# Self-Consistent Sources and Conservation Laws for a Super Broer-Kaup-Kupershmidt Equation Hierarchy

Authors: Hanyu Wei; Tiecheng Xia
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

Based on the matrix Lie superalgebras and supertrace identity, the integrable super
Broer-Kaup-Kupershmidt hierarchy with self-consistent sources is established. Furthermore, we
establish the infinitely many conservation laws for the integrable super Broer-Kaup-Kupershmidt
hierarchy. In the process of computation especially, Fermi variables also play an important role in
super integrable systems.


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## Body

## 1.    Introduction

Soliton theory has achieved great success during the last decades; it is being applied to mathematics, physics, biology, astrophysics, and other potential fields [1–12]. The diversity and complexity of soliton theory enable investigators to do research from different views, such as Hamiltonian structure, self-consistent sources, conservation laws, and various solutions of soliton equations.In recent years, with the development of integrable systems, super integrable systems have attracted much attention. Many scholars and experts do research on the topic and get lots of results. For example, in [13], Ma et al. gave the supertrace identity based on Lie super algebras and its application to super AKNS hierarchy and super Dirac hierarchy, and to get their super Hamiltonian structures, Hu gave an approach to generate superextensions of integrable systems  [14]. Afterwards, super Boussinesq hierarchy [15] and super NLS-mKdV hierarchy [16] as well as their super Hamiltonian structures are presented. The binary nonlinearization of the super classical Boussinesq hierarchy [17], the Bargmann symmetry constraint, and binary nonlinearization of the super Dirac systems were given [18].Soliton equation with self-consistent sources is an important part in soliton theory. They are usually used to describe interactions between different solitary waves, and they are also relevant to some problems related to hydrodynamics, solid state physics, plasma physics, and so forth. Some results have been obtained by some authors [19–21]. Very recently, self-consistent sources for super CKdV equation hierarchy [22] and super G-J hierarchy are presented [23].The conservation laws play an important role in discussing the integrability for soliton hierarchy. An infinite number of conservation laws for KdV equation were first discovered by Miura et al. in 1968 [24], and then lots of methods have been developed to find them. This may be mainly due to the contributions of Wadati and others [25–27]. Conservation laws also play an important role in mathematics and engineering as well. Many papers dealing with symmetries and conservation laws were presented. The direct construction method of multipliers for the conservation laws was presented [28].In this paper, starting from a Lie super algebra, isospectral problems are designed. With the help of variational identity, Yang got super Broer-Kaup-Kupershmidt hierarchy and its Hamiltonian structure [29]. Then, based on the theory of self-consistent sources, the self-consistent sources of super Broer-Kaup-Kupershmidt hierarchy are obtained by us. Furthermore, we present the conservation laws for the super Broer-Kaup-Kupershmidt hierarchy. In the calculation process, extended Fermi quantities u1 and u2 play an important role; namely, u1 and u2 satisfy u12=u22=0 and  u1u2=-u2u1 in the whole paper. Furthermore, the operation between extended Fermi variables satisfies Grassmann algebra conditions.


## 2. A Super Soliton Hierarchy withSelf-Consistent Sources

Based on a Lie superalgebraG,
(1)e1=(1000-10000),e2=(010000000),e3=(000100000),e4=(0010000-10),e5=(000001100)
that is along with the communicative operation [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, and [e5,e5]+=2e3.We consider an auxiliary linear problem(2)(φ1φ2φ3)x=U(u,λ)(φ1φ2φ3),U(u,λ)=R1+∑i=15‍uiei(λ),(φ1φ2φ3)tn=Vn(u,λ)(φ1φ2φ3),
where  u=(u1,…,us)T,Un=R1+u1e1+⋯+u5e5,ui(n,t)=ui(i=1,2,…,5),φi=φ(x,t) are field variables defining x∈R,t∈R;ei=ei(λ)∈s~l(3) and R1 is a pseudoregular element.The compatibility of (2) gives rise to the well-known zero curvature equation as follows:
(3)Unt-Vnx+[Un,Vn]=0,n=1,2,….
If an equation
(4)ut=K(u)
can be worked out through (3), we call (4) a super evolution equation. If there is a super Hamiltonian operator J and a function Hn such that
(5)ut=K(u)=JδHn+1δu,
where
(6)δHnδu=LδHn-1δu=⋯=LnδH0δu,n=1,2,…,δδu=(δδu1,…,δδu5)T,
then (4) possesses a super Hamiltonian equation. If so, we can say that (4) has a super Hamiltonian structure.According to (2), now we consider a new auxiliary linear problem. For N distinct λj,  j=1,2,…,N, the systems of (2) become as follows:
(7)(φ1jφ2jφ3j)x=U(u,λj)(φ1jφ2jφ3j)=∑i=15‍uiei(λ)(φ1jφ2jφ3j),(φ1jφ2jφ3j)tn=Vn(u,λj)(φ1φ2φ3)=[∑m=0n‍Vm(u)λjn-m+Δn(u,λj)](φ1φ2φ3).Based on the result in [30], we can show that the following equation:
(8)δHkδu+∑j=1N‍αjδλjδu=0
holds true, where αj are constants. Equation (8) determines a finite dimensional invariant set for the flows in (6).From (7), we may know that
(9)δλjδui=13Str(Ψj∂U(u,λj)∂ui)=13Str(Ψjeiλj),i=1,2,…5,
where Str denotes the trace of a matrix and
(10)Ψj=(ψ1jψ2j-ψ1j2ψ1jψ3jψ2j2-ψ1jψ2jψ2jψ3jψ2jψ3j-ψ1jψ3j0).
From (8) and (9), a kind of super Hamiltonian soliton equation hierarchy with self-consistent sources is presented as follows:
(11)unt=JδHn+1δui+J∑j=1N‍αjδλjδu=JLnδH1δui+J∑j=1N‍αjδλjδu,n=1,2,….


