高阶非线性薛定谔方程的分步小波方法

用小波变换代替傅里叶变换解高阶非线性薛定谔方程,为高阶薛定谔方程的数值解提供了一种工具,提高了运算速度.本文分析了高阶非线性薛定谔方程分步解法的一般形式,选用Db10小波,得到了小波微分算子和色散算子对应的矩阵,得出了分步小波方法的算法公式.推导了色散算子和时域信号在小波域相乘的近似运算公式,说明了分步傅里叶方法比分步小波方法的复数乘法次数更多,同时说明了提高运算速度必须舍弃一定的运算准确度.最后以分步傅里叶方法为准,分析了分步小波方法的误差,结果表明:对于一阶孤子,分步小波方法与分步傅里叶方法间的相对误差在1.2%左右波动.

Numerical Analysis for High Order Nonlinear Optical Pulse Progration on Slip-step Wavelet Method

Using wavelet transform to replace Fourier transform to solute higher-order nonlinear Schrodinger equation, provides it as another tool, it improves the operation speed.Analyzed the high-order nonlinear Schrodinger equation general solution form.By using Db10 wavelet, obtained the matrix corresponding to differential operator and dispersive operator,also obtained the split-step wavelet method algorithm formula. Derivate the dispersion operator and the signal in wavelet domain multiplied by the approximate calculating formula, the split-step Fourier method need more complex multiplication times than the split-step wavelet method, at the same time that increase the speed of operation cost the computation precision. Finally take the split-step Fourier method as standard, analyzed the split-step wavelet method error, the results show that, for the first order soliton, between the split-step wavelet method and split-step Fourier method relative error fluctuate around 1.2%.