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Zeitschrift für Naturforschung A

A Journal of Physical Sciences

Editor-in-Chief: Holthaus, Martin

Editorial Board: Fetecau, Corina / Kiefer, Claus


IMPACT FACTOR 2017: 1.414

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1865-7109
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Volume 70, Issue 10

Issues

Some 2+1 Dimensional Super-Integrable Systems

Yufeng Zhang / Honwah Tam / Jianqin Mei
Published Online: 2015-08-08 | DOI: https://doi.org/10.1515/zna-2015-0213

Abstract

In the article, we make use of the binormial-residue-representation (BRR) to generate super 2+1 dimensional integrable systems. Using these systems, we can deduce a super 2+1 dimensional AKNS hierarchy, which can be reduced to a super 2+1 dimensional nonlinear Schrödinger equation. In particular, two main results are obtained. One of them is a set of super 2+1 dimensional integrable couplings. The other one is a 2+1 dimensional diffusion equation. The Hamiltonian structure of the super 2+1 dimensional hierarchy is derived by using the super-trace identity.

Keywords: Hamiltonian Structure; Super 2+1 Dimensional Equation; Super Lie Algebra

PACS Numbers:: 05.45.Yv; 02.30.Jr; 02.30.Ik

1 Introduction

It is an interesting task to generate new integrable equations. There are many methods to generate 1+1 dimensional integrable systems. Conversely, the approaches for deriving 2+1 dimensional integrable equations are limited, because the difficulties involved are much more complicated than the 1+1 dimensional case. The Sato theory [1] is a very successful scheme in deriving 2+1 dimensional integrable systems. However, it is not easy to choose proper τ functions in this theory. Ablowitz et al. [2] employed some reductions of the self-dual Yang–Mills equations to exhibit some 2+1 dimensional equations, such as the KP equation, the mKP equation, etc., but the method presented [2] needed to construct more complicated Lax pairs. Tu et al. [3] introduced a residue operator over an associative algebra and proposed a method called the TAH scheme [4] for which the KP equation was derived. However, the TAH scheme did not adopt zero curvature equations to directly derive 2+1 dimensional integrable systems. Therefore, the integrability of the systems obtained could not be ensured. Dorfman and Fokas [5] proposed a method for generating 2+1 dimensional integrable systems through the construction of Hamiltonian operators. Based on this research, Tu et al. [6] presented the BRR method and obtained the 2+1 dimensional AKNS hierarchy. Zhang et al. [7] employed the BRR method to produce a few integrable systems. In the current article, we extend the BRR method to super 2+1 dimensional integrable systems. Specifically, we use the BRR method to generate some new super 2+1 dimensional integrable hierarchies. Subsequently, we obtain a type of super 2+1 dimensional AKNS hierarchy different from the one in [7] and further reduce it to the super 2+1 dimensional nonlinear Schrödinger equation. In particular, the super integrable hierarchy can be reduced to two different super 2+1 dimensional integrable couplings, which are new findings. Another prominent result is a 2+1 dimensional diffusion equation. It is derived by reducing the 2+1 dimensional integrable couplings. In addition, we again generate the Hamiltonian structure of the super 2+1 dimensional hierarchy by using the super-trace identity.

2 The Super Lie Algebra sl(m/n)

The definition of the super Lie algebra sl(m/n) is given by [8–10].

sl(m/n)={X=(ABCD), strX=TrATrD=0},

where A is an m×m matrix, D is an n×n matrix, B is an m×n matrix and C is an n×m matrix. The Lie bracket of sl(m/n) is denoted by

s[X,Y]=XY(1)P(X)P(Y)YX,X,Ysl(m/n),

where the degration of the element X is defined as

P(X)={0,X=(A00D), strX=0,1,X=(0BC0).

