可积系统在周期波背景下怪波和非零背景下初值问题的反散射研究

Nonlinear integrable systems can describe many nonlinear phenomena in physical fields, and it is very important and useful for us to understand the essence of nonlinear phenomena when we solve the nonlinear integrable systems and research the properties of solutions. In this project, the solitons and rogue wave solutions in several types of continuous and semi-discrete nonlinear integrable systems will be investigated. Additionally, the intial value problems of these systems are studied with the non-zero boundary during the inverse scattering transform process. The main contents of this study are as followed: 1) Under the conditions of elliptic function background, the research about the rogue wave (i.e., periodic rogue wave) of Hirota equation, Sasa-Satsuma equation, CID and CCD equation are conducted. The expression of the exact analytic solution are developed, and the shape of these analytic solutions are related to the specified parameters of background wave. 2)The inverse transformation for several NLS-type systems with nonzero boundary condition are presented, including the introduction of the appropriate Riemann surface, the symmetry of Jost function and scattering datas, discrete spectrum. In addition, the general behavior of soliton solutions is discussed. 3) An extended integrable Sasa-Satsuma equation with 5th order dispersion term, semi-discrete FL equation and semi-discrete CCD equation are solved, of which the infinite conservation rate are given. The relationships of FL equation as well as its solution between continuous and semi-discrete classification are analyzed, and so are CCD equation.

非线性可积系统可以描述很多物理领域中非线性现象，求解非线性可积系统并研究解的性质对我们了解非线性现象的本质有重要的理论和应用价值。本项目将研究几类连续的和半离散的非线性可积系统的孤子解和怪波解，以及这些系统在非零边界条件下的初值问题的反散射问题。主要内容：(1)研究Hirota方程、Sasa-Satsuma方程、CID方程和CCD方程在椭圆函数背景下的怪波解，又称周期怪波；给出周期怪波解得精确表达式，分析解的形态与参数之间的关系。(2)研究几个NLS方程推广型系统在非零边界条件下的初值问题的反散射问题，引入合适的黎曼面，研究Jost函数、散射数据的对称性，分析离散谱，给相应的RH问题；研究反射系数为零时，孤子解的性质。(3)求解一个带5阶色散项的Sasa-Satsuma方程的推广型可积系统、半离散的FL方程和半离散的CCD方程；给出这些方程的无穷守恒率；研究这些方程孤子解和怪波解的性质。

非线性可积系统可以描述很多物理领域中非线性现象，求解非线性可积系统并研究解的性质对我们了解非线性现象的本质有重要的理论和应用价值。本项目研了几类非线性可积系统的孤子解和怪波解。 主要内容和结果：求解一个带受激拉曼色散项和五阶非线性项的高阶NLS方程，给出其Lax对，用达布变换的方法求得了系统的零背景以及非零背景条件下的各种类型的解；提出一个可积的逆空时(逆空间-逆时间)非局部Sasa-Satsuma方程和一个新的非局部Yajima-Oikawa系统，研究其在零背景和非零背景条件下解的性质；分别给出了复的无色散方程方程和两分量复的无色散方程对应的半离散方程，给出了其Lax对和达布变换，得到单分量系统的亮孤波，呼吸子解、怪波解和两分量系统的多种类型的共振解。