# Nonlinear Super Integrable Couplings of Super Classical-Boussinesq Hierarchy

Authors: Xiuzhi Xing; Jingzhu Wu; Xianguo Geng
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

Nonlinear integrable couplings of super classical-Boussinesq hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then, its super Hamiltonian structures were established by using super trace identity. As its reduction, nonlinear integrable couplings of the classical integrable hierarchy were obtained.


---

## 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 C-KdV hierarchy, and super classical-Boussinesq hierarchy have been studied [1–4]. There are some interesting results on the super integrable systems, such as Darboux transformation [5], super Hamiltonian structures [6, 7], binary nonlinearization [8], and reciprocal transformation [9].The research of integrable couplings of the well-known integrable hierarchy has received considerable attentions [10–15]. A few approaches to construct linear integrable couplings of the classical soliton equation are presented by permutation, enlarging spectral problem, using matrix Lie algebra constructing new loop Lie algebra [16], and creating semidirect sums of Lie algebra. Recently, Ma and Zhu [17, 18] 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 which 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 algebrasl(2,1) as an example to illustrate the approach for extending Lie super algebra. Based on the enlarged Lie super algebra sl(4,1), we work out nonlinear super integrable Hamiltonian couplings of the super classical-Boussinesq hierarchy. Finally, we will reduce the nonlinear super classical-Boussinesq integrable Hamiltonian couplings to some special cases.


## 2. Enlargement of Lie Superalgebra

Consider the Lie superalgebrasl(2,1). Its basis is
(1)E1=(1000-10000),E2=(010000000),E3=(000100000),E4=(0010000-10),E5=(000001100),
where E1, E2, E3 are even elements and E4, E5 are odd ones. Their nonzero (anti)commutation relations are
(2)[E1,E2]=2E2,[E1,E3]=-2E3,[E2,E3]=E1,[E1,E4]=[E2,E5]=E4,[E1,E5]=[E4,E3]=-E5,[E4,E5]=E1,[E4,E4]=-2E2,[E5,E5]=2E3.Let us enlarge the Lie superalgebrasl(2,1) to the Lie superalgebra sl(4,1) 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, e6 are even and e7, e8 are odd. The generators of the Lie superalgebra sl(4,1), 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]=0,[e4,e7]=[e4,e8]=[e5,e7]=[e5,e8]=[e6,e7]=[e6,e8]=0.
Define a loop superalgebra corresponding to the Lie superalgebra sl(4,1), denoted by
(5)s~l(4,1)=sl(4,1)⊗[λ,λ-1]={eiλm,ei∈sl(4,1),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∈sl(4,1).