The matrix (A00D) is called the even element or the bosonic, while the matrix (0BC0) is called the odd element or called the fermionic. Hu [11] introduced different bases of the super Lie algebra sl(m/n) and obtained some interesting results, including the Hamiltonian structures of the super integrable systems. Geng and Wu [12] constructed a new super KdV equation by using the super algebra as stated. Zhou [13] started from a 3×3 spectral problem to obtain a super integrable hierarchy with bi-super-Hamiltonian structure. Liu and Hu [14, 15] investigated the Darboux transformations for the supersymmetric KdV equation. Zhang and Rui [16] adopted the special super Lie algebra sl(2/1) to obtain some super integrable systems. All the results above are super 1+1 dimensional integrable systems, except f`or the super 2+1 dimensional hierarchy in [16] derived by the TAH scheme. However, we cannot determine the integrability of this 2+1 dimensional hierarchy. In the current paper, we introduce a super loop algebra sl˜(2/1) and deduce a super 2+1 dimensional integrable hierarchy by employing the BRR method. This hierarchy can be reduced to the super 2+1 dimensional nonlinear Schrödinger equation. Furthermore, we obtain two super 2+1 dimensional integrable couplings of the super 2+1 dimensional nonlinear Schrödinger equation.

We briefly recall the BRR method.

  1. The super loop algebra sl˜(2/1) is defined by

    sl˜(2/1)={X(u,λ+ξ)=e0(λ+ξ)+u1e1(λ+ξ)++upep(λ+ξ)},

    where e0, e1, …, ep constitute a set of basis for the super Lie algebra sl(2/1), with u=(u1, …, up) being the potential function. The operator ξ is defined by

    ξf=fξ+fy,fsl˜(2/1).

  2. Construct a couple of matrices

    U=e0(λ+ξ)+u1e1(0)++upep(0)and V=m0Vi,mλm.

    Then solve the stationary matrix equation for V:

    Vx=UVVU=[U,V]

    and obtain a recurrence operator Ψ.

  3. Try to construct a sequence of matrices V(n) so that

    Vx(n)[U,V(n)]Ce1++Cep,

    where C stands for a set of complex numbers.

  4. With the help of the zero curvature equation

    UtnVx(n)+[U,V(n)]=0, (1)(1)

    generate the super integrable hierarchy, or more specifically, the super 2+1 dimensional integrable hierarchy.

  5. Rewrite this integrable hierarchy as

    utn=JΨnb0, (2)(2)

    where J and Ψ are integro-differential operators.

  6. According to the idea given by Magri et al. [17] employ multiple seeds and combine them together to get a hierarchy of the form

    utn=i=0n(ni)ΦniJΨyi, (3)(3)

    where the operator Φ is defined by

    JΨ=ΦJ.

The aforementioned procedure is the BRR method described in Tu et al. [6]. However, it only applies to generating 1+1 dimensional super integrable systems. In the next section, we extend this method to the case of super 2+1 dimensional hierarchies.

3 A New Super 2+1 Dimensional Integrable Hierarchy

Ma et al. [18] once used the super Lie algebra sl(2/1) to generate some super integrable hierarchies and their Hamiltonian structures. In the section, we apply the loop algebra sl˜(2/1) with operator matrices of the super Lie algebra sl(2/1) to generate a super 2+1 dimensional AKNS integrable hierarchy through employing the BRR method.

We set

U=(λ+ξqαrλξβεη0),V=(V1V2V3V4V5V6V7V8V9),

where λ is a spectral parameter, λ, q, r are all bosonic, and α, β, ε, η are all fermionic in U. The V1, V2, V4, V5, V9 are all even, and the V3, V6, V7, V8 are all odd, with V1 + V5V9=0. A solution of the stationary matrix equation

Vx=[U,V]

is given by

V1,x=V1,y+qV4V2r+αV7V3ε,V2,x=2λV2+2V2ξ+V2,y+qV5V1q+αV8V3η,V3,x=λV3+V3ξ+V3,y+qV6+αV9V1αV2β,V4,x=2λV42V4ξV4,y+rV1V5r+βV7V6ε,V5,x=V5,y+rV2V4q+βV8V6η,V6,x=λV6V6ξV6,y+rV3+βV9V4αV5β,V7,x=λV7V7ξV8r+εV1+ηV4V9ε,V8,x=λV8+V8ξ+εV2+ηV5V7qV9η,V9,x=εV3+ηV6V7αV8β.

