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
Accepted manuscript online:

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 (翟云云), Geng Xian-Guo (耿献国), He Guo-Liang (何国亮) 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

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

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