## 3. Nonlinear Super Integrable Couplings of the Super Classical-Boussinesq Hierarchy

Let us start from an enlarged spectral problem associated withsl(4,1)(7)φx=Uφ,
where
(8)U=-e1(1)-1qe1(0)+re2(0)-e3(0)-u1e4(0)+u2e5(0)+αe7(0)+βe8(0)=(-λ-14qr-u1u2α-1λ+14q0u1β00-λ-14q-u1r+u2000-1λ+14q+u10β-α-βα0),
where r, s, u1, and u2 are even potentials but α and β are odd ones. In order to obtain super integrable couplings of super integrable hierarchy, we first solve the adjoint representation (8)
(9)Vx=[U,V]
with
(10)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, ρ and δ are anticommuting fields, and α, β are anticommuting fields.Substituting(11)A=∑m≥0‍Amλ-m,B=∑m≥0‍Bmλ-m,C=∑m≥0‍Cmλ-m,F=∑m≥0‍Fmλ-m,G=∑m≥0‍Gmλ-m,ρ=∑m≥0‍ρmλ-m,σ=∑m≥0‍σmλ-m
into the above equation gives the following recursive formulas:
(12)Am,x=Bm+rCm-βρm+αδm,Bm,x=-2rAm-2Bm+1-12qBm-2αρm,Cm,x=-2Am+2Cm+1+12qCm+2βδm,Em,x=u2Cm+Fm+rGm+u2Gm-αδm-βρm,Fm,x=-2u2Am-2u1Bm-2rEm-2u2Em-2Fm+1-12qFm-2u1Fm+2αρm,Gm,x=2u1Cm-2Em+2Gm+1+12qGm+2u1Gm-2βδm,ρm,x=-αAm-βBm-ρm+1-14qρm+rδm,δm,x=βAm-αCm-ρm+δm+1+14qδm.
From these equations, we can successively deduce
(13)A0=-1,B0=C0=F0=G0=ρ0=δ0=0,E0=ε=const,A1=0,B1=r,C1=-1,F1=u2-rε-u2ε,E1=0,G1=ε,ρ1=α,δ1=β,A2=-12r+αβ,B2=-12rx-14qr,C2=14q,F2=-12u2x+12εrx+12εu2x-14qu2+14qεu2+14qrε-u1r-u1u2+εu1u2+εru1,E2=-12u2+12εr+12εu2-αβ,G2=u1-14qε-u1ε,ρ2=-αx-14qα,δ2=βx-14qβ,A3=12rx+14qr-12qαβ-βαx+αβx,B3=14rxx+18qxr+14qrx+12r2-rαβ+116q2r+ααx,C3=18qx-12r+αβ-116q2-ββx,E3=(1-ε)[u1u2+u1r+14qu2+14u2x]-αβx+βαx+12qαβ-14qr-14εrx,F3=(1-ε)[14u2xx+18qxu2+14qu2x+12u1xr+u1rxxxxxxxxx+12u1xu2+u1u2x+116q2u2+12qu1rxxxxxxxx+12qu1u2+12u22+ru2+ru12+u12u2]-14εrxx-18εqxr-14εqrx+rαβ-12εr2-116εq2r-ααx,G3=12u1x-18qxε-12u1xε-12qu1-12u2+12rε+12u2ε+116q2ε+12qu1ε-αβ-u12+u12ε+ββx,ρ3=αxx+14qxα+12qαx+12αr+12βrx+12βqr-rβx+116q2α,δ3=βxx-14qxβ-12qβx+12βr+116q2β-αx.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∂-14∂-1q∂r+∂-1r∂-∂-1u1∂u2+∂-1u2∂12∂+12∂-1α∂12β+∂-1β∂1212∂-140-u1-12β000L33L3400001212∂-u1-14q00-β2αβ-2α∂-14q-1α+β∂-2βr-α-β∂2βr-αβ-r-∂-14q),L33=-12∂-14∂-1q∂-∂-1u1∂,L34=r+u2+∂-1r∂+∂-1u2∂.Then, let us consider the spectral problem (8) with the following auxiliary problem:
(16)φtn=V(n)φ
with
(17)V(n)=∑j=0n‍=(AjBjEjFjρjCj-AjGj-Ejδj00Aj+EjBj+Fj000Cj+Gj-Aj-Ej0δj-ρj-δjρj0)λn-j+Cn+1e1(0)+Gn+1e4(0).
From the compatible condition φx,tn=φtn,x according to (8) and (17), we get the zero equation:
(18)Utn-Vx(n)+[U,V(n)]=0,
which gives a nonlinear Lax super integrable hierarchy
(19)utn=(qru1u2αβ)tn=(-4Cn+1,x2Bn+1+2rCn+1-Gn+1,x2Fn+1+2u2Cn+1+2rGn+1+2u2Gn+1-ρn+1-αCn+1δn+1+βCn+1).
The super integrable hierarchy (19) is a nonlinear super integrable coupling for the super classical-Boussinesq hierarchy
(20)u¯tn=(qrαβ)tn=(-4Cn+1,x2Bn+1+2rCn+1-ρn+1-αCn+1δn+1+βCn+1).


