浅水波方程模型与奇非线性行波方程解的动力学行为及精确的参数表示:动力系统方法

Models of Shallow Water Wave Equations,Dynamics and Exact Solutions of Singular Nonlinear Traveling Wave Systems:Dynamical System Approach

首先,介绍在重力作用下,无旋、无黏性和不可压缩的流体的表面波传播的一维(或两维单向)浅水波运动的各种不同模型(PDE),这些模型对应的行波系统一般是奇平面动力系统.其次,用著名的广义Camassa-Holm方程作为例子,通过对应的行波系统的精确解来研究该方程的尖孤子、周期尖波、伪尖孤子、伪周期尖波及有界破缺波解的存在性.第三,应用动力系统分支理论和奇摄动几何理论相结合的方法,建立了奇非线性行波方程研究的理论和方法,介绍奇非线性行波动力学行为的2个主要定理,完整地解决了波的光滑性与非光滑性、完整性和破缺性的判定问题.第四,介绍当伴随正则系统直线解上的奇点是结点时,如何用相轨道识别对应的波形,并研究一个非线性水波方程,获得该系统的各型光滑的孤立波和周期波在不同参数条件下的存在性和精确的参数表示.

In this paper,we first introduce various models(PDE)of the evolution of gravity waves on the surface of the inviscid,incompressible fluid under the additional constraint of irrotationality.The corresponding traveling wave systems of these models are singular nonlinear dynamical systems(ODE).Second,as an example,for the generalized Camassa-Holm equation,we derive the exact explicit solitary wave solutions and peakon,periodic peakon,pseudo-peakons,pseudo-periodic peakons as well as compacton solution families.Third,based on the bifurcation theory of dynamical systems and the method of geometric singular perturbations,we give two main theorems.These results tell us that peakon is a limit solution of a family of periodic peakons or a limit solution of a family of pseudo-peakons under two classes of limit senses.The pseudo-peakon and pseudo-periodic peakon family are smooth classical solutions with two time scales.Fourth,we use a well known nonlinear wave equation model to show that for the associated regular system of the singular traveling wave equations,if in a straight line there exist two symmetric nodes,how to use phase orbits to identify the profiles of the nonlinear wave equation and get exact explicit solutions of this system.