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| Maximal Dimension of Invariant Subspaces to Systems of Nonlinear Evolution Equations |
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Citation： |
Shoufeng SHEN,Changzheng QU,Yongyang JIN,Lina JI.Maximal Dimension of Invariant Subspaces to Systems of Nonlinear Evolution Equations[J].Chinese Annals of Mathematics B,2012,33(2):161~178 |
| Page view： 2242
Net amount： 1728 |
Authors： |
Shoufeng SHEN; Changzheng QU; Yongyang JIN; Lina JI; |
Foundation： |
the National Natural Science Foundation of China for Distinguished Young Schol-ars (No. 10925104), the National Natural Science Foundation of China (No.11001240), the Doctoral Program Foundation of the Ministry of Education of China (No. 20106101110008) and the Zhejiang Provincial Natural Science Foundation of China (Nos. Y6090359, Y6090383). |
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| Abstract： |
In this paper, the dimension of invariant subspaces admitted by nonlinear systems is estimated under certain conditions. It is shown that if the two-component nonlinear vector differential operator $\mathbb{F}=(F^1, F^2)$ with orders $\{k_1, k_2\}\ (k_1\geq k_2)$ preserves the invariant subspace $W_{n_1}^1\times W_{n_2}^2\ (n_1\geq n_2)$, then $n_1-n_2\leq k_2$, $n_1\leq 2(k_1+k_2)+1$, where $W_{n_q}^q$ is the space generated by solutions of a linear ordinary differential equation of order $n_q\ (q=1,2)$. Several examples including the (1+1)-dimensional diffusion system and It\^{o}'s type, Drinfel'd-Sokolov-Wilson's type and Whitham-Broer-Kaup's type equations are presented to illustrate the result. Furthermore, the estimate of dimension for $m$-component 1nonlinear systems is also given. |
Keywords： |
Invariant subspace, Nonlinear PDEs, Exact solution, Symmetry, Dynamical system |
Classification： |
37J15, 37K35, 35Q53, 35K55 |
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