## 3. The Super Broer-Kaup-Kupershmidt Hierarchy with Self-Consistent Sources

The super Broer-Kaup-Kupershmidt spectral problem associated with the Lie super algebra is given in [29]:
(12)φx=Uφ,φt=Vφ,
where
(13)U=(λ+rsu11-λ-ru2u2-u10),V=(ABρC-Aσσ-ρ0),
and  A=∑m≥0Amλ-m,B=∑m≥0Bmλ-m,C=∑m≥0Cmλ-m,ρ=∑m≥0ρmλ-m, and  σ=∑m≥0σmλ-m. As u1 and u2 are Fermi variables, they constitute Grassmann algebra. So, we have u1u2=-u2u1,u12=u22=0.Starting from the stationary zero curvature equation(14)Vx=[U,V],
we have
(15)Amx=sCm+u1σm-Bm+u2ρm,Bmx=2Bm+1+2rBm-2sAm-2u1ρm,Cmx=-2Cm+1-2rCm+2Am+2u2σm,ρmx=ρm+1+rρm+sσm-u1Am-u2Bm,σmx=-σm+1-rσm+ρm-u1Cm+u2Am,B0=C0=ρ0=σ0=0,A0=1,B1=s,C1=r,ρ1=u1,σ1=u2,A1=0,….
Then we consider the auxiliary spectral problem
(16)φtn=V(n)φ=(λnV)+φ,
where
(17)V(n)=∑m=0n‍(AmBmρmCm-Amσmσm-ρm0)λn-m,
considering
(18)V(n)=V+(n)+Δn,Δn=-Cm+1e1.
Substituting (18) into the zero curvature equation
(19)Utn-Vx(n)+[U,V(n)]=0,
we get the super Broer-Kaup-Kupershmidt hierarchy
(20)utn=(rsu1u2)t=(0∂00∂0u1-u20u10-120-u2-120)(-2An+1-Cn+12σn+1-2ρn+1)=J(-2An+1-Cn+12σn+1-2ρn+1)=JPn+1,
where
(21)Pn+1=LPn,L=(12∂-∂-1r∂-s-∂-1s∂∂-1u1∂+12u1∂-1u2∂-12u212-12∂-r12u20-u22u1-r-∂-1u1-u2∂2su2s+12u1u2∂-r).According to super trace identity on Lie super algebras, a direct calculation reads as(22)δHnδu=(-2An+1,-Cn+1,2σn+1,-2ρn+1)T,Hn=∫‍2An+2n+1dx,n≥0.When we taken=2, the hierarchy (20) can be reduced to super nonlinear integrable couplings equations
(23)rt2=-12rxx+12sx-2rrx+(u1u2)x+(u1u2x)x,st2=12sxx-2(rs)x+2u1u1x+2su2u2x,u1t2=u1xx-32rxu1+12sxu2-2ru1x+(s+u1u2)u2x,u2t2=-u2xx-12rxu2-2ru2x-u1x.Next, we will construct the super Broer-Kaup-Kupershmidt hierarchy with self-consistent sources. Consider the linear system(24)(φ1jφ2jφ3j)x=U(φ1jφ2jφ3j),(φ1jφ2jφ3j)t=V(φ1jφ2jφ3j).
From (8), for the system (12), we set
(25)δHnδu=∑j=1N‍δλjδu
and obtain the following  δλj/δu:
(26)∑j=1N‍δλjδu=∑j=1N‍(Str(ΨjδUδq)Str(ΨjδUδr)Str(ΨjδUδα)Str(ΨjδUδβ))=(2〈Φ1,Φ2〉〈Φ2,Φ2〉-2〈Φ2,Φ3〉2〈Φ1,Φ3〉),
where Φi=(φi1,…,φiN)T,i=1,2,3.According to (11), the integrable super Broer-Kaup-Kupershmidt hierarchy with self-consistent sources is proposed as follows:
(27)utn=(rsu1tu2t)tn=J(-2An+1-Cn+12σn+1-2ρn+1)+J(2〈Φ1,Φ2〉〈Φ2,Φ2〉-2〈Φ2,Φ3〉2〈Φ1,Φ3〉),
where Φi=(φi1,…,φiN)T, i=1,2,3,  satisfy
(28)φ1jx=(λ+r)φ1j+sφ2j+u1φ3j,φ2jx=φ1j-(λ+r)φ2j+u2φ3j,φ3jx=u2φ1j-u1φ2j,j=1,…,N.
For n=2, we obtain the super Broer-Kaup-Kupershmidt equation with self-consistent sources as follows:
(29)rt2=-12rxx+12sx-2rrx+(u1u2)x+(u1u2x)x+∂∑j=1N‍φ2j2,st2=12sxx-2(rs)x+2u1u1x+2su2u2x+2∂∑j=1N‍φ1jφ2j-2u1∑j=1N‍φ2jφ3j-2u2∑j=1N‍φ1jφ3j,u1t2=u1xx-32rxu1+12sxu2-2ru1x+(s+u1u2)u2x+u1∑j=1N‍φ2j2-∑j=1N‍φ1jφ3j,u2t2=-u2xx-12rxu2-2ru2x-u1x-u2∑j=1N‍φ2j2+∑j=1N‍φ2jφ3j,
where Φi=(φi1,…,φiN)T,i=1,2,3, satisfy
(30)φ1jx=(λ+r)φ1j+sφ2j+u1φ3j,φ2jx=φ1j-(λ+r)φ2j+u2φ3j,φ3jx=u2φ1j-u1φ2j,j=1,…,N.


