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  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.  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  needed to construct more complicated Lax pairs. Tu et al.  introduced a residue operator over an associative algebra and proposed a method called the TAH scheme  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  proposed a method for generating 2+1 dimensional integrable systems through the construction of Hamiltonian operators. Based on this research, Tu et al.  presented the BRR method and obtained the 2+1 dimensional AKNS hierarchy. Zhang et al.  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  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].
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
where the degration of the element X is defined as
The matrix is called the even element or the bosonic, while the matrix is called the odd element or called the fermionic. Hu  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  constructed a new super KdV equation by using the super algebra as stated. Zhou  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  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  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 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.
The super loop algebra is defined by
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
Construct a couple of matrices
Then solve the stationary matrix equation for V:
and obtain a recurrence operator Ψ.
Try to construct a sequence of matrices V(n) so that
where C stands for a set of complex numbers.
With the help of the zero curvature equation
generate the super integrable hierarchy, or more specifically, the super 2+1 dimensional integrable hierarchy.
Rewrite this integrable hierarchy as
where J and Ψ are integro-differential operators.
According to the idea given by Magri et al.  employ multiple seeds and combine them together to get a hierarchy of the form
where the operator Φ is defined by
The aforementioned procedure is the BRR method described in Tu et al. . 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.  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 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.
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 + V5 − V9=0. A solution of the stationary matrix equation
is given by
in these equations, one infers that
a direct calculation yields that
Let Then (1) is satisfied by
We next write the super 2+1 dimensional integrable hierarchy as a binormial-residue representation by using (3). Denote
Case 1: Take
In terms of (4), we have
Substituting (6) into (4) yields
Using (4) and (7), we get
Inserting (7) and (8) into (4) again, we arrive at
Case 2: Take
Substituting (15) into (4) gives
Inserting (16) and (17) into (4) leads to
Case 3: Take
Inserting (24) into (4) produces
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
When n=2, one infers that
If we let α=β=ε=η=0, t2=t, (26) reduces to
which is a generalized 2+1 dimensional nonlinear Schrödinger equation. In particular, when r=0, (27) becomes the 2+1 dimensional diffusion equation
When α=β=0, (26) reduces to a super 2+1 dimensional integrable coupling of (27):
When ε=η=0, (26) reduces to another super 2+1 dimensional integrable coupling of (27):
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:
In what follows, we consider the Hamiltonian structure of the super-integrable hierarchy (5) by using the super-trace identity [11, 18]:
where γ is a constant to be determined. Through direct calculation, we have
Substituting (33) into (32), we get
Setting and comparing the coefficients of λ–m–1, one infers that
When n=1, we see from (6) and (8) that γ=0. Therefore the super-integrable hierarchy (5) can be written as
where J is the Hamiltonian operator
and Hn+1 is the conserved densities of the hierarchy (5) given by
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.
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.
Y. Ohta, J. Satsuma, D. Takahashi, and T. Tokihiro, Prog.Theor. Phys. (Suppl. 94), 210 (1998).Google Scholar
M. J. Ablowitz, S. Chakravarty, and R. Halburd, Commun. Math. Phys. 158, 289 (1993).Google Scholar
G. Z. Tu, R. I. Andrushkiw, and X. C. Huang, J. Math. Phys. 32, 1900 (1991).Google Scholar
G. Z. Tu, B. L. Feng, and Y. F. Zhang, J. Weifang Univ. 14, 1 (2014).Google Scholar
Y. F. Zhang, L. X. Wu, and W. J. Rui, Commun. Theor. Phys. 63, 535 (2015).Google Scholar
M. Schnnet, The Theory of Superalgebras, Lect. Notes Math., Springer 1978.Google Scholar
X. B. Hu, Integrable Systems and Related Problems. Doct. Thesis, Academia Sinica, China 1990.Google Scholar
X. B. Hu, J. Phys. A 30, 619 (1997).Google Scholar
R. G. Zhou, Phys. Lett. A 378, 1816 (2014).Google Scholar
Q. P. Liu, Lett. Math. Phys. 35, 115 (1995).Google Scholar
Q. P. Liu and X. B. Hu, J. Phys. A 38, 6371 (2005).Google Scholar
Y. F. Zhang and W. J. Rui, Rep. Math. Phys. 75, 231 (2015).Google Scholar
F. Magri, C. Morosi, and G. Tondo, Commun. Math. Phys. 115, 457 (1988).Google Scholar
W. X. Ma, J. S. He, and Z. Y. Qin, J. Math. Phys. 49, 033511 (2008).Google Scholar
C. Tian, Lie Group and its Applications in Differential Equations, Science Press 2001 (in Chinese).Google Scholar
S.Y. Lou, Phys. Lett. A 151, 133 (1990).Google Scholar
S.Y. Lou, Phys. Rev. Lett. 71, 4099 (1993).Google Scholar
S. Y. Lou, J. Phys. A 26, 4387 (1993).Google Scholar
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Published Online: 2015-08-08
Published in Print: 2015-10-01