A New Framework of Convergence Analysis for Solving the General Nonlinear Schrodinger Equation using the Fourier Pseudo-Spectral Method in Two Dimensions  

作　　者：Jialing Wang Tingchun Wang Yushun Wang 

机构地区：[1]School of Mathematics and Statistics,Nanjing University of Information Science&Technology,Nanjing,Jiangsu 210044,China [2]Jiangsu Key Laboratory of NSLSCS,School of Mathematics Science,Nanjing Normal University,Nanjing,Jiangsu 210023,China

出　　处：《Advances in Applied Mathematics and Mechanics》2023年第3期786-813,共28页应用数学与力学进展（英文）

基　　金：Jialing Wang’s work is supported by the National Natural Science Foundation of China(Grant No.11801277);Tingchun Wang’s work is supported by the National Natural Science Foundation of China(Grant No.11571181);the Natural Science Foundation of Jiangsu Province(Grant No.BK20171454);Qing Lan Project.Yushun Wang’s work is supported by the National Natural Science Foundation of China(Grant Nos.11771213 and 12171245).

摘　　要：This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the nonlinear term is mere cubic.The new framework of convergence analysis consists of two steps.In the first step,by truncating the nonlinear term into a global Lipschitz function,an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete L2 norm;followed in the second step,the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique,as implies that the two numerical methods are the same one.Under our framework of convergence analysis,with neither any restriction on the grid ratio nor any requirement of the small initial value,we establish the error estimate of the proposed conservative Fourier pseudo-spectral method,while previous work requires the certain restriction for the focusing case.The error bound is proved to be of O(h^(r)+t^(2))with grid size h and time step t.In fact,the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schr¨odinger-type equations.Numerical results are conducted to indicate the accuracy and efficiency of the proposed method,and investigate the effect of the nonlinear term and initial data on the blow-up solution.

关 键 词：Framework of convergence analysis general nonlinear Schr¨odinger equation Fourier pseudo-spectral method conservation laws unconditional convergence blow-up solution 

分 类 号：O175[理学—数学]