# A Integrable Generalized Super-NLS-mKdV Hierarchy, Its Self-Consistent Sources, and Conservation Laws

Authors: Hanyu Wei; Tiecheng Xia
Journal: Advances in Mathematical Physics
 (2018)

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## Abstract

A generalized super-NLS-mKdV hierarchy is proposed related to Lie superalgebraB(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.


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## Body

## 1. 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 Section3, 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 superalgebraB(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].Whenn=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.Forn=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 expandK 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.Expandingδ 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.



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