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# Zeitschrift für Naturforschung A

### A Journal of Physical Sciences

Editor-in-Chief: Holthaus, Martin

Editorial Board: Fetecau, Corina / Kiefer, Claus

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# 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.

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].

$sl(m/n) = {X = (ABCD), strX = TrA − TrD = 0},$

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

$s[X, Y] = XY − (−1)P(X)P(Y)YX, ∀X, Y ∈ sl(m/n),$

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

$ξf = fξ + fy, ∀f ∈ sl˜(2/1).$

2. Construct a couple of matrices

$U = e0(λ + ξ) + u1e1(0) + … + upep(0) and V = ∑m ≥ 0Vi, mλ−m.$

Then solve the stationary matrix equation for V:

$Vx = UV − VU = [U, V]$

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

$Utn − Vx(n) + [U, V(n)] = 0, (1)$(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

$utn = ∑i = 0n(ni)Φn − iJΨ∂yi, (3)$(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

$U = (λ + ξqαr− λ − ξβεη0), V = (V1V2V3V4V5V6V7V8V9),$

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

$V1, x = V1, y + qV4 − V2r + αV7 − V3ε,V2, x = 2λV2 + 2V2ξ + V2, y + qV5 − V1q + αV8 − V3η,V3, x = λV3 + V3ξ + V3, y + qV6 + αV9 − V1α − V2β,V4, x = −2λV4 − 2V4ξ − V4, y + rV1 − V5r + βV7 − V6ε,V5, x = −V5, y + rV2 − V4q + βV8 − V6η,V6, x = −λV6 − V6ξ − V6, y + rV3 + βV9 − V4α − V5β,V7, x = −λV7 − V7ξ − V8r + εV1 + ηV4 − V9ε,V8, x = λV8 + V8ξ + εV2 + ηV5 − V7q − V9η,V9, x = εV3 + ηV6 − V7α − V8β.$

Setting

$Vi = ∑m ≥ 0Vi, m λ−m, i = 1, 2, …, 9, ∂± = ∂x ± ∂y,$

in these equations, one infers that

${∂−V1, m = qV4, m − V2, mr + αV7, m − V3, mε,2V2, m+1 = (V2, m)x − 2V2, mξ − (V2, m)y − qV5, m + V1, mq − αV8, m + V3, mη,V3, m+1 = (V3, m)x − V3, mξ − (V3, m)y − qV6, m − αV9, m + V1, mα + V2, mβ,2V4, m+1 = − (V4, m)x − 2V4, mξ − (V4, m)y + rV1, m − V5, m r + βV7, m − V6, mε,∂+V5, m = rV2, m − V4, mq + βV8, m − V6, mη,V6, m+1 = − (V6, m)x − V6, mξ − (V6, m)y + rV3, m + βV9, m − V4, mα − V5, mβ,V7, m+1 = − (V7, m)x − V7, mξ − V8, mr + εV1, m + ηV4, m − V9, mε,V8, m+1 = (V8, m)x − V8, mξ + V7, mq − εV2, m − ηV5, m + V9, mη,(V9, m)x = εV3, m + ηV6, m − V7, mα − V8, mβ. (4)$(4)

Denoting

$V+(n) = ∑i = 0nVi, mλ− m = λnV − V−(n), Vi, m = (V1, imV2, imV3, imV4, imV5, imV6, imV7, imV8, imV9, im),$

a direct calculation yields that

$−V+x(n) + [U, V+(n)] = V−x(n) − [U, V−(n)] = (0−2V2, n + 1−V3, n + 12V4, n + 10V6, n + 1V7, n + 1−V8, n + 10).$

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

$σ = (1000000−100000012000000−12000000−1200000012),$

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

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

Substituting (6) into (4) yields

${V2, 02 = 12(qx − qy) − qξ,V3, 02 = αx − αy − αξ,V4, 02 = − 12(rx + ry) − rξ,V6, 02 = − βx − βy − βξ,V7, 02 = − εx − εξ,V8, 02 = ηx − ηξ. (7)$(7)

Using (4) and (7), we get

${∂−V1, 02 = qV4, 02 − V2, 02r + αV7, 02 − V3, 02 ε ⇒ V1, 02 = − 12qr − αε,∂+V5, 02 = rV2, 02 − V4, 02q + βV8, 02 − V6, 02 η ⇒ V5, 02 = 12qr + βη,(V9, 02)x = εV3, 02 + ηV6, 02 − V7, 02 α − V8, 02 β ⇒ V9, 02 = εα − ηβ. (8)$(8)

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

$V2, 03 = 14qxx − 12qxy + 14qyy + (qy − qx)ξ + qξ2 − 12q2r − 12qβη − 12αεq+ 12(αxη + ηxα + ηαy + ηyα), (9)$(9)

$V3, 03 = αxx − 2αxy + αyy + (2αy − 2αx)ξ + αξ2 + qβx + αηβ − 12qrβ − αβε + 12qxβ − 12qyβ, (10)$(10)