Setting

Vi=m0Vi,mλm,i=1,2,,9,  ±=x±y,

in these equations, one infers that

{V1,m=qV4,mV2,mr+αV7,mV3,mε,2V2,m+1=(V2,m)x2V2,mξ(V2,m)yqV5,m+V1,mqαV8,m+V3,mη,V3,m+1=(V3,m)xV3,mξ(V3,m)yqV6,mαV9,m+V1,mα+V2,mβ,2V4,m+1=(V4,m)x2V4,mξ(V4,m)y+rV1,mV5,mr+βV7,mV6,mε,+V5,m=rV2,mV4,mq+βV8,mV6,mη,V6,m+1=(V6,m)xV6,mξ(V6,m)y+rV3,m+βV9,mV4,mαV5,mβ,V7,m+1=(V7,m)xV7,mξV8,mr+εV1,m+ηV4,mV9,mε,V8,m+1=(V8,m)xV8,mξ+V7,mqεV2,mηV5,m+V9,mη,(V9,m)x=εV3,m+ηV6,mV7,mαV8,mβ. (4)(4)

Denoting

V+(n)=i=0nVi,mλm=λnVV(n),Vi,m=(V1,imV2,imV3,imV4,imV5,imV6,imV7,imV8,imV9,im),

a direct calculation yields that

V+x(n)+[U,V+(n)]=Vx(n)[U,V(n)]=(02V2,n+1V3,n+12V4,n+10V6,n+1V7,n+1V8,n+10).

Let V(n)=V+(n). Then (1) is satisfied by

utn=[q,r,α,β,ε,η]T=[2V2,n+1,2V4,n+1,V3,n+1,V6,n+1,V7,n+1,V8,n+1]T=2σ[V2,n+1,V4,n+1,V3,n+1,V6,n+1,V7,n+1,V8,n+1]T, (5)(5)

where

σ=(1000000100000012000000120000001200000012),

We next write the super 2+1 dimensional integrable hierarchy as a binormial-residue representation by using (3). Denote

Vj0=σξj,j=0,1,2,,Vji=(V1,jiV2,jiV3,jiV4,jiV5,jiV6,jiV7,jiV8,jiV9,ji).

Case 1: Take

V1,00=1,V5,00=1,V9,00=V2,00=V3,00=V4,00=V6,00=V7,00=V8,00=0.

In terms of (4), we have

{V2,01=q,  V3,01=α,  V4,01=r,V6,01=β,  V7,01=ε,  V8,01=η,V1,01=0V1,01=0,+V5,01=rqqr+βη+ηβ=0V5,01=0,(V9,01)x=εα+ηβ+αε+βη=0V9,01=0. (6)(6)

Substituting (6) into (4) yields

{V2,02=12(qxqy)qξ,V3,02=αxαyαξ,V4,02=12(rx+ry)rξ,V6,02=βxβyβξ,V7,02=εxεξ,V8,02=ηxηξ. (7)(7)

Using (4) and (7), we get

{V1,02=qV4,02V2,02r+αV7,02V3,02εV1,02=12qrαε,+V5,02=rV2,02V4,02q+βV8,02V6,02ηV5,02=12qr+βη,(V9,02)x=εV3,02+ηV6,02V7,02αV8,02βV9,02=εαηβ. (8)(8)

Inserting (7) and (8) into (4) again, we arrive at

V2,03=14qxx12qxy+14qyy+(qyqx)ξ+qξ212q2r12qβη12αεq+12(αxη+ηxα+ηαy+ηyα), (9)(9)

V3,03=αxx2αxy+αyy+(2αy2αx)ξ+αξ2+qβx+αηβ12qrβαβε+12qxβ12qyβ, (10)(10)

V4,03=14rxx+12rxy+14ryy+(rx+ry)ξ+rξ2+12(qr2rαεβηrβεxεβx+βyε+βεy), (11)(11)

