非线性薛定谔型方程的怪波解

项目名称： 非线性薛定谔型方程的怪波解

项目编号： No.11271210

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 贺劲松

作者单位： 宁波大学

项目金额： 68万元

中文摘要： 怪波是一种短时间存在的局域的大振幅的波动。最初,人们在海洋中发现了这种突然出现的大振幅波动，它导致了很多海难,对海上航行的船、舰艇和石油平台等有极大威胁。2007年,物理学家在实验中首次观察到光学怪波，这极大地推动了怪波的研究。由于非线性薛定鄂方程的一种特殊的（准）有理解可以很好地描述怪波,故这种(准)有理解也被称为怪波解。本项目拟研究非线性薛定鄂型方程（即非线性薛定鄂方程及其各种修正、推广的方程）的怪波解的构造、初值问题、相互作用、渐进行为、形成机制和可能的物理应用,主要的研究手段是反散射方法、Hirota方法、Darboux变换方法等。由于怪波解在无穷远处的渐进行为是非零的，这给用上述方法求解相关方程带来很大的难度。这些选题是当前可积系统研究的国际前沿和核心课题之一,有重要的科学价值和极大的潜在应用价值。本项目的研究会在怪波的初值问题、性质、应用等方面做出创新性的重要结果。

中文关键词： 怪波；周期解；非线性薛定鄂型方程；达布变换；Hirota方法

英文摘要： The rogue wave is one kind of short lived local large-amplitude wave. Originally, the kind of suddenly appearing large amplitude wave was observed in the ocean, which is responsible for many perils of the sea and an enormous threat of the ships, vessels and oil drilling platform in the ocean. In 2007, the optical rogue wave has been observed in an experiment by physicists. This result has enforced the study of the rogue wave. In this project, we shall study the construction, initial value problems, interaction, asymptotic behavior, generation mechanism and possible applications of the rogue waves for the equations of the nonlinear Schrodinger type. These equations include nonlinear Schrodinger equation and several versions of its modification. The main tools for research are inverse scattering method, Hirota method and Darboux transformation. The non-vanishing property of the rogue wave at infinite leads to the considerable difficulty. The selected topic in this project is one of the international frontiers and core subjects of the research of the integrable systems, which has important value of science and extreme value of potential applications. The research of the project will get novel and important results on the initial value problem, properties and applications of the rogue waves.

英文关键词： Rogue wave；Periodic solution；Nonlinear Schrodinger type equation；Darboux transformation；Hirota method