$V4, 03 = 14rxx + 12rxy + 14ryy + (rx + ry)ξ + rξ2 + 12(−qr2 − rαε − βηr − βεx − εβx + βyε + βεy), (11)$(11)

$V6, 03 = βxx + 2βxy + βyy + (2βx + 2βy)ξ + βξ2 + rαx − rαy + βεα + 12(rx + ry)α − 12qrβ, (12)$(12)

$V7, 03 = εxx + 2εxξ + εξ2 − ηxr + ηry − 12εqr − 12η(rx + ry) − εα + ηβ, (13)$(13)

$V8, 03 = ηxx − 2ηxξ + ηξ2 − εxq − εqy − 12ε(qx − qy) − 12ηqr + εαη. (14)$(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)

${∂−V1, 11 = qV4, 11 − V2, 11r + αV7, 11 − V3, 11 ε ⇒ V1, 11 = ∂−−1(εα − 12qr)y,∂+V5, 11 = rV2, 11 − V4, 11q + βV8, 11 − V6, 11 η ⇒ V5, 11 = − ∂+−1(12qr + βη)y,(V9, 11)x = εV3, 11 + ηV6, 11 − V7, 11α − V8, 11β ⇒ V9, 11 = 0. (17)$(17)

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

$V2, 12 ​= 14qxy − 14qyy + 12(qx − 2qy)ξ − qξ2 + 12(qV5, 11 + qV1, 11 + αηy + αyη), (18)$(18)

$V3, 12 = αxy − αyy + (αx − 2αy)ξ − αξ2 + 12qyβ + V1, 11α, (19)$(19)

$V4, 12 = − 14rxy − 14ryy − 12(rx + 2ry)ξ − rξ2 + 12(rV1, 11 − rV5, 11 − (βε)y), (20)$(20)

$V6, 12 = − βxy − βyy − (βx + 2βy)ξ − βξ2 − 12ryα − V5, 11β, (21)$(21)

$V7, 12 = − εxξ − εξ2 − 12ηry + εV1, 11, (22)$(22)

$V8, 12 = ηxξ − ηξ2 + 12εqy − ηV5, 11. (23)$(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

${qt2 = 12qxx + 12qyy − q2r − qβη − αεq + αxη − ηαy + 2qV5, 11 + 2qV1, 11,rt2 = − 12rxx − 12ryy + qr2 + rαε + βηr + βεx + εβx + 12(βε)y − rV1, 11 + rV5, 11,αt2 = αxx + αηβ − 12qrβ − αβε + 12qxβ + 12qyβ + 2αV1, 11,βt2 = − βxx − 2rαx + 2rαy − 2βεα − rxα + qrβ + 2βV5, 11,εt2 = − εxx + ηxr + 12εqr + 12η(rx + ry) + εα − ηβ − 2εV1, 11,ηt2 = ηxx − εxq − 12ε(qx − qy) − 12ηqr + εαη − 2ηV5, 11. (26)$(26)

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

${qt = 12qxx + 12qyy − q2r − q∂−−1(qr)y − q∂+−1(qr)y,rt = − 12rxx − 12ryy + qr2 − 12r∂−−1(qr)y − 12r∂+−1(qr)y, (27)$(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):

${qt = 12(qxx + qyy) − q2r − q∂+−1(qr)y − q∂−−1(qr)y,rt = − 12rxx − 12ryy + qr2 − 12r∂−−1(qr)y − 12r∂+−1(qr)y,εt = − εxx + ηxr + 12εqr + 12η(rx + ry) + ε∂−−1(qr)y,ηt = ηxx − εxq − 12ε(qx − qy) − 12ηqr + η∂+−1(qr)y. (29)$(29)

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

${qt = 12(qxx + qyy) − q2r − q∂+−1(qr)y − q∂−−1(qr)y,rt = − 12rxx − 12ryy + qr2 − 12r∂−−1(qr)y − 12r∂+−1(qr)y,αt = αxx − 12qrβ + 12(qx + qy)β − α∂−−1(qr)y,βt = − βxx − 2rαx + 2rαy − rxα + qrβ − β∂+−1(qr)y. (30)$(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:

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

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

$δδu∫str(V∂U∂λ) = λ−γ∂∂λ[λγstr(∂U∂uV)], (32)$(32)

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

$str(VUλ) = V1 − V5, str(UqV) = V4, str(UrV) = V2, str(UαV) = V7, str(UβV) = V8, str(UϵV) = V3, str(UηV) = V6. (33)$(33)

Substituting (33) into (32), we get

$δδu∫(V1 − V5)dx = λ − γ∂∂λλγ(V4V2V7V8V3V6). (34)$(34)

Setting $Vi = ∑m ≥ 0Vi, mλ−m, i = 1, 2, …, 8$ and comparing the coefficients of λm–1, one infers that

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

$J = (020000− 20000000001000000−100−1000000100),$

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

$Hn = − ∫V1, n + 1−V5, n + 1n dx.$

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

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,

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