Constructing super D-Kaup-Newell hierarchy and its nonlinear integrable coupling with self-consistent sources

Abstract

How to construct new super integrable equation hierarchy is an important problem. In this paper, a new Lax pair is proposed and the super D-Kaup-Newell hierarchy is generated, then a nonlinear integrable coupling of the super D-Kaup-Newell hierarchy is constructed. The super Hamiltonian structures of coupling equation hierarchy is derived with the aid of the super variational identity. Finally, the self-consistent sources of super integrable coupling hierarchy is established. It is indicated that this method is a straight- forward and efficient way to construct the super integrable equation hierarchy.

Keywords

Super D-Kaup-Newell hierarchy / nonlinear integrable coupling / super Hamiltonian structures / self-consistent sources

References

[1] Ablowitz M J, Fokas A S. Complex Variables: Introduction and Applications. Cambridge: Cambridge Univ Press, 2003

[2] Afanasieva E S, Ryazanov V I, Salimov R R. On mappings in the Orlicz-Sobolev classes on Riemannian manifolds. J Math Sci, 2012, 181(1): 1–17

[3] Fan E G, Zhang H Q. New exact solutions to a system of coupled KdV equations. Phys Lett A, 1998, 245(5): 389–392

[4] Geng X G, Ma W X. A generalized Kaup-Newell spectral problem, soliton equations and finite-dimensional integrable systems. Nuov Cim A, 1995, 108(4): 477–486

[5] Hu X B. An approach to generate superextensions of integrable systems. J Phys A, 1997, 30(2): 619–633

[6] Ma W X. A new hierarchy of Liouville integrable generalized Hamiltonian equations and its reduction. Chinese J Contemp Math, 1992, 13(1): 79–89

[7] Ma W X. Variational identities and Hamiltonian structures. AIP Conference Proceedings, 2010, 1212(1): 1–27

[8] Ma W X. Nonlinear continuous integrable Hamiltonian couplings. Appl Math Comput, 2011, 217(17): 7238–7244

[9] Ma W X. Integrable couplings and matrix loop algebras. AIP Conference Proceedings, 2013, 1562(1): 105–122

[10] Ma W X. Interaction solutions to Hirota-Satsuma-Ito equation in (2+ 1)-dimensions. Front Math China, 2019, 14(3): 619–629

[11] Ma W X. A search for lump solutions to a combined fourth-order nonlinear PDE in (2+ 1)-dimensions. J Appl Anal Comput, 2019, 9: 1319–1332

[12] Ma W X, Chen M. Hamiltonian and quasi-Hamiltonian structures associated with semi- direct sums of Lie algebras. J Phys A, 2006, 39(34): 10787–10801

[13] Ma W X, He J S, Qin Z Y. A supertrace identity and its applications to super integrable systems. J Math Phys, 2008, 49(3): 033511

[14] Ma W X, Li J, Khalique C M. A study on lump solutions to a generalized Hirota- Satsuma-Ito equation in (2+ 1)-dimensions. Complexity, 2018, 2018: 9059858

[15] Ma W X, Meng J H, Zhang M S. Nonlinear bi-integrable couplings with Hamiltonian structures. Math Comput Simulation, 2016, 127: 166–177

[16] Ma W X, Zhou Y. Lump solutions to nonlinear partial differential equations via Hirota bilinear forms. J Differential Equations, 2018, 264(4): 2633–2659

[17] McAnally M, Ma W X. An integrable generalization of the D-Kaup-Newell soliton hierarchy and its bi-Hamiltonian reduced hierarchy. Appl Math Comput, 2018, 323: 220–227

[18] Shchesnovich V S, Doktorov E V. Modified Manakov system with self-consistent source. Phys Lett A, 1996, 213(1): 23–31

[19] Tu G Z. The trace identity, a powerful tool for constructing the Hamiltonian structure of integrable systems. J Math Phys, 1989, 30(2): 330–338

[20] Tu G Z. An extension of a theorem on gradients of conserved densities of integrable systems. Northeastern Math J, 1990, 6(1): 28–32

[21] Wang H, Xia T C. Super Jaulent-Miodek hierarchy and its super Hamiltonian structure, conservation laws and its self-consistent sources. Front Math China, 2014, 9(6): 1367–1379

[22] Wei H Y, Xia T C. A new six-component super soliton hierarchy and its self-consistent sources and conservation laws. Chin Phys B, 2016, 25(1): 010201

[23] Wei H Y, Xia T C. Constructing variable coefficient nonlinear integrable coupling super AKNS hierarchy and its self-consistent sources. Math Methods Appl Sci, 2018, 41(16): 6883–6894

[24] Yu F J. Nonautonomous rogue waves and ‘catch’ dynamics for the combined Hirota- LPD equation with variable coefficients. Commun Nonlinear Sci Numer Simul, 2016, 34: 142–153

[25] Zhang J F. Multiple soliton solutions of the dispersive long-wave equations. Chin Phys Lett, 1999, 16(1): 4–5

[26] Zhang Y F, Wu L X, Rui W J. A corresponding Lie algebra of a reductive homogeneous group and its applications. Commun Theor Phys (Beijing), 2015, 63(5): 535–548

RIGHTS & PERMISSIONS

2019 Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature