Super Jaulent-Miodek hierarchy and its super Hamiltonian structure, conservation laws and its self-consistent sources

doi:10.1007/s11464-014-0419-x

Front. Math. China

2014, Vol. 9

Issue (6) : 1367-1379 https://doi.org/10.1007/s11464-014-0419-x

RESEARCH ARTICLE

Super Jaulent-Miodek hierarchy and its super Hamiltonian structure, conservation laws and its self-consistent sources

Hui WANG^1,^2,^*(

),Tiecheng XIA²

1. College of Art and Sciences, Shanghai Maritime University, Shanghai 201306, China
2. Department of Mathematics, Shanghai University, Shanghai 200444, China

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Abstract

A super Jaulent-Miodek hierarchy and its super Hamiltonian structures are constructed by means of a kind of Lie super algebras and super trace identity. Moreover, the self-consistent sources of the super Jaulent-Miodek hierarchy is presented based on the theory of self-consistent sources. Furthermore, the infinite conservation laws of the super Jaulent-Miodek hierarchy are also obtained. It is worth noting that as even variables are boson variables, odd variables are fermi variables in the spectral problem, the commutator is different from the ordinary one.

Keywords Super Jaulent-Miodek hierarchy self-consistent sources fermi variables conservation law

Corresponding Author(s): Hui WANG

Issue Date: 29 October 2014

Cite this article:

Hui WANG,Tiecheng XIA. Super Jaulent-Miodek hierarchy and its super Hamiltonian structure, conservation laws and its self-consistent sources[J]. Front. Math. China, 2014, 9(6): 1367-1379.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-014-0419-x
https://academic.hep.com.cn/fmc/EN/Y2014/V9/I6/1367

