Nonlinear Super Integrable Couplings of Super Classical-Boussinesq Hierarchy

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 438741, 7 pages http://dx.doi.org/10.1155/2014/438741 Research Article Nonlinear Super Integrable Couplings of Super Classical-Boussinesq Hierarchy Xiuzhi Xing,1 Jingzhu Wu,1 and Xianguo Geng2 1 2 Department of Mathematics, Zhoukou Normal University, Zhoukou 466000, China Department of Mathematics, Zhengzhou University, Zhengzhou 450052, China Correspondence should be addressed to Xiuzhi Xing; Received 15 December 2013; Accepted 26 February 2014; Published 7 April 2014 Academic Editor: Chein-Shan Liu Copyright © 2014 Xiuzhi Xing et al. 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. Nonlinear integrable couplings of super classical-Boussinesq hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then, its super Hamiltonian structures were established by using super trace identity. As its reduction, nonlinear integrable couplings of the classical integrable hierarchy were obtained. 1. Introduction 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 C-KdV hierarchy, and super classicalBoussinesq hierarchy have been studied [1–4]. There are some interesting results on the super integrable systems, such as Darboux transformation [5], super Hamiltonian structures [6, 7], binary nonlinearization [8], and reciprocal transformation [9]. The research of integrable couplings of the well-known integrable hierarchy has received considerable attentions [10–15]. 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 [16], and creating semidirect sums of Lie algebra. Recently, Ma and Zhu [17, 18] 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 which 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 sl(2, 1) as an example to illustrate the approach for extending Lie super algebra. Based on the enlarged Lie super algebra sl(4, 1), we work out nonlinear super integrable Hamiltonian couplings of the super classical-Boussinesq hierarchy. Finally, we will reduce the nonlinear super classical-Boussinesq integrable Hamiltonian couplings to some special cases. 