Attractor for the nonlinear Schr?dinger equation with a nonlocal nonlinear term

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• 作者：Chaosheng Zhu ; Chunlai Mu and Zhilin Pu
• 关键词：Nonlinear Schr?dinger equation ; global attractor ; regularity ; orthogonal decomposition
• 刊名：Journal of Dynamical and Control Systems
• 出版年：2010
• 出版时间：October 2010
• 年：2010
• 卷：16
• 期：4
• 页码：585-603
• 全文大小：553.3 KB

文摘

In this paper, we consider the long-time behavior of solutions of the dissipative 1D nonlinear Schr?dinger (NLS) equation with nonlocal integral term and with periodic boundary conditions. We prove the existence of the global attractor A \mathcal{A} for the nonlocal equation in the strong topology of H 1(Ω). We also prove that the global attractor is regular, i.e., A ¨¬ H2( W) \mathcal{A} \subset {H^2}\left( \Omega \right) , assuming that f(x) is of class C 2. Furthermore, we estimate the number of the determining modes for this equation.