Maximal Dimension of Invariant Subspaces to Systems of Nonlinear Evolution Equations |
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Authors: | Shoufeng SHEN Changzheng QU Yongyang JIN and Lina JI |
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Institution: | 1. Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, China 2. Corresponding author. Department of Mathematics, Ningbo University, Ningbo 315211, Zhejiang,China 3. Department of Information and Computational Science, Henan Agricultural University, Zhengzhou 450002, China |
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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
\mathbbF = (F1 ,F2 )\mathbb{F} = (F^1 ,F^2 ) with orders {k
1, k
2} (k
1 ≥ k
2) preserves the invariant subspace
Wn1 1 ×Wn2 2 (n1 \geqslant n2 )W_{n_1 }^1 \times W_{n_2 }^2 (n_1 \geqslant n_2 ), then n
1 − n
2 ≤ k
2, n
1 ≤ 2(k
1 + k
2) + 1, where Wnq qW_{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?’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. |
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Keywords: | Invariant subspace Nonlinear PDEs Exact solution Symmetry Dynamical system |
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