Research Article | Open Access
Sixing Tao, Tiecheng Xia, "Nonlinear Super Integrable Couplings of Super Broer-Kaup-Kupershmidt Hierarchy and Its Super Hamiltonian Structures", Advances in Mathematical Physics, vol. 2013, Article ID 520765, 8 pages, 2013. https://doi.org/10.1155/2013/520765
Nonlinear Super Integrable Couplings of Super Broer-Kaup-Kupershmidt Hierarchy and Its Super Hamiltonian Structures
Nonlinear integrable couplings of super Broer-Kaup-Kupershmidt hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity, and the conserved functionals were proved to be in involution in pairs under the defined Poisson bracket. As its reduction, nonlinear integrable couplings of the classical integrable hierarchy were obtained.
With the development of soliton theory, super integrable systems associated with Lie super algebra have aroused growing attentions by many mathematicians and physicists. It was known that super integrable systems contained the odd variables, which would provide more prolific fields for mathematical researchers and physical ones. Several super integrable systems, including super AKNS hierarchy, super KdV hierarchy, super NLS-MKdV hierarchy, super Tu hierarchy, super Broer-Kaup-Kupershmidt hierarchy, have been studied in [1–8]. There are some interesting results on the super integrable systems, such as Darboux transformation in , super Hamiltonian structures in [10–12] binary nonlinearization in , and reciprocal transformation in .
The research of integrable couplings of the well-known integrable hierarchy has received considerable attention in [15–23] A few approaches to construct linear integrable couplings of the classical soliton equation are presented by permutation, enlarging spectral problem, using matrix Lie algebra  constructing new loop Lie algebra and creating semidirect sums of Lie algebra. Recently, Ma  and Ma and Zhu  presented a scheme for constructing nonlinear continuous and discrete integrable couplings using the block type matrix algebra. However, there is one interesting question for us is how to generate nonlinear super integrable couplings for the super integrable hierarchy.
In this paper, we would like to construct nonlinear super integrable couplings of the super soliton equations through enlarging matrix Lie super algebra. We take the Lie algebra as an example to illustrate the approach for extending Lie super algebras. Based on the enlarged Lie super algebra , we work out nonlinear super integrable Hamiltonian couplings of the super Broer-Kaup-Kupershmidt hierarchy. Finally, we will reduce the nonlinear super Broer-Kaup-Kupershmidt integrable Hamiltonian couplings to some special cases.
2. Enlargement of Lie Super Algebra
Consider the Lie super algebra . Its basis is where , , and are even elements and , and are odd elements. Their nonzero (anti)commutation relations are Let us enlarged the Lie super algebra to the Lie super algebra with a basis where , , , , , and are even, and , and are odd.
The generator of Lie super algebra , , , satisfy the following (anti)commutation relations: Define a loop super algebra corresponding to the Lie superalgebra , denote by The corresponding (anti)commutative relations are given as
3. Nonlinear Super Integrable Couplings of Super Broer-Kaup-Kupershmidt Hierarchy
Let us start from an enlarged spectral problem associated with : where , , , and are even potentials but and are odd ones.
In ordr to obtain super integrable couplings of super integrable hierarchy, we first solve the adjoint representation of (7), with where , , , , , and are commuting fields and , and are anticommuting fields. Then we obtain Substituting into the previous equation gives the following recursive formulas: From these equations, we can successively deduce Equations (12) can be written as where
with Then, let us consider the spectral problem (7) with the following auxiliary spectral problem: with
From the compatible condition, , according to (7) and (17), we get the zero curvature equation: which gives a nonlinear Lax super integrable hierarchy: The super integrable hierarchy (20) is a nonlinear super integrable couplings for the super BKK hierarchy:
4. Super Hamiltonian Structures
A direct calculation reads
It then follows that the nonlinear super integrable couplings (20) possess the following super Hamiltonian form where is a super Hamiltonian operator and () are Hamiltonian functions.
It can be verified that where and are conjugate operators of and , respectively.
Suppose are arbitrary nonnegative integers consider By using the properties of and in (29), we can obtain Ascending the subscript of left factor , then we have By using the antisymmetric law of Poisson bracket (31), we have So the conserved functionals are in involution in pairs under the Poisson bracket (30).
Taking , the hierarchy (27) reduces to a nonlinear integrable couplings of the Broer-Kaup-Kupershmidt hierarchy.
When in (27), we obtain the nonlinear super couplings of the second super Broer-Kaup-Kupershmidt equations: Particularly, taking in (37), we can get the nonlinear integrable couplings of the second order Broer-Kaup-Kupershmidt equations: Let in (37); we have If setting , , and in (37), we obtain the second order super Broer-Kaup-Kupershmidt equations:
In this paper, we introduced an approach for constructing nonlinear integrable couplings of super integrable hierarchy. Zhang  once employed two kinds of explicit Lie algebra and to obtain the nonlinear integrable couplings of the GJ hierarchy and Yang hierarchy, respectively. It is easy to see that Lie algebra given in  is isomorphic to the Lie algebra span in . So we can obtain nonlinear integrable couplings of super GJ and Yang hierarchy easily. The method in this paper can be applied to other super integrable systems for constructing their integrable couplings.
This work was supported by the National Natural Science Foundation of China (no. 61072147), the Natural Science Foundation of Henan Province (no. 132300410202), the Science and Technology Key Research Foundation of the Education Department of Henan Province (no. 12A110017) and the Youth Research Foundation of Shangqiu Normal University (no. 2011QN12).
- B. A. Kupershmidt, “Odd and even Poisson brackets in dynamical systems,” Letters in Mathematical Physics, vol. 9, no. 4, pp. 323–330, 1985.
