A generalized super-NLS-mKdV hierarchy is proposed related to Lie superalgebra B(0,1); the resulting supersoliton hierarchy is put into super bi-Hamiltonian form with the aid of supertrace identity. Then, the super-NLS-mKdV hierarchy with self-consistent sources is set up. Finally, the infinitely many conservation laws of integrable super-NLS-mKdV hierarchy are presented.
National Natural Science Foundation of China115471751127100811601055Henan Provincial Institutions of Higher Education of China2017GGJS1451. Introduction
The superintegrable systems have aroused strong interest in recent years; many experts and scholars do research on the field and obtain lots of results [1, 2]. In [3], the supertrace identity and the proof of its constant γ are given by Ma et al. As an application, super-Dirac hierarchy and super-AKNS hierarchy and its super-Hamiltonian structures have been furnished. Then, like the super-C-KdV hierarchy, the super-Tu hierarchy, the multicomponent super-Yang hierarchy, and so on were proposed [4–9]. In [10], the binary nonlinearization and Bargmann symmetry constraints of the super-Dirac hierarchy were given.
Soliton equations with self-consistent sources have important applications in soliton theory. They are often used to describe interactions between different solitary waves, and they can provide variety of dynamics of physical models; some important results have been got by some scholars [11–18]. Conservation laws play an important role in mathematical physics. Since Miura et al. discovery of conservation laws for KdV equation in 1968 [19], lots of methods have been presented to find them [20–23].
In this work, a generalized super-NLS-mKdV hierarchy is constructed. Then, we present the super bi-Hamiltonian form for the generalized super-NLS-mKdV hierarchy with the help of the supertrace identity. In Section 3, we consider the generalized super-NLS-mKdV hierarchy with self-consistent sources based on the theory of self-consistent sources. Finally, the conservation laws of the generalized super-NLS-mKdV hierarchy are given.
2. The Generalized Superequation Hierarchy
Based on the Lie superalgebra B(0,1)(1)e1=1000-10000,e2=010-100000,e3=010100000,e4=0010000-10,e5=000001100;that is, along with the communicative operation(2)e1,e2=2e3,e1,e3=2e2,e1,e4=e2,e5=e3,e5=e4,e4,e2=e5,e1=e3,e4=e5,e2,e3=2e1,e4,e4+=-e2-e3,e4,e5+=e1,e5,e5+=e3-e2.
To set up the generalized super-NLS-mKdV hierarchy, the spectral problem is given as follows:(3)φx=Uφ,φt=Vφ,with(4)U=λ+wu1+u2u3u1-u2-λ-wu4u4-u30,V=AB+CσB-C-Aρρ-σ0,where w=ε1/2u12-1/2u22+u3u4 with ε being an arbitrary even constant, λ is the spectral parameter, u1 and u2 are even potentials, and u3 and u4 are odd potentials. Note that spectral problem (3) with ε=0 is reduced to super-NLS-mKdV hierarchy case [24].
Setting(5)A=∑m≥0amλ-m,B=∑m≥0bmλ-m,C=∑m≥0cmλ-m,σ=∑m≥0σmλ-m,ρ=∑m≥0ρmλ-m,solving the equation Vx=[U,V], we have(6)amx=2u2bm-2u1cm+u3ρm+u4σm,bmx=2cm+1-2u2am-u3σm+u4ρm+2wcm,cmx=2bm+1-2u1am-u3σm-u4ρm+2wbm,σmx=σm+1+wσm+u1+u2ρm-u3am-u4bm-u4cm,ρmx=-ρm+1-wρm+u1σm-u2σm-u3bm+u3cm+u4am,and, from the above recursion relationship, we can get the recursion operator L which meets the following:(7)bm+1-cm+1ρm+1-σm+1=Lbm-cmρm-σm,m⩾0,where the recursive operator L is given as follows:(8)L=-w+2u1∂-1u2-12∂+2u1∂-1u112u4+u1∂-1u3-12u3-u1∂-1u4-12∂-2u2∂-1u2-w+2u2∂-1u112u4-u2∂-1u312u3+u2∂-1u4-u3+2u4∂-1u2-u3+2u4∂-1u1-∂-w+u4∂-1u3-u1+u2-u4∂-1u4-u4-2u3∂-1u2u4-2u3∂-1u1u1+u2-u3∂-1u3∂-w+u3∂-1u4.
