In this paper, the dimension of invariant subspaces admitted by nonlinear sys- tems is estimated under certain conditions. It is shown that if the two-component nonlinear vector differential operator F = (F1, F2) wi...In this paper, the dimension of invariant subspaces admitted by nonlinear sys- tems is estimated under certain conditions. It is shown that if the two-component nonlinear vector differential operator F = (F1, F2) with orders {k1, k2} (k1≥ k2) preserves the invariant subspace Wn1^1× Wn2^2 (n1 ≥ n2), then n1 - n2 ≤ k2, n1 ≤2(k1 + k2) + 1, where Wnq^q is the space generated by solutions of a linear ordinary differential equation of order nq (q = 1, 2). Several examples including the (1+1)-dimensional diffusion system and Ito'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 nonlinear systems is also given.展开更多
基金Project supported by the National Natural Science Foundation of China for Distinguished Young Scholars (No.10925104)the National Natural Science Foundation of China (No.11001240)+1 种基金the Doctoral Program Foundation of the Ministry of Education of China (No.20106101110008)the Zhejiang Provincial Natural Science Foundation of China (Nos.Y6090359,Y6090383)
文摘In this paper, the dimension of invariant subspaces admitted by nonlinear sys- tems is estimated under certain conditions. It is shown that if the two-component nonlinear vector differential operator F = (F1, F2) with orders {k1, k2} (k1≥ k2) preserves the invariant subspace Wn1^1× Wn2^2 (n1 ≥ n2), then n1 - n2 ≤ k2, n1 ≤2(k1 + k2) + 1, where Wnq^q is the space generated by solutions of a linear ordinary differential equation of order nq (q = 1, 2). Several examples including the (1+1)-dimensional diffusion system and Ito'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 nonlinear systems is also given.