基于分数阶薛定谔方程的光束动力学和光孤子特性的研究

The application of fractional Schrodinger equation in optics has flourished to become a research area of great interest in nonlinear science. This research can extend its application in the theory of the beam propagation dynamics and the optical soliton solution...The main work in the project are as follows: First, the research on the beam propagation dynamics under the framework of the fractional linear Schrodinger equation. Through the Fourier transform, the Adomian decomposition method, the finite difference method etc., study the beam dynamics and the physical parameter modulation analytically and numerically. Second, we discuss the methods of soliton solution in the nonlinear fractional Schrodinger equation and extend the application of the analytical and numerical methods such as the extended symmetry reduction method, Alice-Bob method, differential algorithm and so on. Third, analysis on the constraints in the generation of soliton is discussed. The stability of soliton is analyzed based on the numerical method, i.e., the method of eigenvalue stability analysis. Last, the research on the parameter modulation and dynamics of soliton solution in the nonlinear fractional Schrodinger equation based on the numerical simulation such as the variational method and fast split step Fourier method. This project can promote the physical and mathematical theory in fractional Schrodinger equation, which is of great scientific value in the fields of nonlinear optics and optical communication.

分数阶薛定谔方程在光学领域的应用是目前非线性科学的前沿课题。本研究力求拓展其在光束传输动力学和光孤子理论的应用，具体内容涵括四个方面：一是分数阶线性薛定谔方程框架下的多种光束传输动力学的研究，通过傅里叶变换、Adomian分解法、有限差分法等解析和数值方法研究光束的传输特性和参量调控；二是探讨分数阶非线性薛定谔方程的光孤子求解方法，拓展对称（群）约化法、Alice-Bob思想方法、差分算法等解析和数值求解手段的应用范围；三是讨论孤子解存在的物理参数条件，利用本征值稳定性分析法等数值方法对孤子的稳定性进行分析。最后，结合变分法、快速分步傅里叶算法等数值方法，对光孤子解的传输动力学和参量调控进行研究。本项目对于分数阶薛定谔方程相关内容的研究能完善分数阶微积分数学物理理论，并且在非线性光学和光通信等领域上面具有重要的科学价值。

分数阶薛定谔方程在光学领域的应用是目前非线性科学的前沿课题。本研究可以拓展其在光束传输动力学和光孤子理论的应用。首先，本项目研究根据不同介质条件建立分数阶薛定谔方程。结合解析或者准解析的求解方法，以及数值方法来研究了其中的光束传输动力学和光学孤子的求解方法，如分数阶映射方程方法、分数阶双函数方法，傅里叶变换、有限差分算法等。其次，讨论孤子解存在受到物理参数的影响，利用线性稳定性方法等对孤子的稳定性进行分析。最后，结合快速分步傅里叶算法等数值方法研究了光孤子解的传输动力学演化情况和参量调控。本项目对于分数阶薛定谔方程相关内容的研究能完善分数阶微积分数学物理理论，并且在非线性光学和光通信等领域上面具有重要的科学价值。