Choosing the initial data(9)a0=1,b0=c0=σ0=ρ0=0,from the recursion relations in (6), we can obtain(10)a1=0,b1=u1,c1=u2,σ1=u3,ρ1=u4,a2=-12u12+12u22-u3u4,b2=12u2x-wu1,c2=12u1x-wu2,σ2=u3x-wu3,ρ2=-u4x-wu4,a3=12u2u1x-u1u2x+u3u4x+u4u3x+wu12-u22+2u3u4,b3=14u1xx-12wxu2+12u1u22-12u13+12u3u3x-12u4u4x-u1u3u4-wu2x+w2u1,c3=14u2xx-12wxu1+12u23-12u2u12+12u3u3x+12u4u4x-u2u3u4-wu1x+w2u2,σ3=u3xx+12u3u22-12u3u12+12u4u1x+12u4u2x+u1u4x+u2u4x-wxu3+w2u3-2wu3x,ρ3=u4xx+12u4u22-12u4u12+12u3u1x-12u3u2x+u1u3x-u2u3x+wxu4+w2u4+2wu4x.
Then we consider the auxiliary spectral problem(11)φtn=Vnφ,with (12)Vn=∑m=0nambm+cmσmbm-cm-amρmρm-σm0λn-m+Δn,n≥0.
Suppose(13)Δn=ab+ceb-c-aff-e0,substituting V(n) into the zero curvature equation(14)Utn-Vxn+U,Vn=0,where n≥0. Making use of (6), we have(15)wtn=ax,b=c=e=f=0,u1tn=bnx-2wcn+2u2an+u3σn-u4ρn+bx-2wc+u3e-u4f+2u2a=2cn+1+2u2a,u2tn=cnx-2wbn+2u1a+u3σn+u4ρn+cx+u3e+u4f+2u1a-2wb=2bn+1+2u1a,u3tn=σnx-wσn-u1ρn-u2ρn+u3an+u4bn+u4cn+ex-we-u1f-u2f+u3a+u4b+u4c=σn+1+u3a,u4tn=ρnx+wρn-u1σn+u2σn+u3bn-u3cn-u4an-u4a+fx+wf-u1e+u2e+u3b-u3c=-ρn+1-u4a,which guarantees that the following identity holds true:(16)12u22-12u12+u4u3tn=2u2bn+1-2u1cn+1-ρn+1u3+u4σn+1=an+1x.Choosing a=-εan+1, we arrive at the following generalized super-NLS-mKdV hierarchy:(17)utn=u1u2u3u4tn=2cn+1-2εu2an+12bn+1-2εu1an+1σn+1-εu3an+1-ρn+1+εu4an+1,where n≥0. The case of (17) with ε=0 is exactly the standard supersoliton hierarchy [24].
When n=1 in (17), the flow is trivial. Taking n=2, we can obtain second-order generalized super-NLS-mKdV equations(18)u1t2=12u2xx+u23-u2u12+u3u3x+u4u4x-2u2u3u4+ε2u1u2u2x-2u12u1x+u4xu3u1-u3xu4u1-2u3u4u1x-2u2u3u4x-2u2u4u3x-2u2u12u3u4+2u23u3u4+2ε2u212u12-12u22+u3u42+2ε2u212u12-12u22+u3u4u22-u12,u2t2=12u1xx-u13+u1u22-2u1u3u4+u3u3x-u4u4x+ε2u22u2x-2u1u2u1x-u3xu4u2+u4xu3u2-2u3u4u2x-2u1u3u4x-2u1u4u3x-2ε2u112u12-12u22+u3u4u12-u22+2ε212u12-12u22+u3u42u1-2ε2u1u3u12-u22u4,u3t2=u3xx+12u3u22-12u3u12+12u4u1x+12u4u2x+u1u4x+u2u4x+ε12u3u1u2x-12u3u2u1x+u22u3x-u12u3x+u2u2xu3-u1u1xu3-2u3u4u3x+ε212u12-12u22+u3u42u3-ε2u312u12-12u22+u3u4u12-u22,u4t2=-u4xx-12u4u22+12u4u12+12u3u2x-12u3u1x-u1u3x+u2u3x+ε12u4u2u1x-12u4u1u2x-u1u1xu4+u2u2xu4-u12u4x+u22u4x-2u3u4u4x-ε212u12-12u22+u3u42u4-ε2u412u12-12u22+u3u4u12-u22.
