某些非线性波方程的周期波的动力学性质、精确参数表示和极限形式研究

项目名称： 某些非线性波方程的周期波的动力学性质、精确参数表示和极限形式研究

项目编号： No.11461022

项目类型： 地区科学基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 何斌

作者单位： 红河学院

项目金额： 36万元

中文摘要： 非线性波方程至今仍然是世界性的热门研究课题之一，非线性波方程的各种周期波的动力学性质、精确参数表示和极限形式在流体力学、凝聚态物理、固体物理、生物数学以及光纤通讯等诸多领域都有着广泛的应用。本项目以各类广义Schr?dinger方程、广义短脉冲方程、广义Hunter-Saxton方程、广义Broer-Kaup系统和广义sine-Gordon方程为主要研究对象，以动力系统分支方法为主要研究方法并将它与对称分析方法、首次积分方法和符号计算方法有机结合，研究各种周期波，特别是一些较复杂的周期波如周期尖孤立波、周期紧孤立波、双周期波、混合型周期波等的动力学性质、精确参数表示和极限形式。揭示各种周期波随参数变化的动态特征，进一步明确以上非线性波方程的平面动力系统的各种相轨道与各类非线性波的对应关系，特别是各类非封闭、非连续的周期轨道与各种周期波的对应关系。

中文关键词： 非线性波方程；周期波；动力学性质；精确参数表示；极限形式

英文摘要： Nonlinear wave equation is one of the most popular research topics worldwide, the dynamical properties and the exact parametric representations of the various types of nonlinear wave solutions and their limit forms have applications to a wide variety of fields, from fluid mechanics, condensed matter physics, solid state physics, to biomathematics, fiber optic communications and many others. Our main research topics consist of some nonlinear wave equations, such as different kinds of generalized Schr?dinger equations, generalized short pulse equations, generalized Hunter-Saxton equations，generalized Broer-Kaup systems and generalized sine-Gordon equations etc. Incorporating techniques such as the symmetry analysis method, the first integral method and symbolic methods, we mainly implement the bifurcation technique of dynamical systems to study the properties, which includes dynamics, the exact parametric representations and limit forms, of various types of periodic waves, especially the complex ones, such as periodic peakon, periodic compacton，double periodic wave and compounded periodic wave etc. We shall also reveal the dynamical characteristics of all kinds of periodic wave depending on the parameters, further clarify the correspondence relationship between the orbits of the various planar dynamical systems and the types of nonlinear waves, especially between the non-closed, discontinuous periodic orbits and periodic waves.

英文关键词： nonlinear wave equation;periodic wave;dynamical property;exact parametric representation;limit form