1C¯-β1A¯-β1B¯=0,-δ¯x-λρ¯+sρ¯-rδ¯+12β1G¯-α1A¯+α1B¯+12β1C¯=0,-τ¯x-λε¯+β2A¯+β2B¯-12α2C¯-12α2G¯-sε¯+rτ¯=0,-ε¯x+λτ¯+12β2C¯-12β2G¯+α2A¯-α2B¯-rε¯-sτ¯=0.
From (23), we obtain
(24)(-G¯24-B¯2+C¯24+A¯2-ϵ¯ρ¯+δ¯τ¯)x=0.
Integrating (24) with respect to x, we get
(25)-G¯24-B¯2+C¯24+A¯2-ϵ¯ρ¯+δ¯τ¯=-1-g024.
Substituting (22) into (25) and equating the coefficients of λ-j on both sides of the equation, we have
(26)B¯1=G¯1=0,(27)B¯j-14g0G¯j=12∑l=1j-1‍A¯lA¯j-l+14∑l=1j-1‍C¯lC¯j-l+12∑l=1j-1‍ρ-lϵ-j-l-12∑l=1j-1‍τ-lδ-j-l.
We point that the left side of (27) involves an arbitrary constant g0. Based on (27) and (9), it is easy to find that B¯i,G¯i,A¯i,C¯i,ϵ-i,ρ-i,δ-i, and τ-i are all differential polynomials in six variables r,s,α1,α2,β1, and β2. So the hierarchy of the super integrable equations (17) is pure differential equations.


## 4. The Super Hamiltonian Structures

In this section, we will establish the super Hamiltonian structure of the generalized super Dirac hierarchy (18) by super trace identity [2, 11]:
(28)δδu∫Str(V∂U∂λ)dx=λ-γ∂∂λλγStr(V∂U∂u),
where u=(r,s,α1,β1,α2,β2)T and the constant γ is determined by
(29)γ=-λ2ddλln⁡⁡|Str(VV)|.Through direct calculations, we have(30)Str(V∂U∂r)=C,Str(V∂U∂α1)=-ε,Str(V∂U∂s)=2A,Str(V∂U∂α2)=δ,Str(V∂U∂λ)=-2B,Str(V∂U∂β1)=-τ,Str(V∂U∂β2)=ρ.
Substituting the above results into the super trace identity (28) yields that
(31)δδu∫-2Bdx=λ-γ∂∂λλγ(C,2A,-ε,-τ,δ,ρ)T.
Comparing the coefficients of λ-m-2 on both sides of (31) gives rise to
(32)δδu∫-2Bm+2dx=(γ-m-1)(Cm+12Am+1-εm+1-τm+1δm+1ρm+1),ddddddddddddddddm≥0.
By employing the computing formula (29) on the constant γ, we obtain γ=0. Thus, we have
(33)δℋmδu=(Cm+12Am+1-εm+1-τm+1δm+1ρm+1),ℋm=∫2m+1Bm+1dx,ddddddddddddddddddddddddddddm≥0.
It then follows that the super integrable hierarchy (17) possesses the following super Hamiltonian form:
(34)utn=Kn(u)=Jδℋnδu,n≥0,
where the super Hamiltonian operator J is given by
(35)J=(010000-10000000001000000-1001000000-100).


## 5. Concluding Remarks

In this paper, based on a3×3 matrix super spectral problem, we considered the related four Fermi component super Dirac-type systems and also proved the localness of the whole super soliton hierarchy. We obtained the super Hamiltonian structure and different reductions for the super integrable equations. Our computation reflects how to choose appropriate Lie algebras to generate soliton hierarchies [12], and the generating procedure can be applied to the other super soliton hierarchies. Let us notice that the super integrable hierarchy (17) allows for an arbitrary constant g0 and such system is interesting since different specifications of g0 lead to different super Dirac type equations. The nonlinearization approach for integrable systems is a powerful tool to generate finite-dimensional integrable Hamiltonian systems. The super integrable system (17) may admit nonlinearization. Moreover, the super Dirac system (17) may inherit various other integrable characteristics, such as first-degree time-dependent symmetries [13] and Ba¨cklund transformation [14]. In particular, it is of interest to study multi-integrable couplings and the corresponding super Hamiltonian structures of the super Dirac system (17) by the super variational identity [15]. These related issues may be considered in further publication.



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