广义非线性薛定谔类方程周期背景波上的非线性波解及其动力学性质

周期背景波上的非线性波解更能刻画相关物理现象的理论机制，一直是非线性数学物理的研究重点与难点。为更好地理解周期背景波上非线性波的产生机制及动力学性质，本项目拟开展以下四个方面的研究：1. 构造高阶及耦合广义薛定谔类方程周期背景波上的非线性波解，研究非线性波解的态转化关系及动力学性质，并分析与平坦背景波情形之间的约化；2. 将连续情形的构造方法与结果推广到离散情形，分析离散与连续情形周期背景波上非线性波解之间的对应关系；3. 将经典方程的方法及结论推广到非局域情形，分析其动力学性质并建立与经典情形之间的约化关系；4. 利用数值模拟研究周期背景波上非线性波解的演化、相互作用及鲁棒性等动力学性质。本课题的研究成果有望预测新的物理实验并应用于非线性光学与玻色-爱因斯坦凝聚等非线性物理领域，以揭示相关物理现象的理论机制。

The nonlinear wave solution on the periodic background can better describe the theoretical mechanism of the related physical phenomena, which is always the research focus and difficulty of nonlinear mathematical physics. In order to better understand the generation mechanism and dynamic properties of nonlinear wave on periodic background, this project will devote to the following investigations: 1. To construct the nonlinear wave solutions on periodic background of higher order and coupled generalized nonlinear Schrödinger equations, and study the relationships and dynamic properties of various types of nonlinear waves, and analyze the reduction relationships with uniform background case; 2. To extend the methods and results of continuous case to the discrete case, and analyze the relationships of the nonlinear wave solutions on the periodic background between discrete and continuous cases; 3. To further develop the relevant conclusions to the nonlocal case, and examine the dynamic properties and reduction relationships with classical case; 4. To investigate the evolution, interaction and robustness of nonlinear wave solutions on periodic background by numerical simulation. The results of this project may raise the possibility of related experiments and potential applications in the nonlinear physical fields, such as nonlinear optics and Bose–Einstein condensates, and reveal the theoretical mechanism of related physical phenomena.