若干时空调制的广义非线性薛定谔方程的波参数调控与动力学分析

This project mainly investigates the variable-coefficient (coupled) Nonlinear Schrödinger equations with space- and time-modulated higher-order nonlinear terms, higher-order dispersion terms and potentials in the inhomogeneous medium. Based on the symbolic computation and numerical computation, the complex nonlinear wave solutions of the equations will be constructed in this project by using and improving the Darboux transformation method and Hirota bilinear method in the classical integrable system. It will be realized through the following specific goals: (1) To construct the nonlinear wave solutions of the higher-order Nonlinear Schrödinger equations and their coupled forms. To build the integrable constrain conditions of the variable-coefficient (coupled) Nonlinear Schrödinger equations, and give the analytic functions of the nonlinear wave solutions; (2) To investigate the dynamic properties of the solutions, and study the roles of the coefficients in the modulation and interaction of the nonlinear wave solutions, whose validity will be verified by numerical simulation; (3) To develop the symbolic computation program package to construct the rogue wave solutions automatically. The results of this project will explain the physical mechanism of the experimental results in the nonlinear optics and Bose-Einstein condensates and provide theoretical foundations to predict new experiments.

本项目以非均匀介质中含时空调制的高阶非线性项、高阶色散项及外势的变系数（耦合）薛定谔方程为研究对象，以符号计算与数值计算为工具，利用并改进经典可积系统中的Darboux变换与Hirota双线性方法，构造方程的复杂非线性波解。具体研究以下三方面的内容：（1）构造高阶薛定谔方程及其耦合形式的非线性波解，给出变系数(耦合)薛定谔方程的可积约束条件，并给出复杂非线性波解的解析表达式；（2）分析非线性波解的动力学性质，建立方程各系数对非线性波的调制、相互作用等解析关系式，并给出数值模拟以验证相关结论；（3）开发自动推导非线性发展方程怪波解的符号计算软件包。本课题的研究可为非线性光学、Bose-Einstein凝聚态等物理领域的有关实验结果给出理论解释，并未预测新的物理实验及物理现象提供理论依据。

本项目以非线性光学、玻色爱因斯坦凝聚态等物理领域的广义非线性可积方程为研究对象，以符号计算和数值计算为工具，基于Darboux变换与伪势方法，构造了方程的复杂非线性波解并分析了其动力学性质。具体研究了以下五个方面的内容：（1）高阶非局域广义非线性薛定谔类方程复杂非线性波解的构造及动力学性质分析；（2）离散Ablowitz-Ladik方程复杂非线性波解的构造及动力学性质分析；（3）非线性薛定谔方程周期背景波上的非线性波解及局域化;（4）含高阶色散项与非线性项非线性发展方程的相互作用解；（5）五阶Korteweg–de Vries方程初边值问题及解的适定性和渐近性态。