多分量非线性薛定谔型方程的怪波特性与物理机制

The project is mainly devoted to study rogue wave characteristics and physical mechanisms for the multi-component nonlinear Schrödinger-type equations. The models we would like to investigate are related to the complicated higher-order spectral problems, mainly include the two-component Sasa-Satsuma equation, two-component derivative nonlinear Schrödinger equation, two-component Kundu-Eckhaus equation and so on. We will mainly focus on: 1. Construct the generalized Darboux transformation and the general expression of rogue wave solutions with arbitrary order; 2. Study the complete classification and the relevant mathematical physical properties of the rogue waves; 3. Research the state transition, asymptotic behaviors of interactional solutions between rogue waves and other nonlinear waves, and the production mechanisms and propagation mechanisms for the vector rogue waves under periodic backgrounds. The research results will provide theoretical supports for revealing the mathematical expression of rogue wave solutions with arbitrary order, the dynamical structures and properties of rogue waves, the discovery of new nonlinear waves, and the physical mechanisms of rogue waves in nonlinear optics and deep water wave associated with multi-component integrable models.

本项目主要对多分量非线性薛定谔型方程的怪波特性与物理机制展开研究。所涉及的模型主要是与复杂的高阶谱问题相联系的两分量Sasa-Satsuma方程、两分量导数非线性薛定谔方程和两分量Kundu-Eckhaus方程等。研究内容包括：1. 构造多分量非线性薛定谔型方程的推广的Darboux变换和任意阶怪波解的一般表达式；2. 研究多分量非线性薛定谔型方程中怪波的完全分类问题和相关的数学物理性质；3. 探索多分量非线性薛定谔型方程中怪波与其它非线性波的态转换、相互作用解的渐近行为、以及周期波背景下矢量怪波的产生机制与传播机理。研究结果为进一步揭示多分量可积模型任意阶怪波解的数学表示、怪波的动力学结构与性质、新的非线性波的发现、以及非线性光学和深水波中怪波的物理机制提供理论支持。

本项目主要对可积系统怪波解的构造、形态转换及相应的数学物理性质展开研究。基于推广的Darboux变换方法和调制不稳定性分析，得到了如下的结果：1. 研究了多分量可积系统的怪波解、呼吸子解和周期解，以及怪波与W型孤子之间的形态转换问题；2. 构造了高维可积系统的Darboux变换，得到了共振孤子解、怪波解和Lump解；3. 讨论了高阶可积系统的局域波解的形态转换性质和物理机制。研究结果一方面为揭示可积系统任意阶怪波解的数学结构建立了一个统一的构造模式，另一方面对可积系统中怪波的激发机理、参数调控及新的非线性波的发现提供理论支持。