非线性KdV-Schrdinger方程Fourier谱逼近的大时间性态

Long-time Behavior of Fourier Spectral Approximation to Nonlinear KdV-Schrdinger Equation

近几年来,对具弱阻尼的非线性发展方程的研究越来越受到人们的关注.大部分情况下,由于精确解无法得到,我们只有通过求数值解来研究方程解的性质.本文讨论具弱阻尼的非线性KdV-Schrdinger方程Fourier谱逼近的大时间性态问题.我们构造了方程的Fourier近似谱格式,并对方程的近似解作了相应的先验估计及方程近似解与精确解之间的误差估计.最后,证明了近似吸引子A_N的存在性及其弱上半连续性d~ω（A_N,A）→0.

Recently,more and more attentions are paid to the study of nonlinear evolution equations with weakly damped.Mostly,the equations＇ exact solution can＇t be gotten,all we can do is to investigate the equations＇ properties through their numerical simulations. The main purpose of this work is to investigate the long-time behaviors of Fourier spectral approximation to nonlinear KdV-Schrdinger equation. Firstly,we introduce some orthogonal systems,on which we develop some results of projected operator,and several basic inequalities. Secondly,for the equation considered in this paper,we constitute Fourier spectral scheme and relevant priori estimates.Following which,we give the error estimates between the approximate solution and the exact solution. At last,we prove the existence and convergence of the attractor.