2. Enlargement of Lie Superalgebra Consider the Lie superalgebra sl(2, 1). Its basis is 1 0 0 𝐸1 = (0 −1 0) , 0 0 0 0 1 0 𝐸2 = (0 0 0) , 0 0 0 0 0 0 𝐸3 = (1 0 0) , 0 0 0 0 0 1 𝐸4 = (0 0 0) , 0 −1 0 0 0 0 𝐸5 = (0 0 1) , 1 0 0 (1) 2 Journal of Applied Mathematics [𝑒8 , 𝑒8 ] = 2𝑒3 − 2𝑒6 , where 𝐸1 , 𝐸2 , 𝐸3 are even elements and 𝐸4 , 𝐸5 are odd ones. Their nonzero (anti)commutation relations are [𝐸1 , 𝐸2 ] = 2𝐸2 , [𝐸2 , 𝐸3 ] = 𝐸1 , [𝑒1 , 𝑒4 ] = [𝑒2 , 𝑒5 ] = [𝑒2 , 𝑒7 ] = [𝑒3 , 𝑒6 ] = [𝑒3 , 𝑒8 ] = 0, [𝐸1 , 𝐸3 ] = −2𝐸3 , [𝐸1 , 𝐸4 ] = [𝐸2 , 𝐸5 ] = 𝐸4 , [𝐸1 , 𝐸5 ] = [𝐸4 , 𝐸3 ] = −𝐸5 , [𝐸4 , 𝐸4 ] = −2𝐸2 , [𝐸4 , 𝐸5 ] = 𝐸1 , [𝑒4 , 𝑒7 ] = [𝑒4 , 𝑒8 ] = [𝑒5 , 𝑒7 ] = [𝑒5 , 𝑒8 ] = [𝑒6 , 𝑒7 ] = [𝑒6 , 𝑒8 ] = 0. (2) [𝐸5 , 𝐸5 ] = 2𝐸3 . Let us enlarge the Lie superalgebra sl(2, 1) to the Lie superalgebra sl(4, 1) with a basis 1 0 0 0 0 0 −1 0 0 0 𝑒1 = (0 0 1 0 0) , 0 0 0 −1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 𝑒2 = (0 0 0 1 0) , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 𝑒3 = (0 0 0 0 0) , 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 −1 0 𝑒4 = (0 0 1 0 0) , 0 0 0 −1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 𝑒5 = (0 0 0 1 0) , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 𝑒6 = (0 0 0 0 0) , 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 𝑒7 = (0 0 0 0 0) , 0 0 0 0 0 0 −1 0 1 0 0 0 0 0 0 0 0 0 0 1 𝑒8 = (0 0 0 0 0) , 0 0 0 0 0 1 0 −1 0 0 (3) (4) Define a loop superalgebra corresponding to the Lie superalgebra sl(4, 1), denoted by ̃sl (4, 1) = sl (4, 1) ⊗ [𝜆, 𝜆−1 ] = {𝑒𝑖 𝜆𝑚 , 𝑒𝑖 ∈ sl (4, 1) , 𝑖 = 1, 2, . . . , 8; 𝑚 = 0, ±1, ±2, . . .