## 4.  Conservation Laws for the SuperBroer-Kaup-Kupershmidt Hierarchy

In the following, we will construct conservation laws of the super Broer-Kaup-Kupershmidt hierarchy. We introduce the variables(31)E=φ2φ1,K=φ3φ1.
From (7) and (12), we have
(32)Ex=1-2λE-2rE+u2K-sE2-u1EK,Kx=u2-λK-u1E-rK-sKE-u1K2.
Expand E,  K in the power of λ as follows:
(33)E=∑j=1∞‍ejλ-j,K=∑j=1∞‍kjλ-j.
Substituting (33) into (32) and comparing the coefficients of the same power of λ, we obtain
(34)e1=12,k1=u2,e2=-12r,k2=-u2x-12u1-ru2,e3=14rx-12u2u2x+12r2-12u1u2-18s,k3=u2xx+rxu2+2ru2x+12u1x+ru1+r2u2-12su2,…
and a recursion formula for en and kn(35)en+1=-12en,x-ren+12u2kn-12s∑l=1n-1‍elen-l-12u1∑l=1n-1‍elkn-l,kn+1=-kn,x-u1en-rkn-s∑l=1n-1‍klen-l-u1∑l=1n-1‍klkn-l,
because of
(36)∂∂t[λ+r+sE+u1K]=∂∂x[A+BE+ρK],
where
(37)A=m0λ2+m1λ+12m0s-m0u1u2,B=m0sλ+12m0sx-m0rs+m1s,ρ=m0u1λ+m0u1x-m0ru1+m1u1.Assume thatδ=λ+r+sE+u1K,θ=A+BE+ρK. Then (36) can be written as δt=θx, which is the right form of conservation laws. We expand δ and θ as series in powers of λ with the coefficients, which are called conserved densities and currents, respectively,
(38)δ=λ+r+∑j=1∞‍δjλ-j,θ=m0λ2+m1λ+m0s+∑j=1∞‍θjλ-j,
where m0,m1 are constants of integration. The first two conserved densities and currents are read as follows:
(39)δ1=12s+u1u2,θ1=m0(14sx-sr-u1u2x+u2u1x-2ru1u2)+m1(12s+u1u2),δ2=-12sr-u1u2x-ru1u2,θ2=m0(14srx-12su2u2x+sr2-su1u2-18s2-14rsx+u1u2xx+rxu1u2+3ru1u2x+2r2u1u2-u1xu2x-ru1xu214)-m1(12sr+u1u2x+ru1u2).
The recursion relation for δn and θn are
(40)δn=sen+u1kn,θn=m0(sn+1+12sxen-rsen+u1kn+1+u1xkn-ru1kn12)+m1(sen+u1kn),
where en and kn can be calculated from (35). The infinitely many conservation laws of (20) can be easily obtained from (32)–(40), respectively.


## 5.  Conclusions

Starting from Lie super algebras, we may get super equation hierarchy. With the help of variational identity, the Hamiltonian structure can also be presented. Based on Lie super algebra, the self-consistent sources of super Broer-Kaup-Kupershmidt hierarchy can be obtained. It enriched the content of self-consistent sources of super soliton hierarchy. Finally, we also get the conservation laws of the super Broer-Kaup-Kupershmidt hierarchy. It is worth to note that the coupling terms of super integrable hierarchies involve fermi variables; they satisfy the Grassmann algebra which is different from the ordinary one.



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*Source: 913758-2013-06-18.xml*