V6,03=βxx+2βxy+βyy+(2βx+2βy)ξ+βξ2+rαxrαy+βεα+12(rx+ry)α12qrβ, (12)(12)

V7,03=εxx+2εxξ+εξ2ηxr+ηry12εqr12η(rx+ry)εα+ηβ, (13)(13)

V8,03=ηxx2ηxξ+ηξ2εxqεqy12ε(qxqy)12ηqr+εαη. (14)(14)

Case 2: Take

V1,10=ξ,V5,10=ξ,V9,10=V2,10=V3,10=V4,10=V6,10=V7,10=V8,10=0. (15)(15)

Substituting (15) into (4) gives

V2,11=12qy+qξ,V3,11=αξ+αy,V4,11=12ry+rξ,V6,11=βξ+βy,V7,11=εξ,  V8,11=ηξ, (16)(16)

{V1,11=qV4,11V2,11r+αV7,11V3,11εV1,11=1(εα12qr)y,+V5,11=rV2,11V4,11q+βV8,11V6,11ηV5,11=+1(12qr+βη)y,(V9,11)x=εV3,11+ηV6,11V7,11αV8,11βV9,11=0. (17)(17)

Inserting (16) and (17) into (4) leads to

V2,12=14qxy14qyy+12(qx2qy)ξqξ2+12(qV5,11+qV1,11+αηy+αyη), (18)(18)

V3,12=αxyαyy+(αx2αy)ξαξ2+12qyβ+V1,11α, (19)(19)

V4,12=14rxy14ryy12(rx+2ry)ξrξ2+12(rV1,11rV5,11(βε)y), (20)(20)

V6,12=βxyβyy(βx+2βy)ξβξ212ryαV5,11β, (21)(21)

V7,12=εxξεξ212ηry+εV1,11, (22)(22)

V8,12=ηxξηξ2+12εqyηV5,11. (23)(23)

Case 3: Take

V1,20=ξ2,V5,20=ξ2,V9,20=V2,20=V3,20=V4,20=V6,20=V7,20=V8,20=0. (24)(24)

Inserting (24) into (4) produces

{V2,21=qξ2+qyξ+12qyy,V3,21=αξ2+2αyξ+αyy,V4,21=rξ2+ryξ+12ryy,V6,21=βξ2+2βyξ+βyy,V7,21=εξ2,V8,21=ηξ2. (25)(25)

In what follows, we make use of (3) to rewrite (5) into a binormial-residue representation. Firstly, we discuss some of its reductions by using (6)–(25).

When n=1, we have

{qt1=2(V2,02+V2,11)=qx,rt1=2(V4,02+V4,11)=rx,αt1=αx,βt1=βx,εt1=εx,ηt1=ηx.

When n=2, one infers that

{qt2=12qxx+12qyyq2rqβηαεq+αxηηαy+2qV5,11 +2qV1,11,rt2=12rxx12ryy+qr2+rαε+βηr+βεx+εβx +12(βε)yrV1,11+rV5,11,αt2=αxx+αηβ12qrβαβε+12qxβ+12qyβ+2αV1,11,βt2=βxx2rαx+2rαy2βεαrxα+qrβ+2βV5,11,εt2=εxx+ηxr+12εqr+12η(rx+ry)+εαηβ2εV1,11,ηt2=ηxxεxq12ε(qxqy)12ηqr+εαη2ηV5,11. (26)(26)

If we let α=β=ε=η=0, t2=t, (26) reduces to

{qt=12qxx+12qyyq2rq1(qr)yq+1(qr)y,rt=12rxx12ryy+qr212r1(qr)y12r+1(qr)y, (27)(27)

which is a generalized 2+1 dimensional nonlinear Schrödinger equation. In particular, when r=0, (27) becomes the 2+1 dimensional diffusion equation

qt=12(qxx+qyy). (28)(28)

When α=β=0, (26) reduces to a super 2+1 dimensional integrable coupling of (27):