1	Chowdhury A R, Roy S. On the Backlund transformation and Hamiltonian properties of superevaluation equations. J Math Phys, 1986, 27: 2464 https://doi.org/10.1063/1.527309
2	Dong H H, Wang X Z. Lie algebras and Lie super algebra for the integrable couplings of NLS-MKdV hierarchy. Commun Nonlinear Sci Numer Simul, 2009, 14: 4071-4077 https://doi.org/10.1016/j.cnsns.2009.03.010
3	Ge J Y, Xia T C. A New integrable couplings of Classical-Boussinesq hierarchy with self-consistent sources. Commun Theor Phys, 2010, 54: 1-6 https://doi.org/10.1088/0253-6102/54/1/01
4	Guo F K. A hierarchy of integrable Hamiltonian equations. Math Appl Sin, 2000, 23(2): 181-187
5	He J S, Yu J, Cheng Y. Binary nonlinearization of the super AKNS system. Modern Phys Lett B, 2008, 22: 275-288 https://doi.org/10.1142/S0217984908014778
6	Hu X B. An approach to generate superextensions of integrable systems. J Phys A: Math Gen, 1997, 30: 619-632 https://doi.org/10.1088/0305-4470/30/2/023
7	Jaulent M, Miodek K. Nonlinear evolution equations associated with enegry-dependent Schr?dinger potentials. Lett Math Phys, 1976, 1: 243 https://doi.org/10.1007/BF00417611
8	Li Z. Super-Burgers soliton hierarchy and it super-Hamiltonian structure. Modern Phys Lett B, 2009, 23: 2907-2914 https://doi.org/10.1142/S0217984909020990
9	Ma W X. Integrable couplings of soliton equations by perturbations I. A general theory and application to the KdV hierarchy. Methods Appl Anal, 2000, 7: 21-56
10	Ma W X. Variational identities and applications to Hamiltonian structures of soliton equations. Nonlinear Anal, 2009, 71: 1716-1726 https://doi.org/10.1016/j.na.2009.02.045
11	Ma W X, Fuchssteiner B. Integrable theory of the perturbation equations. Chaos Solitons Fractals, 1996, 7: 1227-1250 https://doi.org/10.1016/0960-0779(95)00104-2
12	Ma W X, Fuchssteiner B. The bi-Hamiltonian structures of the perturbation equ<?Pub Caret?>ations of KdV hierarchy. Phys Lett A, 1996, 213: 49-55 https://doi.org/10.1016/0375-9601(96)00112-0
13	Ma W X, He J S, Qin Z Y. A supertrace identity and its applications to superintegrable systems. J Math Phys, 2008, 49: 033511 https://doi.org/10.1063/1.2897036
14	Mel’nikov V K. Integration of the nonlinear Schr?dinger equation with a source. Inverse Problems, 1992, 8: 133 https://doi.org/10.1088/0266-5611/8/1/009
15	Miua R M, Gardner C S, Gardner M D. The KdV equation has infinitely many integrals of motion conservation laws and constants of motion. J Math Phys, 1968, 9: 1204-1209
16	Shchesnovich V S, Doktorov E V. Modified Manakov system with self-consistent source. Phys Lett A, 1996, 213: 23-31 https://doi.org/10.1016/0375-9601(96)00090-4
17	Tao S X, Xia T C. Lie algebra and Lie super algebra for integrable couplings of C-KdV hierarchy. Chin Phys Lett, 2010, 27: 040202 https://doi.org/10.1088/0256-307X/27/4/040202
18	Tu G Z. On Liouville integrability of zero-curvature equations and the Yang hierarchy. J Phys A: Math Gen, 1989, 22: 2375-2392 https://doi.org/10.1088/0305-4470/22/13/031
19	Tu G Z. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J Math Phys, 1989, 30: 330-338 https://doi.org/10.1063/1.528449
20	Tu G Z. A trace identity and its application to the theory of discrete integrable systems. J Phys A, 1990, 23: 3903-3922 https://doi.org/10.1088/0305-4470/23/17/020
21	Tu G Z. An extension of a theorem on gradients conserved densities of integrable system. Northeastern Math J, 1990, 6: 26
22	Wadati M, Sanuki H, Konno K. Relationships among inverse method, B?ckland transformation and an infinite number of conservation laws. Progr Theoret Phys, 1975, 53: 419-436 https://doi.org/10.1143/PTP.53.419
23	Wang H, Xia T C. Conservation laws for a super G-J hierarchy with self-consistent sources. Commun Nonlinear Sci Numer Simul, 2012, 17: 566-572 https://doi.org/10.1016/j.cnsns.2011.06.007
24	Wang H, Xia T C. Conservation laws and self-consistent sources for a super KN hierarchy. Appl Math Comput, 2013, 219: 5458-5464 https://doi.org/10.1016/j.amc.2012.11.042
25	Wang H, Xia T C. The fractional supertrace identity and its application to the super Ablowitz-Kaup-Newell-Segur hierarchy. J Math Phys, 2013, 54: 043505 https://doi.org/10.1063/1.4799914
26	Wang H, Xia T C. The fractional supertrace identity and its application to the super Jaulent-Miodek hierarchy. Commun Nonlinear Sci Numer Simul, 2013, 18: 2859-2867 https://doi.org/10.1016/j.cnsns.2013.02.005
27	Wang X Z, Dong H H. A Lie superalgebra and corresponding hierarchy of evolution equations. Modern Phys Lett B, 2009, 23: 3387-3396 https://doi.org/10.1142/S0217984909021429
28	Xia T C. Two new integrable couplings of the soliton hierarchies with self-consistent sources. Chin Phys B, 2010, 19: 100303 https://doi.org/10.1088/1674-1056/19/10/100303
29	Yang H X, Sun Y P. Hamiltonian and super-Hamiltonian extensions related to Broer-Kaup-Kupershmidt System. Int J Theor Phys, 2010, 49: 349-364 https://doi.org/10.1007/s10773-009-0208-6
30	Yu J, He J S, Ma W X, Cheng Y. The Bargmann symmetry constraint and binary nonlinearization of the super Dirac systems. Chin Ann Math Ser B, 2010, 31: 361-372 https://doi.org/10.1007/s11401-009-0032-6
31	Zeng Y B. New factorization of the Kaup-Newell hierarchy. Phys D, 1994, 73: 171-188 https://doi.org/10.1016/0167-2789(94)90155-4
32	Zeng Y B. The integrable system associated with higher-order constraint. Acta Math Sinica (Chin Ser), 1995, 38: 642-652 (in Chinese)
33	Zhang Y F. Lie algebras for constructing nonlinear integrable couplings. Commun Theor Phys, 2011, 56(5): 805-812 https://doi.org/10.1088/0253-6102/56/5/03
34	Zhang Y F, Hon Y C. Some evolution hierarchies derived from self-dual Yang-Mills equations. Commun Theor Phys, 2011, 56(5): 856-872 https://doi.org/10.1088/0253-6102/56/5/12
35	Zhang Y F, Tam H, Feng B. A generalized Zakharov-Shabat equation with finite-band solutions and a soliton-equation hierarchy with an arbitrary parameter. Chaos Solitons Fractals, 2011, 44(11): 968-976 https://doi.org/10.1016/j.chaos.2011.07.014
36	Zhou R G. Lax representation, r-matrix method, and separation of variables for the Neumann-type restricted flow. J Math Phys, 1998, 39: 2848-2858 https://doi.org/10.1063/1.532424

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