一种分步基督وف方法用于近似有理偏微分方程解
A split-step Christov method for approximating rational PDE solutions
AI总结：
本文提出一种分步基督وف方法，用于近似有理偏微分方程解，通过傅里叶级数和快速傅里叶变换计算频谱，并探讨了非线性薛定谔方程中的孤子波解。
摘要：
有理偏微分方程（PDEs）的解在使用谱傅里叶方法进行近似时，因其代数衰减率缓慢而 notoriously 非常困难。在本工作中，我们讨论在正交函数基（称为傅里叶级数）中近似有理PDE解，允许通过快速傅里叶变换计算其频谱。推导了谱微分矩阵。推导了几种显式四阶分步积分器，并进行了比较。作为应用，探讨了非线性薛定谔方程族中的孤子波解。发现扰动常数背景会生成类似孤子波的结构。还检查了更高阶色散和广义非线性效应的影响。
英文摘要：
Rational solutions of partial differential equations (PDEs) are notoriously difficult to approximate via spectral Fourier methods due to their algebraically slow decay rate. In this work we discuss approximating rational PDE solutions in a basis of orthogonal functions known as the Fourier series, allowing for the computation of its spectrum via the fast Fourier transform. Spectral differentiation matrices are derived. Several explicit fourth-order split-step integrators are derived and their performance compared. As an application, rogue wave solutions in a family of nonlinear Schrödinger equations are explored. Perturbing the constant background is found to generate rogue wave-like structures. The effects of higher-order dispersion and generalized nonlinearities are also examined.
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