A novel hierarchy of differential–integral equations and their generalized bi-Hamiltonian structures

doi:10.1088/1674-1056/23/6/060201

Abstract With the aid of the zero-curvature equation, a novel integrable hierarchy of nonlinear evolution equations associated with a 3×3 matrix spectral problem is proposed. By using the trace identity, the bi-Hamiltonian structures of the hierarchy are established with two skew-symmetric operators. Based on two linear spectral problems, we obtain the infinite many conservation laws of the first member in the hierarchy.

Keywords: spectral problem nonlinear evolution equations bi-Hamiltonian structure conservation laws

Received: 11 November 2013
Revised: 10 December 2013
Published: 15 June 2014

PACS:	02.30.Jr	(Partial differential equations)
	02.30.Ik	(Integrable systems)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11331008 and 11171312).

Corresponding Authors: Zhai Yun-Yun
E-mail: zhai1226@163.com

Cite this article:

Zhai Yun-Yun, Geng Xian-Guo, He Guo-Liang A novel hierarchy of differential–integral equations and their generalized bi-Hamiltonian structures 2014 Chin. Phys. B 23 060201

[1]	Ablowitz M J and Clarkson P A 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge: Cambridge University Press)
[2]	Ablowitz M J, Kaup D J, Newell A C and Segur H 1974 Studies Appl. Math. 53 249
[3]	Dubrovin B A 1981 Russian Math. Surveys 36 11
[4]	Gesztesy F and Holden H 2003 Soliton Equations and Their Algebro-Geometric Solutions (Cambridge: Cambridge University Press)
[5]	Belokolos E D, Bobenko A I, Enolskii V Z, Its A R and Matveev V B 1994 Algebro-Geometric Approach to Nonlinear Integrable Equations (Berlin: Springer)
[6]	Geng X G, Wu L H and He G L 2013 J. Nonlinear Sci. 23 527
[7]	Matveev V B and Salle M A 1991 Darboux Transformations and Solitons (Berlin: Springer)
[8]	Levi D and Ragnisco O 1988 Inverse Problems 4 815
[9]	Geng X G and Lü Y Y 2012 Nonlinear Dyn. 69 1621
[10]	Hu H C, Lou S Y and Liu Q P 2003 Chin. Phys. Lett. 20 1413
[11]	Zhu J Y and Geng X G 2013 Chin. Phys. Lett. 30 080204
[12]	Su T, Dai H H and Geng X G 2013 Chin. Phys. Lett. 30 060201
[13]	Qu C Z, Yao R X and Li Z B 2004 Chin. Phys. Lett. 21 2077
[14]	Lenells J 2012 Physica D 241 857
[15]	Geng X G and Xue B 2011 Adv. Math. 226 827
[16]	Lundmark H and Szmigielski J 2003 Inverse Problems 19 1241
[17]	He G L and Geng X G 2012 Chin. Phys. B 21 070205
[18]	He G L, Zhai Y Y and Geng X G 2013 J. Math. Phys. 54 083509
[19]	Huang D J, Zhou S G and Yang Q M 2011 Chin. Phys. B 20 070202
[20]	Yan Z Y and Zhang H Q 2001 Acta Phys. Sin. 50 1232 (in Chinese)
[21]	Tu G Z 1990 J. Phys. A: Math. Gen. 23 3903

[1]	Two integrable generalizations of WKI and FL equations: Positive and negative flows, and conservation laws Xian-Guo Geng(耿献国), Fei-Ying Guo(郭飞英), Yun-Yun Zhai(翟云云). Chin. Phys. B, 2020, 29(5): 050201.
[2]	An extension of integrable equations related to AKNS and WKI spectral problems and their reductions Xian-Guo Geng(耿献国), Yun-Yun Zhai(翟云云). Chin. Phys. B, 2018, 27(4): 040201.
[3]	A local energy-preserving scheme for Zakharov system Qi Hong(洪旗), Jia-ling Wang(汪佳玲), Yu-Shun Wang(王雨顺). Chin. Phys. B, 2018, 27(2): 020202.
[4]	A new six-component super soliton hierarchy and its self-consistent sources and conservation laws Han-yu Wei(魏含玉) and Tie-cheng Xia(夏铁成). Chin. Phys. B, 2016, 25(1): 010201.
[5]	Multi-symplectic variational integrators for nonlinear Schrödinger equations with variable coefficients Cui-Cui Liao(廖翠萃), Jin-Chao Cui(崔金超), Jiu-Zhen Liang(梁久祯), Xiao-Hua Ding(丁效华). Chin. Phys. B, 2016, 25(1): 010205.
[6]	The Wronskian technique for nonlinear evolution equations Jian-Jun Cheng(成建军) and Hong-Qing Zhang(张鸿庆). Chin. Phys. B, 2016, 25(1): 010506.
[7]	Symmetries and conservation laws of one Blaszak–Marciniak four-field lattice equation Wang Xin, Chen Yong, Dong Zhong-Zhou . Chin. Phys. B, 2014, 23(1): 010201.
[8]	Multi-symplectic scheme for the coupled Schrödinger–Boussinesq equations Huang Lang-Yang, Jiao Yan-Dong, Liang De-Min. Chin. Phys. B, 2013, 22(7): 070201.
[9]	Integrability of extended (2+1)-dimensional shallow water wave equation with Bell polynomials Wang Yun-Hu, Chen Yong. Chin. Phys. B, 2013, 22(5): 050509.
[10]	Noether symmetry and conserved quantities of the analytical dynamics of a Cosserat thin elastic rod Wang Peng, Xue Yun, Liu Yu-Lu. Chin. Phys. B, 2013, 22(10): 104503.
[11]	Lattice soliton equation hierarchy and associated properties Zheng Xin-Qing, Liu Jin-Yuan. Chin. Phys. B, 2012, 21(9): 090202.
[12]	Mei symmetry and conserved quantities in Kirchhoff thin elastic rod statics Wang Peng, Xue Yun, Liu Yu-Lu. Chin. Phys. B, 2012, 21(7): 070203.
[13]	An extension of the modified Sawada–Kotera equation and conservation laws He Guo-Liang, Geng Xian-Guo. Chin. Phys. B, 2012, 21(7): 070205.
[14]	Conservation laws for variable coefficient nonlinear wave equations with power nonlinearities Huang Ding-Jiang, Zhou Shui-Geng, Yang Qin-Min. Chin. Phys. B, 2011, 20(7): 070202.
[15]	A note on teleparallel Killing vector fields in Bianchi type VIII and IX space–times in teleparallel theory of gravitation Ghulam Shabbir, Amjad Ali, Suhail Khan. Chin. Phys. B, 2011, 20(7): 070401.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Shared

Discussed

Metrics
Related Articles