{qt=12(qxx+qyy)q2rq+1(qr)yq1(qr)y,rt=12rxx12ryy+qr212r1(qr)y12r+1(qr)y,εt=εxx+ηxr+12εqr+12η(rx+ry)+ε1(qr)y,ηt=ηxxεxq12ε(qxqy)12ηqr+η+1(qr)y. (29)(29)

When ε=η=0, (26) reduces to another super 2+1 dimensional integrable coupling of (27):

{qt=12(qxx+qyy)q2rq+1(qr)yq1(qr)y,rt=12rxx12ryy+qr212r1(qr)y12r+1(qr)y,αt=αxx12qrβ+12(qx+qy)βα1(qr)y,βt=βxx2rαx+2rαyrxα+qrββ+1(qr)y. (30)(30)

Obviously, the super integrable coupling (29) is different from the super integrable coupling (30). To our best knowledge, these results are not found in previous research on integrable systems.

The super 2+1 dimensional integrable hierarchy (5) can be rewritten into a general binormial representation:

{qtn=2i=0n(ni)V2,n+1i,rtn=2i=0n(ni)V4,n+1i,αtn=i=0n(ni)V3,n+1i,βtn=i=0n(ni)V6,n+1i,εtn=i=0n(ni)V7,n+1i,ηtn=i=0n(ni)V8,n+1i. (31)(31)

In what follows, we consider the Hamiltonian structure of the super-integrable hierarchy (5) by using the super-trace identity [11, 18]:

δδustr(VUλ)=λγλ[λγstr(UuV)], (32)(32)

where γ is a constant to be determined. Through direct calculation, we have

str(VUλ)=V1V5, str(UqV)=V4, str(UrV)=V2,str(UαV)=V7, str(UβV)=V8, str(UϵV)=V3,str(UηV)=V6. (33)(33)

Substituting (33) into (32), we get

δδu(V1V5)dx=λγλλγ(V4V2V7V8V3V6). (34)(34)

Setting Vi=m0Vi,mλm,i=1,2,,8 and comparing the coefficients of λm–1, one infers that

δδu(V1.m+1V5,m+1)=(m+γ)(V4mV2mV7mV8mV3mV6m). (35)(35)

When n=1, we see from (6) and (8) that γ=0. Therefore the super-integrable hierarchy (5) can be written as

utn=(q,r,α,β,ϵ,η)T=J(V4,n+1V2,n+1V7,n+1V8,n+1V3,n+1V6,n+1)=JδHn+1δu, (36)(36)

where J is the Hamiltonian operator

J=(020000200000000010000001001000000100),

and Hn+1 is the conserved densities of the hierarchy (5) given by

Hn=V1,n+1V5,n+1ndx.

4 Remark

It is important to investigate the symmetry analyses of (27), (29), and (30) according to [19–22]. However, since there exist operators (∂±)–1 in these partial differential equations, it may be difficult to discuss their Lie group properties. Can we follow the ways for generating symmetries of the KP equation to discuss the symmetries of (27), (29) and (30)? We shall consider this problem in other articles.

Acknowledgments

This work was supported by the Innovation Team of Jiangsu Province hosted by the Chinese University of Mining and Technology (2014), the National Natural Science Foundation of China (Grant No. 11371361), the Fundamental Research Funds for the Central Universities (Grant No. 2013XK03), the Natural Science Foundation of Shandong Province (Grant No. ZR2013AL016), and Hong Kong Research Grant Council (Grant No. HKBU202512).

Yufeng Zhang is grateful to the reviewers for their helpful suggestions.

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About the article

Corresponding author: Yufeng Zhang, College of Sciences, China University of Mining and Technology, Xuzhou 221116, China, E-mail: zyfxz@cumt.edu.cn


Received: 2015-05-02

Accepted: 2015-07-15

Published Online: 2015-08-08

Published in Print: 2015-10-01


Citation Information: Zeitschrift für Naturforschung A, Volume 70, Issue 10, Pages 791–796, ISSN (Online) 1865-7109, ISSN (Print) 0932-0784, DOI: https://doi.org/10.1515/zna-2015-0213.

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