## 4. Super Hamiltonian Structures

A direct calculation reads(21)Str(Uλ,V)=-4A-2E,Str(Uq,V)=-A-12E,Str(Ur,V)=2C+G,Str(Uu1,V)=-2A-2E,Str(Uu2,V)=C+G,Str(Uα,V)=2δ,Str(Uβ,V)=-2ρ.
Substituting the above results into the super trace identity in [6, 7]
(22)δδu∫‍Str(δUδλV)dx=λ-γ∂∂λλγStr(δUδuV)
yields that
(23)δδu∫‍(-4A-2E)dx=λ-γ∂∂λλγ(-A-12E2C+G-2A-2EC+G2δ-2ρ).
Comparing the coefficients of λ-n-1 on both sides of (23),
(24)δδu∫‍(-4An+1-2En+1)dx=λ-γ∂∂λλγ(-An-12En2Cn+Gn-2An-2EnCn+Gn2δn-2ρn),hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhn≥0.
From the initial values in (12), we obtain γ=0. Thus, we have
(25)δHnδu=(-An+1-12En+12Cn+1+Gn+1-2An+1-2En+1Cn+1+Gn+12δn+1-2ρn+1),Hn=∫‍4An+2+2En+2n+1dx,hhhhhhhhhhhhhhhhhn≥0.
It then follows that the nonlinear super integrable couplings (22) possess the following super Hamiltonian form:
(26)utn=Kn(u)=JδHnδu,
where
(27)J=(0-4∂04∂00-4∂0∂0-αβ0∂0-2∂004∂0-2∂0-αβ0α0-α0120-β0β120)
is a super Hamiltonian operator and Hn(n≥0) are Hamiltonian functions.


## 5. Reductions

Takingα=β=0, the hierarchy (26) reduces to a nonlinear integrable coupling of the super classical-Boussinesq hierarchy.Whenn=2 in (26), we obtain the nonlinear super integrable couplings of the second-order super classical-Boussinesq equations
(28)qt2=-12qxx+2rx-4αxβ-4αβx+12qqx+4ββxx+4βx2,rt2=12rxx+12qxr+12qrx+2ααx-2rββx,u1,t2=12(1-ε)[14qqxu2x+qxu1+qu1x+2u1u1x-u1xxxxxxxxxxx+rx-14qxx+14qqx]-12rx+18qxx-18qqx+αxβ+αβx-ββxx,ut2=(1-ε)[2u1xr+2u1xu2+2u1u2x+12u2xxxxxxxxxx+12qxu2+12qu2x+12qrxxxxxxxxx+12qxr+12rxx+2rxu1]+2rββx-12qxr-12qrx-12rxx-2ααxαt2=-αxx-12qαx-38qxα-12βrx+αββx,βt2=βxx-18qxβ-12qβx-αx.
Taking α=β=0 in (28), we obtain the nonlinear super integrable couplings of the second-order super classical-Boussinesq equation
(29)qt2=-12qxx+2rx+12qqx,rt2=12rxx+12qxr+12qrx,u1,t2=12(1-ε)[14u2x+qxu1+qu1x+2u1u1xxxxxxxxxxx-u1xx+rx-14qxx+14qqx]-12rx+18qxx-18qqx,ut2=(1-ε)[2u1xr+2u1xu2+2u1u2x+12u2xx+12qxu2xxxxxxxx+12qu2x+12qrx+12qxr+12rxx+2rxu1]-12qxr-12qrx-12rxx.Lettingε=0 in (28), we have
(30)qt2=-12qxx+2rx-4αxβ-4αβx+12qqx+4ββxx+4βx2,rt2=12rxx+12qxr+12qrx+2ααx-2rββx,u1,t2=12[u2x+qxu1+qu1x+2u1u1x-u1xx]+αxβ+αβx-ββxx,ut2=[2u1xr+2u1xu2+2u1u2x+12u2xx+12qxu2+12qu2x+2rxu1]+2rββx-2ααx,αt2=-αxx-12qαx-38qxα-12βrx+αββx,βt2=βxx-18qxβ-12qβx-αx.When settingε=1, u1=-(1/4)q, u2=r in (28), we obtain the second-order super classical-Boussinesq equations
(31)qt2=-12qxx+2rx-4αxβ-4αβx+12qqx+4ββxx+4βx2,rt2=12rxx+12qxr+12qrx+2ααx-2rββx,αt2=-αxx-12qαx-38qxα-12βrx+αββx,βt2=βxx-18qxβ-12qβx-αx.



---

*Source: 438741-2014-04-07.xml*