From (3) and (11), one infers the following:(19)V2=V112V122V132V212-V112V232V232-V1320,with(20)V112=λ2+12u22-12u12-u3u4-ε12u2u1x-12u1u2x+u4u3x+u3u4x-2ε212u12-12u22+u3u42,V122=u1+u2λ+12u1+u2x-ε12u12-12u22+u3u4u1+u2,V132=u3λ+u3x-ε12u12-12u22+u3u4u3,V212=u1-u2λ+12u2-u1x+ε12u12-12u22+u3u4u2-u1,V232=u4λ-u4x-ε12u12-12u22+u3u4u4.
In what follows, we shall set up super-Hamiltonian structures for the generalized super-NLS-mKdV hierarchy.
Through calculations, we obtain(21)StrV∂U∂λ=2A,Str∂U∂u1V=2B+εu1A,Str∂U∂u2V=-2C+εu2A,Str∂U∂u3V=2ρ+εu4A,Str∂U∂u4V=-2σ+εu3A.
Substituting the above results into the supertrace identity [3] and balancing the coefficients of λn+2, we obtain(22)δδu∫an+2dx=γ-n-1bn+1+εu1an+1-cn+1-εu2an+1ρn+1+εu4an+1-σn+1-εu3an+1,n≥0.Thus, we have(23)Hn+1=∫-an+2n+1dx,δHn+1δu=bn+1+εu1an+1-cn+1-εu2an+1ρn+1+εu4an+1-σn+1-εu3an+1,n⩾0.Moreover, it is easy to find that(24)bn+1-cn+1ρn+1-σn+1=R1bn+1+εu1an+1-cn+1-εu2an+1ρn+1+εu4an+1-σn+1-εu3an+1,n⩾0,where R1 is given by(25)R1=1-2εu1∂-1u2-2εu1∂-1u1εu1∂-1u3εu1∂-1u42εu2∂-1u21+2εu2∂-1u1εu2∂-1u3-εu2∂-1u4-2εu4∂-1u2-2εu4∂-1u11-εu4∂-1u3εu4∂-1u42εu3∂-1u22εu3∂-1u1εu3∂-1u31-εu3∂-1u4.
Therefore, superintegrable hierarchy (17) possesses the following form:(26)utn=R2bn+1-cn+1ρn+1-σn+1=R2R1bn+1+εu1an+1-cn+1-εu2an+1ρn+1+εu4an+1-σn+1-εu3an+1=JδHn+1δu,n≥0,where(27)R2=-4εu2∂-1u2-2-4εu2∂-1u1-2εu2∂-1u32εu2∂-1u42-4εu1∂-1u2-4εu1∂-1u1-2εu1∂-1u32εu1∂-1u4-4εu3∂-1u2-4εu3∂-1u1-2εu3∂-1u3-1+2εu3∂-1u42εu4∂-1u22εu4∂-1u1-1+εu4∂-1u3-εu4∂-1u4,and super-Hamiltonian operator J is given by(28)J=R2R1=-8εu2∂-1u2-2-8εu2∂-1u1-4εu2∂-1u34εu2∂-1u42-8εu1∂-1u2-8εu1∂-1u1-4εu1∂-1u34εu1∂-1u4-6εu3∂-1u2-6εu3∂-1u1-3εu3∂-1u3-1+3εu3∂-1u44εu4∂-1u24εu4∂-1u1-1+2εu4∂-1u3-2εu4∂-1u4.
