# Nonlinear Super Integrable Couplings of Super Broer-Kaup-Kupershmidt Hierarchy and Its Super Hamiltonian Structures

Authors: Sixing Tao; Tiecheng Xia
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

Nonlinear integrable couplings of super Broer-Kaup-Kupershmidt hierarchy based upon an
enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures
were established by using super trace identity, and the conserved functionals were proved to be
in involution in pairs under the defined Poisson bracket. As its reduction, nonlinear integrable
couplings of the classical integrable hierarchy were obtained.


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## Body

## 1. Introduction

With the development of soliton theory, super integrable systems associated with Lie super algebra have aroused growing attentions by many mathematicians and physicists. It was known that super integrable systems contained the odd variables, which would provide more prolific fields for mathematical researchers and physical ones. Several super integrable systems, including super AKNS hierarchy, super KdV hierarchy, super NLS-MKdV hierarchy, super Tu hierarchy, super Broer-Kaup-Kupershmidt hierarchy, have been studied in [1–8]. There are some interesting results on the super integrable systems, such as Darboux transformation in [9], super Hamiltonian structures in [10–12] binary nonlinearization in [13], and reciprocal transformation in [14].The research of integrable couplings of the well-known integrable hierarchy has received considerable attention in [15–23] A few approaches to construct linear integrable couplings of the classical soliton equation are presented by permutation, enlarging spectral problem, using matrix Lie algebra [24] constructing new loop Lie algebra and creating semidirect sums of Lie algebra. Recently, Ma [25] and Ma and Zhu [26] presented a scheme for constructing nonlinear continuous and discrete integrable couplings using the block type matrix algebra. However, there is one interesting question for us is how to generate nonlinear super integrable couplings for the super integrable hierarchy.In this paper, we would like to construct nonlinear super integrable couplings of the super soliton equations through enlarging matrix Lie super algebra. We take the Lie algebraB(0,1) as an example to illustrate the approach for extending Lie super algebras. Based on the enlarged Lie super algebra gl(6,2), we work out nonlinear super integrable Hamiltonian couplings of the super Broer-Kaup-Kupershmidt hierarchy. Finally, we will reduce the nonlinear super Broer-Kaup-Kupershmidt integrable Hamiltonian couplings to some special cases.


## 2. Enlargement of Lie Super AlgebraB(0,1)

Consider the Lie super algebraB(0,1). Its basis is
(1)E1=(1000-10000),E2=(010000000),E3=(000100000),E4=(0010000-10),E5=(000001100),
where E1, E2, and E3 are even elements and E4, and E5 are odd elements. Their nonzero (anti)commutation relations are
(2)[E1,E2]=2E2,[E1,E3]=-2E3,[E1,E4]=E4,[E1,E5]=-E5,[E2,E3]=E1,[E2,E5]=E4,[E3,E4]=E5,[E4,E4]=-2E2,[E4,E5]=E1,[E5,E5]=2E3.
Let us enlarged the Lie super algebra B(0,1) to the Lie super algebra gl(6,2) with a basis
(3)e1=(100000-100000100000-1000000),e2=(0100000000000100000000000),e3=(0000010000000000010000000),e4=(00100000-1000100000-1000000),e5=(0001000000000100000000000),e6=(0000000100000000010000000),e7=(000010000000000000000-1010),e8=(0000000001000000000010-100),
where e1, e2, e3, e4, e5, and e6 are even, and e7, and e8 are odd.The generator of Lie super algebragl(6,2), ei, 0≤i≤8, satisfy the following (anti)commutation relations:
(4)[e1,e2]=2e2,[e1,e3]=-2e3,[e1,e5]=2e5,[e1,e6]=-2e6,[e1,e7]=e7,[e1,e8]=-e8,[e2,e3]=e1,[e2,e4]=-2e5,[e2,e6]=e4,[e2,e8]=e7,[e3,e4]=2e6,[e3,e5]=-e4,[e3,e7]=e8,[e4,e5]=2e5,[e4,e6]=-2e6,[e5,e6]=e4,[e7,e8]=e1-e4,[e7,e7]=2e5-2e2,[e8,e8]=2e3-2e6,[e1,e4]=[e2,e5]=[e2,e7]=[e3,e6]=[e3,e8]=[e4,e7]=[e4,e8]=[e5,e7]=[e5,e8]=[e6,e7]=[e6,e8]=0.
Define a loop super algebra corresponding to the Lie superalgebra gl(6,2), denote by
(5)g~l(6,2)=gl(6,2)⊗ℂ[λ,λ-1]={eiλm,ei∈gl(6,2),i=1,2,…,8,m=0,±1,±2,…}.
The corresponding (anti)commutative relations are given as
(6)[eiλm,ejλn]=[ei,ej]λm+n,∀ei,ej∈gl(6,2).


