非线性薛定谔类型方程的可积系统研究方法

The project is mainly devoted to study integrable nonlinear Schrödinger type equation. The models that we would like to investigate mainly include multi-component nonlinear Schrödinger equations, multi-component derivative nonlinear Schrödinger equations, semi-classical nonlinear Schrödinger equation, Sata-Sastuma equation and so on. In this project, we will deeply study the dark soliton of multi-component nonlinear Schrödinger type equations and its interaction, stability of high order soliton of nonlinear Schrödinger equation,the relation between rogue wave solutions and rogue wave phenomenon of semi-classical nonlinear Schrödinger equation. We would like to use the Darboux transformation theory, Riemann-Hilbert approach, Deift-Zhou method, combining with limit technique to analysis these models. Thus the results in mathematical theories will be obtained. The contents of this project are not only important in theories but also have great values in practices.

本项目主要对可积的非线性薛定谔类型方程展开研究。研究的物理模型主要包括多分量的非线性薛定谔方程，多分量的导数薛定谔方程，半经典的非线性薛定谔方程，Sasa-Satsuma方程等。本项目将深入讨论多分量薛定谔类型方程暗孤立子解的Darboux变换以及多孤立子相互作用规律、非线性薛定谔方程的高阶孤立子解的稳定性以及“怪波”解同半经典非线性薛定谔方程的“怪波”现象之间的联系。我们将利用Darboux 变换、Riemann-Hilbert 方法、Deift-Zhou方法等理论，并结合极限技巧来系统的研究非线性薛定谔类型的方程，从而得到相关的数学理论结果。所研究的问题不仅具有重要的理论价值，而且还具有广泛的应用价值。

非线性薛定谔类型方程在非线性科学研究中有着强有力的应用背景和价值。本项目主要利用 Darboux 变换方法研究可积非线性薛定谔类型方程相关性质，发展 Darboux 变换理论在非线性薛定谔类型方程中的应用。利用矩阵分析和极限技巧推广了 Darboux 变换在非零背景条件下的理论，得到了暗孤子解，呼吸子解以及怪波解以及调制不稳定性之间的关系。这些结果有助于进一步加深对非线性薛定谔系统的理解和认识，并可能用于指导具体的物理实验操作。