离散与连续非线性耦合薛定谔系统的矢量畸形波解的构造及相互作用研究

Discrete and continuous nonlinear coupled Schr?dinger systems are the integrable models and arise in many engineering settings and natural science including ocean and nonlinear optics. As one specific type of nonlinear waves, rogue waves have attracted considerable interest for their significant practical applicability in ocean and nonlinear optics fields. This research project firstly solves how to construct the solutions of high-order vector rational rogue wave in discrete and continuous nonlinear coupled Schr?dinger systems. Furthermore, via computerized symbolic computation, we will propose the algebra arithmetic about analyzing the mathematical structure of vector rogue wave solutions and discussing their interactions in discrete and continuous nonlinear coupled Schr?dinger systems. The basic properties and nonlinear dynamics of vector rogue waves can be analyzed based on the obtained rational solutions and algebra arithmetic. Considering the physical applied setting of disctete and continuous nonlinear coupled Schr?dinger systems, we will also explore and investigate the practical applications of vector rogue waves in natural science and engineering fields by combining the analytic study and numerical calculation methods.

离散与连续非线性耦合薛定谔系统是海洋和非线性光学等工程技术和自然科学领域中具有广泛应用背景的可积非线性模型。作为一种特殊的非线性波- - 畸形波，由于它在非线性光学和海洋等领域拥有非常重要的实际应用价值，从而最近对畸形波的研究引起了极大关注。本项目首先解决如何构建离散与连续非线性耦合薛定谔系统的高阶矢量有理畸形波解的问题。其次，借助计算机符号计算，提出讨论该系统矢量畸形波解的数学结构以及畸形波之间相互作用的代数算法，从而可以弄清畸形波的基本特性和动力学机制。结合离散与连续非线性耦合薛定谔系统的物理应用背景，采用解析研究和数值计算分析相结合的方法来探索研究矢量畸形波在自然科学和工程技术中的应用。

本项目基于孤子与可积系统中的解析方法研究了非线性薛定谔型方程以及耦合系统的畸形波解的构造和动力学性质。取得了以下研究成果：(1)发展达布变换和双线性方法研究了非线性薛定谔型方程以及耦合系统畸形波有理解。(2)基于畸形波有理解，分析了畸形波的生成、演化和发展时空特征。(3)研究了有理畸形波解的性质，主要有递推性质、积分性质、分解性质和不变性质。(4)进一步探索研究了高维、变系数等非线性薛定谔型模型的畸形波以及动力学特征。项目共发表SCI收录核心版英文期刊论文十篇，研究生的学位论文“耦合非线性薛定谔方程的有理解与畸形波”被评为2014年上海市研究生优秀学位论文。