奇异孤子的动力学研究

In this project, the exact traveling wave solutions, the bifurcations and dynamical behavior of traveling wave solutions are studied for several classes of extensively used classical nonlinear wave equations, including the generalized Camassa-Holm equation, Degasperis-Procesi equation and nonlinearly dispersive K(m,n) equation. Based on the bifurcation theory of dynamical systems, the effect of double singular straight lines on the singular soliton solutions of nonlinear wave equations is discussed and the relationships among the singular homoclinic (hetereclinic) orbits, compactons and compact kinks are considered. By applying the qualitative theory of differential equation, the dynamical properties of singular solitons are investigated, the conversions among the smooth solitary waves, periodic cusp waves and peakons are discussed and the explicit representations of singular solitons are obtained through different approaches.

本项目研究几类经典的非线性波方程，广义Camassa-Holm方程，Degasperis-Procesi方程以及非线性色散K(m,n)方程的精确行波解、行波解分支及其动力学行为。借助动力系统分支理论分析双奇异直线对非线性波方程的奇异孤子的影响以及奇异同宿异宿轨道和紧孤子与紧扭子的联系。利用定性理论研究奇异孤子的动力学性质，探索光滑孤立波、周期尖波与尖孤子之间的演化机制，并通过不同途径获得各种奇异孤子的精确表达式。

本项目从动力学系统定性理论和分支理论的角度来研究非线性波方程的精确行波解、行波解分支及其动力学行为。首先，我们研究了一类C(3,2,2)方程的尖峰孤立波、钟状孤立波和周期尖峰孤立波的精确表达式及其分支，发现尖峰孤立波实际上是钟状孤立波和周期尖峰孤立波的极限形式。其次，我们研究了可积Novikov方程的奇异行波解分支，得到关于奇异行波解的一个新的有趣现象：即两个尖峰孤立波解 (peakons或cuspons) 在相同波速下是可以共存的，另外我们获得了Novikov方程的静态和周期尖峰波解。通过对非线性波方程的求解和定性分析的研究，有助于人们弄清系统在非线性作用下的运动变化规律，合理解释相关自然现象，更加深刻地描述系统的本质特征。