## 3. Nonlinear Super Integrable Couplings of Super Broer-Kaup-Kupershmidt Hierarchy

Let us start from an enlarged spectral problem associated withgl(6,2):
(7)ϕx=U(u,λ)ϕ,U=e1(1)+re1(0)+se2(0)+e3(0)+u1e4(0)+u2e5(0)+αe7(0)+βe8(0)=(λ+rsu1u2α1-λ-r0-u1β00λ+r+u1s+u20001-λ-r-u10β-α-βα0),
where r, s, u1, and u2 are even potentials but α and β are odd ones.In ordr to obtain super integrable couplings of super integrable hierarchy, we first solve the adjoint representation of (7),
(8)Vx=[U,V],
with
(9)V=Ae1(0)+Be2(0)+Ce3(0)+Ee4(0)+Fe5(0)+Ge6(0)+ρe7(0)+δe8(0)=(ABEFρC-AG-Eδ00A+EB+F000C+G-A-E0δ-ρ-δρ0),
where A, B, C, E, F, and G are commuting fields and ρ, and δ are anticommuting fields. Then we obtain
(10)Ax=-B+sC+βρ+αδ,Bx=-2sA+2λB+2rB-2αρ,Cx=2A-2λC-2rC+2βδ,Ex=u2C-F+sG+u2G-αδ-βρ,Fx=-2u2A+2u1B-2sE-2u2E+2λF+2rF+2u1F+2αρ,Gx=-2u1C+2E-2λG-2rG-2u1G-2βδ,ρx=-αA-βB+λρ+rρ+sδ,δx=βA-αC+ρ-λδ-rδ.
Substituting
(11)A=∑m≥0‍Amλ-m,B=∑m≥0‍Bmλ-m,C=∑m≥0‍Cmλ-m,E=∑m≥0‍Emλ-m,F=∑m≥0‍Fmλ-m,G=∑m≥0‍Gmλ-m,ρ=∑m≥0‍ρmλ-m,δ=∑m≥0‍δmλ-m,
into the previous equation gives the following recursive formulas:
(12)Am,x=-Bm+sCm+βρm+αδm,Bm,x=-2sAm+2Bm+1+2rBm-2αρm,Cm,x=2Am-2Cm+1-2rCm+2βδm,Em,x=u2Cm-Fm+sGm+u2Gm-αδm-βρm,Fm,x=-2u2Am+2u1Bm-2sEm-2u2Em+2Fm+1+2rFm+2u1Fm+2αρm,Gm,x=-2u1Cm+2Em-2Gm+1-2rGm-2u1Gm-2βδm,ρm,x=-αAm-βBm+ρm+1+rρm+sδm,δm,x=βAm-αCm+ρm-δm+1-rδm.
From these equations, we can successively deduce
(13)A0=1,B0=C0=F0=G0=ρ0=δ0=0,E0=ɛ=constant,A1=0,B1=s,C1=1,E1=0,F1=u2+ɛu2+ɛs,G1=ɛ,ρ1=α,δ1=β,A2=-12s-αβ,B2=12sx-rs,C2=-r,E2=-12u2-12ɛu2-12ɛs+αβ,F2=12u2x+12ɛu2x+12ɛsx-su1-ru2-ɛru2-ɛrs-u1u2-ɛu1u2-ɛsu1,G2=-u1-ɛu1-ɛr,ρ2=αx-rα,δ2=-βx-rβ,A3=-14sx+rs+αβx-αxβ+2rαβ,B3=14sxx-12rxs-rsx-12s2-sαβ+r2s+ααx,C3=12rx-12s-αβ+r2-ββx,E3=(ɛ+1)(ru2+su1+u1u2-14u2x)+ɛrs-14ɛsx-αβx+αxβ-2rαβ,F3=(ɛ+1)×(-12su1x-sxu1-12rxu2-ru2x-12u1xu2-u1u2x+14u2xx-su2-12u22+r2u2+su12+u12u2+2ru1u2+2rsu1{12})+sαβ+14ɛsxx-12ɛs2-ɛrsx-12ɛrxs+ɛr2s-ααx,G3=12u1x+12ɛu1x+12ɛrx+2ru1-12u2-12ɛu2-12ɛs+αβ+2ɛru1+ɛr2+u12+ɛu12+ββx,ρ3=αxx-2rαx-rxα-12sα+12sxβ+sβx+r2α,δ3=βxx+rxβ+2rβx-12sβ+αx+r2β.
Equations (12) can be written as
(14)(4Am+1+2Em+12Cm+1+Gm+12Am+1+2Em+1Cm+1+Gm+12δm+1-2ρm+1)=L(4Am+2Em2Cm+Gm2Am+2EmCm+Gm2δm-2ρm),
where(15)L=(12∂-∂-1r∂-s-∂-1s∂-∂-1u1∂-u2-∂-1u2∂-12α-∂-1α∂12β-∂-1β∂12-12∂-r0-2u112β000L33L34000012-12∂-r-u100β-2α-βα-∂-r-1-α+β∂-2sβα-β∂2sβs+αβ∂-r).with(16)L33=12∂-∂-1r∂-∂-1u1∂,L34=-s-u2-∂-1s∂-∂-1u2∂.
Then, let us consider the spectral problem (7) with the following auxiliary spectral problem:
(17)ϕtn=V(n)ϕ,
with
(18)V(n)=∑j=0n‍(AjBjEjFjρjCj-AjGj-Ejδj00Aj+EjBj+Fj000Cj+Gj-Aj-Ej0δj-ρj-δjρj0)λn-j+(-Cn+10-Gn+1000Cn+10Gn+1000-Cn+1-Gn+100000Cn+1+Gn+1000000).From the compatible condition,ϕx,tn=ϕtn,x, according to (7) and (17), we get the zero curvature equation:
(19)Utn-Vx(n)+[U,V(n)]=0,
which gives a nonlinear Lax super integrable hierarchy:(20)utn=(rsu1u2αβ)tn=(-Cn+1,x2Bn+1-2sCn+1-Gn+1,x-2u2Cn+1+2Fn+1-2sGn+1-2u2Gn+1ρn+1-αCn+1-δn+1+βCn+1).
The super integrable hierarchy (20) is a nonlinear super integrable couplings for the super BKK hierarchy:
(21)u~tn=(rsαβ)tn=(-Cn+1,x2Bn+1-2sCn+1ρn+1-αCn+1-δn+1+βCn+1).


