高维非线性Schrdinger方程的Fourier谱方法

其中i=(-1)(1/2),△为Laplace算子,q(·)为实变量实值函数,u_0(x)和u(x,t)分别为关于x以2π为周期的已知和未知复值函数,J=(0,T](T>0),β为一实常数,e_j为R~m的第j个单位向量,x=(x_1,…,x_m)∈R~m. 方程(1.1)在非线性光学、等离子体物理、流体动力学及非相对论量子场论中用得很

THE FOURIER SPECTRAL METHODS FOR MULTI-DIMENSIONAL AND NONLINEAR SCHRDINGER EQUATIONS

Fourier spectral methods of Galerkin and collocation type for multi-dimensional andnonlinear Schrodinger equations are presented. Convergence with spectral accuracy is provedfor both Galerkin and collocation approximations. Stable discretizations in time by one-stepand two-step methods are analyzed. An algorithm for the discretizations is proposed. The nu-mber of operations for the pseudo-spectral technique by FFT is computed. Finally, some nu-merical results are given, which are compared with those of other methods.