|
|
|
Second-order nonlinear differential operators possessing invariant subspaces of submaximal dimension |
| Zhu Chun-Rong(朱春蓉)† |
| College of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, China |
|
|
|
|
Abstract The invariant subspace method is used to construct the explicit solution of a nonlinear evolution equation. The second-order nonlinear differential operators that possess invariant subspaces of submaximal dimension are described. There are second-order nonlinear differential operators, including cubic operators and quadratic operators, which preserve an invariant subspace of submaximal dimension. A full description of the second-order cubic operators with constant coefficients admitting a four-dimensional invariant subspace is given. It is shown that the maximal dimension of invariant subspaces preserved by a second-order cubic operator is four. Several examples are given for the construction of the exact solutions to nonlinear evolution equations with cubic nonlinearities. These solutions blow up in a finite time.
|
Received: 12 May 2010
Revised: 25 August 2010
Accepted manuscript online:
|
|
PACS:
|
02.20.-a
|
(Group theory)
|
| |
02.30.Jr
|
(Partial differential equations)
|
| |
02.30.Tb
|
(Operator theory)
|
|
| Fund: Project supported by the National Natural Science Foundation of China (Grant No. 10926082), the Natural Science Foundation of Anhui Province of China (Grant No. KJ2010A128), and the Fund for Youth of Anhui Normal University, China (Grant No. 2009xqn55). |
Cite this article:
Zhu Chun-Rong(朱春蓉) Second-order nonlinear differential operators possessing invariant subspaces of submaximal dimension 2011 Chin. Phys. B 20 010201
|
| [1] |
Titov S S 1988 Aerodynamics of Plane and Axis-Symmetric Flows of Liquids (Saratov: Saratov University) p104 (in Russsian)
|
| [2] |
Galaktionov V A 1995 em Proc. R. Soc. Endinburg A bf125 225
|
| [3] |
Fokas A S and Liu Q M 1994 Phys. Rev. Lett. bf72 3293
|
| [4] |
Zhdanov R Z 1995 J. Phys. A: Math. Gen. bf28 3841
|
| [5] |
Zhdanov R Z and Lahno V I 1998 Physica D bf122 178
|
| [6] |
Kaptsov O V 1995 J. Nonlinear Math. Phys. bf2 283
|
| [7] |
Kaptsov O V 1998 Sb. Math. bf189 1839
|
| [8] |
Kaptsov O V and Verevkin I V 2003 J. Phys. A: Math. Gen. bf36 1401
|
| [9] |
Zhang S L, Lou S Y and Qu C Z 2006 Chin. Phys. bf15 2765
|
| [10] |
Galaktionov V A and Svirshchevskii S R 2007 Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics (London: Chapman and Hall/CRC)
|
| [11] |
Kamran N, Milson R and Olver P J 2000 Advances Math. bf156 286
|
| [12] |
Svirshchevskii S R 1995 Theor. Math. Phys. bf105 198
|
| [13] |
Svirshchevskii S R 1993 Modern Group Analysis (Moscow: MPhTI) p75
|
| [14] |
Svirshchevskii S R 1995 Phys. Lett. A bf199 344
|
| [15] |
Svirshchevskii S R 1996 J. Nonlinear Math. Phys. bf3 164
|
| [16] |
Svirshchevskii S R 2004 Commun. Nonlinear Sci. Numer. Simul. bf9 105
|
| [17] |
Zhu C R and Qu C Z 2009 Commun. Theor. Phys. bf52 403
|
| [18] |
Qu C Z and Zhu C R 2009 J. Phys. A: Math. Theor. bf42 475201
|
| [19] |
G'omez-Ullate D, Kamran N and Milson R 2007 Discr. Contin. Dyn. Sys. bf18 85
|
| [20] |
Bernis F, Hulshof J and V'azquez J L 1993 J. Reine Angew. Math. bf435 1 endfootnotesize
|
| 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
|
|
|
