Third-order nonlinear differential operators preserving invariant subspaces of maximal dimension

doi:10.1088/1674-1056/23/11/110202

Abstract In this paper, third-order nonlinear differential operators are studied. It is shown that they are quadratic forms when they preserve invariant subspaces of maximal dimension. A complete description of third-order quadratic operators with constant coefficients is obtained. One example is given to derive special solutions for evolution equations with third-order quadratic operators.

Keywords: nonlinear evolution equation quadratic operator invariant subspace method blow-up solution

Received: 26 December 2013
Revised: 14 May 2014
Accepted manuscript online:

PACS:	02.30.Jr	(Partial differential equations)
	02.20.Sv	(Lie algebras of Lie groups)
	02.30.Tb	(Operator theory)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11371293), the Civil Military Integration Research Foundation of Shaanxi Province, China (Grant No. 13JMR13), the Natural Science Foundation of Shaanxi Province, China (Grant No. 14JK1246), the Mathematical Discipline Foundation of Shaanxi Province, China (Grant No. 14SXZD015), and the Basic Research Project Foundation of Weinan City, China (Grant No. 2013JCYJ-4).

Corresponding Authors: Zhang Shun-Li
E-mail: zhangshunli@nwu.edu.cn

Cite this article:

Qu Gai-Zhu (屈改珠), Zhang Shun-Li (张顺利), Li Yao-Long (李尧龙) Third-order nonlinear differential operators preserving invariant subspaces of maximal dimension 2014 Chin. Phys. B 23 110202

[1]	Lie S 1881 Arch. Math. 6 328
[2]	Bluman G W and Cole J D 1969 J. Math. Mech. 18 1025
[3]	Clarkson P A and Kruskal M D 1989 J. Math. Phys. 30 2201
[4]	Fokas A S and Liu Q M 1994 Theor. Math. Phys. 99 263
[5]	Fokas A S and Liu Q M 1994 Phys. Rev. Lett. 72 3293
[6]	Zhdanov R Z 1995 J. Phys. A: Math. Gen. 28 3841
[7]	Bluman G W and Kumei S 1989 Symmetries and Differential Equations
[8]	Galaktionov V A 1994 Nonlin. Anal. TMA 23 1595
[9]	Galaktionov V A and Posashkov S A 1996 Physica D 99 217
[10]	Galaktionov V A and Posashkov S A 1998 J. Math. Phys. 39 4948
[11]	Titov S S 1988 Aerodynamics of Plane and Axis-Symmetric Flows of Liquids p. 104
[12]	Galaktionov V A 1995 Proc. Royal. Soc. Edinburgh 125 225
[13]	Svirshchevskii S R 1995 Theor. Math. Phys. 105 198
[14]	Svirshchevskii S R 1993 Modern Group Analysis MPhTI Mosco p. 75
[15]	Svirshchevskii S R 1995 Phys. Lett. A 199 344
[16]	Galaktionov V A and Svirshchevskii S R 2007 Exact Solutions Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics (London: Chapman and Hall/CRC)
[17]	Qu C Z and Zhu C R 2009 J. Phys. A: Math. Theor. 42 475201
[18]	Ma W X 2012 Sci. China A: Math. 55 1769
[19]	Zhu C R and Qu C Z 2011 J. Math. Phys. 52 403
[20]	Zhu C R 2011 Chin. Phys. B 20 010201
[21]	Shen S F, Qu C Z, Jin Y Y and Ji L N 2012 Chin. Ann. Math. 33 161
[22]	King J R 1993 Physica D 64 35
[23]	Hu X R, Lou S Y and Chen Y 2012 Phys. Rev. E 85 056607
[24]	Lou S Y, Hu X R and Chen Y 2012 J. Phys. A: Math. Theor. 45 155209
[25]	Chen Y M, Ma S H and Ma Z Y 2013 Chin. Phys. B 22 050510
[26]	Ji F Y and Yang C X 2013 Chin. Phys. B 22 100202
[27]	Xin X P, Miao Q and Chen Y 2014 Chin. Phys. B 23 010203

[1]	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.
[2]	A more general form of lump solution, lumpoff, and instanton/rogue wave solutions of a reduced (3+1)-dimensional nonlinear evolution equation Panfeng Zheng(郑攀峰), Man Jia(贾曼). Chin. Phys. B, 2018, 27(12): 120201.
[3]	Multiple exp-function method for soliton solutions ofnonlinear evolution equations Yakup Yıldırım, Emrullah Yašar. Chin. Phys. B, 2017, 26(7): 070201.
[4]	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.
[5]	A novel hierarchy of differential–integral equations and their generalized bi-Hamiltonian structures Zhai Yun-Yun (翟云云), Geng Xian-Guo (耿献国), He Guo-Liang (何国亮). Chin. Phys. B, 2014, 23(6): 060201.
[6]	Constructing infinite sequence exact solutions of nonlinear evolution equations Taogetusang(套格图桑) and Narenmandula(那仁满都拉) . Chin. Phys. B, 2011, 20(11): 110203.
[7]	Second-order nonlinear differential operators possessing invariant subspaces of submaximal dimension Zhu Chun-Rong(朱春蓉). Chin. Phys. B, 2011, 20(1): 010201.
[8]	New application to Riccati equation Taogetusang(套格图桑), Sirendaoerji(斯仁道尔吉), and Li Shu-Min(李姝敏). Chin. Phys. B, 2010, 19(8): 080303.
[9]	A new expansion method of first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term and its application to mBBM model Pan Jun-Ting(潘军廷) and Gong Lun-Xun(龚伦训). Chin. Phys. B, 2008, 17(2): 399-402.
[10]	An algorithm and its application for obtaining some kind of infinite-dimensional Hamiltonian canonical formulation Ren Wen-Xiu(任文秀) and Alatancang(阿拉坦仓). Chin. Phys. B, 2007, 16(11): 3154-3160.
[11]	New exact solitary wave solutions to generalized mKdV equation and generalized Zakharov--Kuzentsov equation Taogetusang (套格图桑), Sirendaoreji. Chin. Phys. B, 2006, 15(6): 1143-1148.
[12]	A method for constructing exact solutions and application to Benjamin Ono equation Wang Zhen (王振), Li De-Sheng (李德生), Lu Hui-Fang (鲁慧芳), Zhang Hong-Qing (张鸿庆). Chin. Phys. B, 2005, 14(11): 2158-2163.
[13]	New solitary wave solutions for nonlinear evolution equations Yao Ruo-Xia (姚若侠), Li Zhi-Bin (李志斌). Chin. Phys. B, 2002, 11(9): 864-868.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Third-order nonlinear differential operators preserving invariant subspaces of maximal dimension

Cite this article:

Online attention