|
|
Third-order nonlinear differential operators preserving invariant subspaces of maximal dimension |
Qu Gai-Zhu (屈改珠)a b, Zhang Shun-Li (张顺利)a, Li Yao-Long (李尧龙)b |
a Center for Nonlinear Studies, Department of Mathematics, Northwest University, Xi'an 710069, China; b School of Mathematics and Information Science, Weinan Normal University, Weinan 714000, China |
|
|
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.
|
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
|
No Suggested Reading articles found! |
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
Cited |
|
|
|
|
Altmetric
|
blogs
Facebook pages
Wikipedia page
Google+ users
|
Online attention
Altmetric calculates a score based on the online attention an article receives. Each coloured thread in the circle represents a different type of online attention. The number in the centre is the Altmetric score. Social media and mainstream news media are the main sources that calculate the score. Reference managers such as Mendeley are also tracked but do not contribute to the score. Older articles often score higher because they have had more time to get noticed. To account for this, Altmetric has included the context data for other articles of a similar age.
View more on Altmetrics
|
|
|