In addition, generalized super-NLS-mKdV hierarchy (17) also possesses the following super-Hamiltonian form:(29)utn=R2Lbn-cnρn-σn=R2LR1bn+εu1an-cn-εu2anρn+εu4an-σn-εu3an=MδHnδu,n≥0,where M=R2LR1=Mij4×4 is the second super-Hamiltonian operator.
3. Self-Consistent Sources
Consider the linear system(30)φ1jφ2jφ3jx=Uφ1jφ2jφ3j,φ1jφ2jφ3jt=Vφ1jφ2jφ3j.
From the result in [25], we set(31)δλjδui=13StrΨj∂Uu,λjδui,i=1,2,…,4,with Str on behalf of the supertrace, and(32)Ψj=φ1jφ2j-φ1j2φ1jφ3jφ2j2-φ1jφ2jφ2jφ3jφ2jφ3j-φ1jφ3j0,j=1,2,…,N.
From system (30), we get δλj/δu as follows:(33)∑j=1Nδλjδui=∑j=1NStrΨjδUδu1StrΨjδUδu2StrΨjδUδu3StrΨjδUδu4=2εu1Φ1,Φ2-Φ1,Φ1+Φ2,Φ2-2εu2Φ1,Φ2+Φ1,Φ1+Φ2,Φ22εu4Φ1,Φ2-2Φ2,Φ32εu3Φ1,Φ2-2Φ1,Φ3,where Φi=φi1,…,φiNT,i=1,2,3.
So, we obtain the self-consistent sources of generalized super-NLS-mKdV hierarchy (17):(34)utn=u1u2u3u4tn=JδHn+1δui+J∑j=1Nδλjδui=Jbn+1+εu1an+1-cn+1-εu2an+1ρn+1+εu4an+1-σn+1-εu3an+1+J2εu1Φ1,Φ2-Φ1,Φ1+Φ2,Φ2-2εu2Φ1,Φ2+Φ1,Φ1+Φ2,Φ22εu4Φ1,Φ2-2Φ2,Φ32εu3Φ1,Φ2-2Φ1,Φ3.
For n=2, we get supersoliton equation with self-consistent sources as follows:(35)u1t2=12u2xx+u23-u2u12+u3u3x+u4u4x-2u2u3u4+ε2u1u2u2x-2u12u1x+u4xu3u1-u3xu4u1-2u3u4u1x-2u2u3u4x-2u2u4u3x-2u2u12u3u4+2u23u3u4+2ε2u212u12-12u22+u3u42+2ε2u212u12-12u22+u3u4u22-u12+2εu1∑j=1Nφ1jφ2j-∑j=1Nφ1j2-φ2j2,u2t2=12u1xx-u13+u1u22-2u1u3u4+u3u3x-u4u4x+ε2u22u2x-2u1u2u1x-u3xu4u2+u4xu3u2-2u3u4u2x-2u1u3u4x-2u1u4u3x-2ε2u112u12-12u22+u3u4u12-u22+2ε212u12-12u22+u3u42u1-2ε2u1u3u12-u22u4-2εu2∑j=1Nφ1jφ2j+∑j=1Nφ1j2+φ2j2,u3t2=u3xx+12u3u22-12u3u12+12u4u1x+12u4u2x+u1u4x+u2u4x+ε12u3u1u2x-12u3u2u1x+u22u3x-u12u3x+u2u2xu3-u1u1xu3-2u3u4u3x+ε212u12-12u22+u3u42u3-ε2u312u12-12u22+u3u4u12-u22+2εu4∑j=1Nφ1jφ2j-∑j=1Nφ2jφ3j,u4t2=-u4xx-12u4u22+12u4u12+12u3u2x-12u3u1x-u1u3x+u2u3x+ε12u4u2u1x-12u4u1u2x-u1u1xu4+u2u2xu4-u12u4x+u22u4x-2u3u4u4x-ε212u12-12u22+u3u42u4-ε2u412u12-12u22+u3u4u12-u22+2εu3∑j=1Nφ1jφ2j-2∑j=1Nφ1jφ3j,φ1jx=λ+wφ1j+u1+u2φ2j+u3φ3j,φ2jx=u1-u2φ1j-λ+wφ2j+u4φ3j,φ3jx=u4φ1j-u3φ2j,j=1,…,N.
