非线性薛定谔型方程特殊结构解的探究

The nonlinear Schrödinger equation is one of the most important models which can describe the nonlinear phenomena in several fields such as the optical communications. Considering different situations such as inhomogeneous parameters and higher-order dispersion, several nonlinear Schrödinger type equations of different form with variable-coefficient, higher-dimension, higher-order and coupling are presented. In this project, we will discuss the ways to obtain the special structure solutions such as the determinant and Pfaffian solutions for those nonlinear Schrödinger type equation with the help of the corresponding linear system, derive the dark soliton and rogue wave from the determinant solutions, and then discuss the propagation properties and dynamics mechanism for these waves. Also, this project will generalize the solutions of determinant structure to Pfaffian form using the algebraic theory. And we will Pfaffianize the corresponding nonlinear Schrödinger type equation to construct new coupled system. It is hoped that the obtained research results will help for pushing forward the research on the algebraic structure of nonlinear Schrödinger type equation, developing more methods for constructing dark soliton and rogue wave, and providing theoretical basis for the numerical simulation and experimental researches of the nonlinear waves in the fields involving the optical communications.

考虑到如非均匀参数、高阶色散等不同情况，变系数、高维、高阶或耦合等各类不同形式的非线性薛定谔型方程被提出。本项目拟借由非线性薛定谔型方程对应的线性系统，构造非线性薛定谔型方程的具有特殊代数结构如行列式或Pfaffian解，由行列式解约化得到非线性薛定谔型方程的暗孤子和怪波解，讨论解的传播特性和动力学机制，将行列式结构的解利用代数理论进一步推广到Pfaffian 结构。然后将非线性薛定谔型方程Pfaffian 化构造新的耦合系统。希望本项目所得研究结果有助于进一步推动非线性薛定谔型方程的代数结构研究，有助于拓展构造暗孤子解和怪波解的方法，为光纤通信等领域中非线性波动的数值研究和实验观测提供理论依据。

本项目利用Hirota双线性方法、KP梯队约化法、Darboux变换法、Bell多项式方法等解析方法研究了高维浅水波方程、变系数Sasa-Satsuma方程、变系数高阶非线性薛定谔方程、Kundu方程、广义高阶非线性薛定谔方程等方程的行列式结构有理解、呼吸子解、扭结怪波解、高阶怪波解等解析解以及方程的可积性质，并利用解析结果进行非线性波动的传播演化特性探究和动力学行为分析。主要研究成果如下：(1)通过恰当变量变换，利用KP梯队约化法构造出了高维浅水波方程的Gramm行列式结构的有理解，分析了不同参数取值下解的传播演化情况；(2)得到了变系数Sasa-Satsuma方程的无穷多守恒律和自Bäcklund变换，利用Darboux变换求得该方程的扭结怪波解；(3)求得变系数高阶非线性薛定谔方程的呼吸子解，进一步通过对呼吸子解中的参数取极限得到原方程的怪波解；(4)基于广义的(n，N-n)波Darboux变换，得到了Kundu方程的新的精确高阶怪波解，进行了调制不稳定性分析；(5)基于解析和数值差分方法研究了广义高阶非线性薛定谔方程所描述的深海非线性波的传播演化；(6)利用N次迭代Darboux变换得到了Boussinesq方程在非零背景下的行列式形式孤波解并进行了渐近分析和碰撞分析；(7)借助于Bell多项式得到了变系数Gardner-KP方程的多孤波解和Bäcklund变换，利用李对称方法求得高阶Extended KdV方程的相似变换与相似解；(8)将本项目所得解析研究结果应用于实际海洋背景中，研究了海洋内波信号和参数提取的方法。所得结果有望为数值研究和实验观测提供理论依据。