## 4. Super Hamiltonian Structures

A direct calculation reads(22)Str(Uλ,V)=4A+2E,Str(Ur,V)=4A+2E,Str(Us,V)=2C+G,Str(Uu1,V)=C+G,Str(Uu2,V)=B+F,Str(Uα,V)=2δ,Str(Uβ,V)=-2ρ.Substituting the above results into the super trace identity in [11, 12],
(23)δδu∫‍Str(δUδλV)dx=λ-γ∂∂λλγStr(δUδuV)
yields that
(24)δδu∫‍(4A+2E)dx=λ-γ∂∂λλγ(4A+2E2C+G2A+2EC+G2δ-2ρ).Comparing the coefficients ofλ-n-1 on both side of (24),
(25)δδu∫(4An+1+2En+1)dx=λ-γ∂∂λλγ(4An+2En2Cn+Gn2An+2EnCn+Gn2δn-2ρn),n≥0.
From the initial values in (12), we obtain γ=0. Thus we have
(26)ℋnδu=(4An+2En2Cn+Gn2An+2EnCn+Gn2δn-2ρn),ℋn=-∫‍1n+1(4An+1+2En+1)dx,n≥0.It then follows that the nonlinear super integrable couplings (20) possess the following super Hamiltonian form
(27)utn=Kn(u)=Jδℋnδu,
where
(28)J=(0-∂0∂00-∂0∂0α-β0∂0-2∂00∂0-2∂0-αβ0-α0α0-120β0-β-120).
is a super Hamiltonian operator and ℋn (n≥0) are Hamiltonian functions.It can be verified that(29)J*=(0∂0-∂00∂0-∂0-αβ0-∂02∂00-∂02∂0α-β0α0-α0-120-β0β-120),J*L=L*J*,
where J* and L* are conjugate operators of J and L, respectively.If we define the following Poisson bracket(30)[ℋn,ℋm]=(δℋnδu,Jδℋmδu)=∫‍(δℋnδr,δℋnδs,δℋnδu1,δℋnδu2,δℋnδα,δℋnδβ)×J(δℋmδrδℋmδsδℋmδu1δℋmδu2δℋmδαδℋmδβ)dx,
where (·,·) denotes the inner product, as (δℋn/δα)(δℋm/δβ)=-(δℋm/δβ)(δℋn/δα), we can obtain
(31)[ℋn,ℋm]=∫‍(δℋmδr,δℋmδs,δℋmδu1,δℋmδu2,δℋmδα,δℋmδβ)×J*(δℋnδrδℋnδsδℋnδu1δℋnδu2δℋnδαδℋnδβ)dx=-[ℋm,ℋn].
From (14) and (31), we have
(32)δℋm+1δu=Lδℋmδu,δℋmδu=Lmδℋ0δu.Supposek,l are arbitrary nonnegative integers consider
(33)[ℋk,ℋk+l]=(Lkδℋ0δu,JLk+lδℋ0δu).
By using the properties of J and L in (29), we can obtain
(34)[ℋk,ℋk+l]=(Lkδℋ0δu,JLk+lδℋ0δu)=(Lkδℋ0δu,(JL)Lk+l-1δℋ0δu)=((JL)*Lkδℋ0δu,Lk+l-1δℋ0δu)=(L*J*Lkδℋ0δu,Lk+l-1δℋ0δu)=(J*Lk+1δℋ0δu,Lk+l-1δℋ0δu)=(Lk+1δℋ0δu,JLk+l-1δℋ0δu)=(δℋk+1δu,Jδℋk+l-1δu)=[ℋk+1,ℋk+l-1].
Ascending the subscript of left factor ℋk, then we have
(35)[ℋk,ℋk+l]=[ℋk+l,ℋk].
By using the antisymmetric law of Poisson bracket (31), we have
(36)[ℋk,ℋk+l]=0.
So the conserved functionals {ℋn}1∞ are in involution in pairs under the Poisson bracket (30).