} . (5) The corresponding (anti)commutative relations are given as [𝑒𝑖 𝜆𝑚 , 𝑒𝑗 𝜆𝑛 ] = [𝑒𝑖 , 𝑒𝑗 ] 𝜆𝑚+𝑛 , ∀𝑒𝑖 , 𝑒𝑗 ∈ sl (4, 1) . (6) 3. Nonlinear Super Integrable Couplings of the Super Classical-Boussinesq Hierarchy Let us start from an enlarged spectral problem associated with sl(4, 1) 𝜑𝑥 = 𝑈𝜑, (7) where 𝑈 1 = −𝑒1 (1) − 𝑒1 (0) + 𝑟𝑒2 (0) − 𝑒3 (0) − 𝑢1 𝑒4 (0) 𝑞 + 𝑢2 𝑒5 (0) + 𝛼𝑒7 (0) + 𝛽𝑒8 (0) where 𝑒1 , 𝑒2 , 𝑒3 , 𝑒4 , 𝑒5 , 𝑒6 are even and 𝑒7 , 𝑒8 are odd. The generators of the Lie superalgebra sl(4, 1), 𝑒𝑖 , 0 ≤ 𝑖 ≤ 8, satisfy the following (anti)commutation relations: [𝑒1 , 𝑒2 ] = 2𝑒2 , [𝑒1 , 𝑒5 ] = 2𝑒5 , [𝑒1 , 𝑒6 ] = −2𝑒6 , [𝑒1 , 𝑒8 ] = −𝑒8 , [𝑒2 , 𝑒4 ] = −2𝑒5 , [𝑒1 , 𝑒3 ] = −2𝑒3 , [𝑒1 , 𝑒7 ] = 𝑒7 , [𝑒2 , 𝑒3 ] = 𝑒1 , [𝑒2 , 𝑒6 ] = 𝑒4 , [𝑒2 , 𝑒8 ] = 𝑒7 , [𝑒3 , 𝑒4 ] = 2𝑒6 , [𝑒3 , 𝑒5 ] = −𝑒4 , [𝑒3 , 𝑒7 ] = 𝑒8 , [𝑒4 , 𝑒5 ] = 2𝑒5 , [𝑒4 , 𝑒6 ] = −2𝑒6 , [𝑒5 , 𝑒6 ] = 𝑒4 , [𝑒7 , 𝑒8 ] = 𝑒1 − 𝑒4 , [𝑒7 , 𝑒7 ] = 2𝑒5 − 2𝑒2 , 1 −𝜆 − 𝑞 4 𝑟 ( ( ( =( ( ( ( −1 1 𝜆+ 𝑞 4 0 0 0 0 ( 𝛽 −𝛼 −𝑢1 𝑢2 𝛼 𝛽) ) ) ), 1 𝑟 + 𝑢2 0) −𝜆 − 𝑞 − 𝑢1 ) 4 ) 1 −1 𝜆 + 𝑞 + 𝑢1 0 4 −𝛽 𝛼 0) (8) 0 𝑢1 where 𝑟, 𝑠, 𝑢1 , and 𝑢2 are even potentials but 𝛼 and 𝛽 are odd ones. In order to obtain super integrable couplings of super integrable hierarchy, we first solve the adjoint representation (8) 𝑉𝑥 = [𝑈, 𝑉] (9) Journal of Applied Mathematics 3 with From these equations, we can successively deduce 𝐴 0 = −1, 𝑉 = 𝐴𝑒1 (0) + 𝐵𝑒2 (0) + 𝐶𝑒3 (0) + 𝐸𝑒4 (0) + 𝐹𝑒5 (0) 𝐸0 = 𝜀 = const, + 𝐺𝑒6 (0) + 𝜌𝑒7 (0) + 𝛿𝑒8 (0) 𝐵1 = 𝑟, 𝐴 𝐵 𝐸 𝐹 𝜌 𝐶 −𝐴 𝐺 −𝐸 𝛿 = ( 0 0 𝐴 + 𝐸 𝐵 + 𝐹 0) , 0 0 𝐶 + 𝐺 −𝐴 − 𝐸 0 𝜌 0) ( 𝛿 −𝜌 −𝛿 (10) 𝐴 = ∑ 𝐴 𝑚 𝜆−𝑚 , 𝐵 = ∑ 𝐵𝑚 𝜆−𝑚 , −𝑚 −𝑚 𝐶 = ∑ 𝐶𝑚 𝜆 , 𝑚≥0 𝐹1 = 𝑢2 − 𝑟𝜀 − 𝑢2 𝜀, 𝐸1 = 0, 𝐺1 = 𝜀, 𝛿1 = 𝛽, 𝐺 = ∑ 𝐺𝑚 𝜆 , 𝑚≥0 −𝑚 𝜌 = ∑ 𝜌𝑚 𝜆 − 𝑢1 𝑟 − 𝑢1 𝑢2 + 𝜀𝑢1 𝑢2 + 𝜀𝑟𝑢1 , 1 𝐺2 = 𝑢1 − 𝑞𝜀 − 𝑢1 𝜀, 4 , 1 𝜌2 = −𝛼𝑥 − 𝑞𝛼, 4 1 𝛿2 = 𝛽𝑥 − 𝑞𝛽, 4 (11) , 1 1 1 𝐴 3 = 𝑟𝑥 + 𝑞𝑟 − 𝑞𝛼𝛽 − 𝛽𝛼𝑥 + 𝛼𝛽𝑥 , 2 4 2 𝑚≥0 𝜎 = ∑ 𝜎𝑚 𝜆−𝑚 1 1 1 1 1 𝐵3 = 𝑟𝑥𝑥 + 𝑞𝑥 𝑟 + 𝑞𝑟𝑥 + 𝑟2 − 𝑟𝛼𝛽 + 𝑞2 𝑟 + 𝛼𝛼𝑥 , 4 8 4 2 16 𝑚≥0 1 1 1 𝐶3 = 𝑞𝑥 − 𝑟 + 𝛼𝛽 − 𝑞2 − 𝛽𝛽𝑥 , 8 2 16 into the above equation gives the following recursive formulas: 1 1 𝐸3 = (1 − 𝜀) [𝑢1 𝑢2 + 𝑢1 𝑟 + 𝑞𝑢2 + 𝑢2𝑥 ] 4 4 𝐴 𝑚,𝑥 = 𝐵𝑚 + 𝑟𝐶𝑚 − 𝛽𝜌𝑚 + 𝛼𝛿𝑚 , 1 1 1 − 𝛼𝛽𝑥 + 𝛽𝛼𝑥 + 𝑞𝛼𝛽 − 𝑞𝑟 − 𝜀𝑟𝑥 , 2 4 4 1 𝐵𝑚,𝑥 = −2𝑟𝐴 𝑚 − 2𝐵𝑚+1 − 𝑞𝐵𝑚 − 2𝛼𝜌𝑚 , 2 1 1 1 1 𝐹3 = (1 − 𝜀) [ 𝑢2𝑥𝑥 + 𝑞𝑥 𝑢2 + 𝑞𝑢2𝑥 + 𝑢1𝑥 𝑟 + 𝑢1 𝑟𝑥 4 8 4 2 (...truncated)