高阶耦合非线性系统奇异波的理论研究及应用

Rogue wave is a large-amplitude wave existing within short term in a local area, which appears in the field of oceanography and optics. Also known as freak wave, it has such features as serious damage, uncertainty and unpredictability when occurring. Research on rogue wave has been a hot-spot issue in wave theory as well as nonlinear dynamics and control. Time varying system and high-order coupled system are used to describe rogue wave in this project. Combined nonlinear dynamics theory with rogue wave theory, the project is to study the existence conditions including its disappearance, maintenance, reproduction, rapid excitation and additionally its dynamic characteristics, establish theoretical framework and reveal formation mechanism of the rogue wave. The main object aims to investigate rogue wave mechanism of the time varying system by using Darboux transformation. Darboux matrix is deduced and eigenfunction corresponding to the Lax equation is constructed to derive rogue wave solution and analyze the influence of small perturbation parameters on the dynamic behaviors of the rogue wave. Based on the generalized nonhomogeneous nonlinear Schrödinger equation, the effect of system parameters on the dynamic characteristics of rogue wave is studied. The research will provide theoretical development and application in practical engineering.

奇异波，也称畸形波，是短时间存在的局部区域的大振幅波动，在海洋学和光学等领域时有发生，具有突发性、不确定性和不可预测性等特点，危害巨大，近年来奇异波的研究成为波浪理论以及动力学与控制领域一个热点问题。本项目利用时变系统和高阶耦合非线性系统来描述奇异波，基于非线性动力学理论和奇异波理论，研究奇异波的消失、维持、重现以及快速激发等存在条件及动态性质，建立奇异波的非线性理论框架，揭示和完善奇异波的生成机理。主要研究内容包括：利用Darboux变换理论对时变系统的奇异波机理进行研究；推导Darboux矩阵并构造Lax对方程对应的特征函数得到奇异波解，分析小扰动参数对奇异波动力学行为的影响；以光学领域中描述奇异波的广义非均匀非线性薛定谔方程为应用，研究系统参数对奇异波动态特性的影响。本项目的研究可以促进奇异波非线性动力学理论的发展及其在实际工程中的应用。

奇异波，也称畸形波，是短时间存在的局部区域的大振幅波动，在海洋学和光学等领域时有发生，具有突发性、不确定性和不可预测性等特点，危害巨大，近年来奇异波的研究成为波浪理论以及动力学与控制领域一个热点问题。本项目利用时变系统和高阶耦合非线性系统来描述奇异波，基于非线性动力学理论和奇异波理论，研究奇异波的消失、维持、重现以及快速激发等存在条件及动态性质，建立奇异波的非线性理论框架，揭示和完善奇异波的生成机理。主要研究内容包括：利用Darboux变换理论对时变系统的奇异波机理进行研究；推导Darboux矩阵并构造Lax对方程对应的特征函数得到奇异波解，分析小扰动参数对奇异波动力学行为的影响；以光学领域中描述奇异波的广义非均匀非线性薛定谔方程为应用，研究系统参数对奇异波动态特性的影响。本项目的研究可以促进奇异波非线性动力学理论的发展及其在实际工程中的应用。