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

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.

Key words： Super Jaulent-Miodek hierarchy; self-consistent sources; fermi variables; conservation law

References

1 Chowdhury A R, Roy S. On the Backlund transformation and Hamiltonian properties of superevaluation equations. J Math Phys, 1986, 27: 2464

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

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

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

6 Hu X B. An approach to generate superextensions of integrable systems. J Phys A: Math Gen, 1997, 30: 619-632

7 Jaulent M, Miodek K. Nonlinear evolution equations associated with enegry-dependent Schrödinger potentials. Lett Math Phys, 1976, 1: 243

8 Li Z. Super-Burgers soliton hierarchy and it super-Hamiltonian structure. Modern Phys Lett B, 2009, 23: 2907-2914

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

11 Ma W X, Fuchssteiner B. Integrable theory of the perturbation equations. Chaos Solitons Fractals, 1996, 7: 1227-1250

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

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

14 Mel’nikov V K. Integration of the nonlinear Schrödinger equation with a source. Inverse Problems, 1992, 8: 133

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

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

18 Tu G Z. On Liouville integrability of zero-curvature equations and the Yang hierarchy. J Phys A: Math Gen, 1989, 22: 2375-2392

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

20 Tu G Z. A trace identity and its application to the theory of discrete integrable systems. J Phys A, 1990, 23: 3903-3922

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, Bäckland transformation and an infinite number of conservation laws. Progr Theoret Phys, 1975, 53: 419-436

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

24 Wang H, Xia T C. Conservation laws and self-consistent sources for a super KN hierarchy. Appl Math Comput, 2013, 219: 5458-5464

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

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

27 Wang X Z, Dong H H. A Lie superalgebra and corresponding hierarchy of evolution equations. Modern Phys Lett B, 2009, 23: 3387-3396

28 Xia T C. Two new integrable couplings of the soliton hierarchies with self-consistent sources. Chin Phys B, 2010, 19: 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

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

31 Zeng Y B. New factorization of the Kaup-Newell hierarchy. Phys D, 1994, 73: 171-188

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

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

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

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