Second-order nonlinear differential operators possessing invariant subspaces of submaximal dimension

doi:10.1088/1674-1056/20/1/010201

中国物理B ›› 2011, Vol. 20 ›› Issue (1): 10201-010201.doi: 10.1088/1674-1056/20/1/010201

Second-order nonlinear differential operators possessing invariant subspaces of submaximal dimension

朱春蓉

College of Mathematics and Computer Science, Anhui Normal University, Wuhu 241000, China

收稿日期:2010-05-12 修回日期:2010-08-25 出版日期:2011-01-15 发布日期:2011-01-15
基金资助:
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).

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

Received:2010-05-12 Revised:2010-08-25 Online:2011-01-15 Published:2011-01-15
Supported by:
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).

摘要/Abstract

摘要： 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.

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.

Key words: nonlinear evolution equations, cubic operators, invariant subspace method, submaximal dimension, blow-up solution

中图分类号: (Group theory)

02.20.-a

02.30.Jr (Partial differential equations) 02.30.Tb (Operator theory)

朱春蓉. Second-order nonlinear differential operators possessing invariant subspaces of submaximal dimension[J]. 中国物理B, 2011, 20(1): 10201-010201.

Zhu Chun-Rong(朱春蓉). Second-order nonlinear differential operators possessing invariant subspaces of submaximal dimension[J]. Chin. Phys. B, 2011, 20(1): 10201-010201.

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Second-order nonlinear differential operators possessing invariant subspaces of submaximal dimension

Second-order nonlinear differential operators possessing invariant subspaces of submaximal dimension

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