- Y. S. Li and L. N. Zhang, “A note on the super AKNS equations,” Journal of Physics, vol. 21, no. 7, pp. 1549–1552, 1988.
- M.-H. Tu and J.-C. Shaw, “Hamiltonian structures of generalized Manin-Radul super-KdV and constrained super KP hierarchies,” Journal of Mathematical Physics, vol. 40, no. 6, pp. 3021–3034, 1999.
- Q. P. Liu and X.-B. Hu, “Bilinearization of supersymmetric Korteweg-de Vries equation revisited,” Journal of Physics, vol. 38, no. 28, pp. 6371–6378, 2005.
- H.-H. Dong and X. Wang, “Lie algebras and Lie super algebra for the integrable couplings of NLS-MKdV hierarchy,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 12, pp. 4071–4077, 2009.
- X.-Z. Wang and X.-K. Liu, “Two types of Lie super-algebra for the super-integrable Tu-hierarchy and its super-Hamiltonian structure,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 8, pp. 2044–2049, 2010.
- H.-X. Yang and Y.-P. Sun, “Hamiltonian and super-Hamiltonian extensions related to Broer-Kaup-Kupershmidt system,” International Journal of Theoretical Physics, vol. 49, no. 2, pp. 349–364, 2010.
- Q. P. Liu, X.-B. Hu, and M.-X. Zhang, “Supersymmetric modified Korteweg-de Vries equation: bilinear approach,” Nonlinearity, vol. 18, no. 4, pp. 1597–1603, 2005.
- H. Aratyn, E. Nissimov, and S. Pacheva, “Supersymmetric Kadomtsev-Petviashvili hierarchy: “ghost” symmetry structure, reductions, and Darboux-Bäcklund solutions,” Journal of Mathematical Physics, vol. 40, no. 6, pp. 2922–2932, 1999.
- C. Morosi and L. Pizzocchero, “On the bi-Hamiltonian structure of the supersymmetric KdV hierarchies. A Lie superalgebraic approach,” Communications in Mathematical Physics, vol. 158, no. 2, pp. 267–288, 1993.
- X.-B. Hu, “An approach to generate superextensions of integrable systems,” Journal of Physics, vol. 30, no. 2, pp. 619–632, 1997.
- W.-X. Ma, J.-S. He, and Z.-Y. Qin, “A supertrace identity and its applications to superintegrable systems,” Journal of Mathematical Physics, vol. 49, no. 3, Article ID 033511, 2008.
- J. He, J. Yu, Y. Cheng, and R. Zhou, “Binary nonlinearization of the super AKNS system,” Modern Physics Letters B, vol. 22, no. 4, pp. 275–288, 2008.
- Q. P. Liu, Z. Popowicz, and K. Tian, “Supersymmetric reciprocal transformation and its applications,” Journal of Mathematical Physics, vol. 51, no. 9, Article ID 093511, 2010.
- W.-X. Ma, “Enlarging spectral problems to construct integrable couplings of soliton equations,” Physics Letters A, vol. 316, no. 1-2, pp. 72–76, 2003.
- W.-X. Ma, X.-X. Xu, and Y. Zhang, “Semidirect sums of Lie algebras and discrete integrable couplings,” Journal of Mathematical Physics, vol. 47, no. 5, Article ID 053501, 2006.
- W.-X. Ma, X.-X. Xu, and Y. Zhang, “Semi-direct sums of Lie algebras and continuous integrable couplings,” Physics Letters A, vol. 351, no. 3, pp. 125–130, 2006.
- F. Guo and Y. Zhang, “The quadratic-form identity for constructing the Hamiltonian structure of integrable systems,” Journal of Physics, vol. 38, no. 40, pp. 8537–8548, 2005.
- W.-X. Ma and M. Chen, “Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras,” Journal of Physics, vol. 39, no. 34, pp. 10787–10801, 2006.
- T.-C. Xia and F.-C. You, “Generalized multi-component TC hierarchy and its multi-component integrable coupling system,” Communications in Theoretical Physics, vol. 44, no. 5, pp. 793–798, 2005.
- T.-C. Xia and F.-C. You, “Multi-component Dirac equations hierarchy and its multicomponent integrable couplings system,” Chinese Physics, vol. 16, pp. 605–610, 2007.
- Y. Zhang and E. Fan, “Coupling integrable couplings and bi-Hamiltonian structure associated with the Boiti-Pempinelli-Tu hierarchy,” Journal of Mathematical Physics, vol. 51, no. 8, Article ID 083506, 2010.
- Y. Zhang and H. Tam, “Four Lie algebras associated with R6 and their applications,” Journal of Mathematical Physics, vol. 51, no. 9, Article ID 093514, 2010.
- Y. Zhang and H. Zhang, “A direct method for integrable couplings of TD hierarchy,” Journal of Mathematical Physics, vol. 43, no. 1, pp. 466–472, 2002.
- W.-X. Ma, “Nonlinear continuous integrable Hamiltonian couplings,” Applied Mathematics and Computation, vol. 217, no. 17, pp. 7238–7244, 2011.
- W.-X. Ma and Z.-N. Zhu, “Constructing nonlinear discrete integrable Hamiltonian couplings,” Computers & Mathematics with Applications, vol. 60, no. 9, pp. 2601–2608, 2010.
- Y.-F. Zhang, “Lie algebras for constructing nonlinear integrable couplings,” Communications in Theoretical Physics, vol. 56, no. 5, pp. 805–812, 2011.
Copyright © 2013 Sixing Tao and Tiecheng Xia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.