标题： (2+1)D 有扰非线性薛定谔方程的Crank-Nicolson有限差分格式分析 显示英文标题

标题： Analysis of a Crank-Nicolson finite difference scheme for (2+1)D perturbed nonlinear Schrödinger equations with saturable nonlinearity

摘要： 我们分析了带有饱和非线性和三次损耗扰动的(2+1)D非线性薛定谔方程的Crank-Nicolson有限差分离散化。 我们证明了数值解的有界性、存在性和唯一性。 我们建立了误差界以证明数值解的收敛性。 此外，我们在一个温和假设下发现收敛速率在时间步长和空间网格尺寸上均为二阶。 数值方案通过(2+1)D饱和非线性薛定谔模型的大量模拟得到验证。 使用加速虚时演化方案和Crank-Nicolson有限差分方法实现了对行进孤子的模拟。 显示英文摘要

摘要： We analyze a Crank-Nicolson finite difference discretization for the perturbed (2+1)D nonlinear Schr\"odinger equation with saturable nonlinearity and a perturbation of cubic loss. We show the boundedness, the existence and uniqueness of a numerical solution. We establish the error bound to prove the convergence of the numerical solution. Moreover, we find that the convergence rate is at the second order in both time step and spatial mesh size under a mild assumption. The numerical scheme is validated by the extensive simulations of the (2+1)D saturable nonlinear Schr\"odinger model with cubic loss. The simulations for travelling solitons are implemented by using an accelerated imaginary-time evolution scheme and the Crank-Nicolson finite difference method.