## 5. Reductions

Takingα=β=0, the hierarchy (27) reduces to a nonlinear integrable couplings of the Broer-Kaup-Kupershmidt hierarchy.Whenn=2 in (27), we obtain the nonlinear super couplings of the second super Broer-Kaup-Kupershmidt equations:
(37)rt2=-12rxx-2rrx+12sx+αxβ+αβx+ββxx,st2=12sxx-2rxs-2rsx+2ααx+2sββx,u1,t2=(ɛ+1)(-12u1xx-2rxu1-2ru1x+12u2x-2u1u1x-12rxx+12sx-2rrx)+12rxx-12sx-αxβ-αβx+2rrx-ββxx,u2,t2=(ɛ+1)(12u2xx-2sxu1-2su1x-2rxu2-2ru2x-2u1xu2-2u1u2x+12sxx-2rxs-2rsx)-12sxx+2rxs+2rsx-2ααx-2sββx,αt2=αxx-2rαx-32rxα+12sxβ+sβx+αββx,βt2=-βxx-12rxβ-2rβx-αx.
Particularly, taking α=β=0 in (37), we can get the nonlinear integrable couplings of the second order Broer-Kaup-Kupershmidt equations:
(38)rt2=-12rxx-2rrx+12sx,st2=12sxx-2rxs-2rsx,u1,t2=(ɛ+1)(-12u1xx-2rxu1-2ru1x+12u2x-2u1u1x-12rxx+12sx-2rrx)+12rxx-12sx+2rrx,u2,t2=(ɛ+1)×(12u2xx-2sxu1-2su1x-2rxu2-2ru2x-2u1xu2-2u1u2x+12sxx-2rxs-2rsx)-12sxx+2rxs+2rsx.
Let ɛ=0 in (37); we have
(39)rt2=-12rxx-2rrx+12sx+αxβ+αβx+ββxx,st2=12sxx-2rxs-2rsx+2ααx+2sββx,u1,t2=-12u1xx-2rxu1-2ru1x+12u2x-2u1u1x-αxβ-αβx-ββxx,u2,t2=12u2xx-2sxu1-2su1x-2rxu2-2ru2x-2u1xu2-2u1u2x-2ααx-2sββx,αt2=αxx-2rαx-32rxα+12sxβ+sβx+αββx,βt2=-βxx-12rxβ-2rβx-αx.
If setting ɛ=-1, u1=-r, and u2=-s in (37), we obtain the second order super Broer-Kaup-Kupershmidt equations:
(40)rt2=-12rxx-2rrx+12sx+αxβ+αβx+ββxx,st2=12sxx-2rxs-2rsx+2ααx+2sββx,αt2=αxx-2rαx-32rxα+12sxβ+sβx+αββx,βt2=-βxx-12rxβ-2rβx-αx.


## 6. Remarks

In this paper, we introduced an approach for constructing nonlinear integrable couplings of super integrable hierarchy. Zhang [27] once employed two kinds of explicit Lie algebra F and G to obtain the nonlinear integrable couplings of the GJ hierarchy and Yang hierarchy, respectively. It is easy to see that Lie algebra F given in [27] is isomorphic to the Lie algebra span {e1,e2,e3,e4,e5,e6} in gl(6,2). So we can obtain nonlinear integrable couplings of super GJ and Yang hierarchy easily. The method in this paper can be applied to other super integrable systems for constructing their integrable couplings.



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*Source: 520765-2013-09-09.xml*