4. Conservation Laws
In the following, we shall derive the conservation laws of supersoliton hierarchy. Introducing the variables(36)K=φ2φ1,G=φ3φ1,then we obtain(37)Kx=u1-u2-2λ+wK+u4G-u1+u2K2-u3KG,Gx=u4-u3K-λ+wG-u1+u2GK-u3G2.
Next, we expand K and G as series of the spectral parameter λ(38)K=∑j=1∞kjλj,G=∑j=1∞gjλj.Substituting (38) into (37), we raise the recursion formulas for kj and gj:(39)kj+1=-12kjx-wkj+12u4gj-12u1+u2∑l=1j-1klkj-l-12u3∑l=1j-1klgj-l,gj+1=-gjx-u3kj-wgj-u1+u2∑l=1j-1glkj-l-u3∑l=1j-1glgj-l,j≥2.We write the first few terms of kj and gj:(40)k1=12u1-u2,g1=u4,k2=-14u1-u2x-12εu1-u212u12-12u22+u3u4,g2=-u4x-12u1-u2u3+εu1+εu2,k3=18u1-u2xx-18u1+u2u1-u22+14wxu1-u2+12u1-u2xw+12u1-u2w2-12u4u4x,g3=u4xx+12u1-u2u3x-12u12-u22u4+34u1-u2xu3+wxu4+2wu4x+wu1-u2u3+w2u4,….
Note that(41)∂∂tλ+w+u1+u2K+u3G=∂∂xA+B+CK+σG,setting δ=λ+w+(u1+u2)K+u3G, θ=A+(B+C)K+σG, which admitted that the conservation laws is δt=θx. For (19), one infers that(42)A=λ2+12u22-12u12-u3u4,B=u1λ+12u2x-εu112u12-12u22+u3u4,C=u2λ+12u1x-εu212u12-12u22+u3u4,σ=u3λ+u3x-12εu3u12-u22.
Expanding δ and θ as(43)δ=λ+w+∑j=1∞δjλ-j,θ=λ2+12u22-12u12-u3u4+∑j=1∞θjλ-j,conserved densities and currents are the coefficients δj,θj, respectively. The first two conserved densities and currents are read:(44)δ1=12u12-u22+u3u4,θ1=-14u1+u2u1-u2x+14u1-u2u1+u2x-14εu12-u2212u12-12u22+u3u4-12u12-u2212u12-12u22+u3u4-12εu12-u22u3-12εu12-u22u3u4-u3u4x+u3xu4,δ2=-14u1+u2u2-u1x-12εu12-u2212u12-12u22+u3+u3u4-u3u4x,θ2=u1+u2k3-14u1+u2x-εu1+u212u12-12u22+u3u412u1-u2x+u1-u2×12u12-12u22+u3u4+u3g3-u3x-12εu12-u22u3u4x+12u1-u2u3+εu1+εu2,where k3 and g3 are given by (40). The recursion relationship for δj and θj is as follows:(45)δj=u1+u2kj+u3gj,θj=u1+u2kj+1+12u1+u2x-εu1+u212u12-12u22+u3u4kj+u3gj+1+u3x-12εu12-u22u3gj,where kj and gj can be recursively calculated from (39). We can display the first two conservation laws of (18) as(46)δ1t=θ1x,δ2t=θ2x,where δ1,θ1,δ2, and θ2 are defined in (44). Then we can obtain the infinitely many conservation laws of (17) from (37)–(46).
5. Conclusions
In this work, we construct the generalized super-NLS-mKdV hierarchy with bi-Hamiltonian forms with the help of variational identity. Self-consistent sources and conservation laws are also set up. In [26–29], the nonlinearization of AKNS hierarchy and binary nonlinearization of super-AKNS hierarchy were given. Can we do the binary nonlinearization for hierarchy (17)? The question may be investigated in further work.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The project is supported by the National Natural Science Foundation of China (Grant nos. 11547175, 11271008, and 11601055) and the Aid Project for the Mainstay Young Teachers in Henan Provincial Institutions of Higher Education of China (Grant no. 2017GGJS145).
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