一类非线性波方程的光滑与非光滑行波解

The Smooth and Nonsmooth Travelling Wave Solutions in a Nonlinear Wave Equation

讨论了一类偏微分方程的行波解· 该方程的行波方程对应于一个平面三次多项式系统 ,因而可将行波解的研究化为对平面系统所定义的相轨线的拓扑分类研究· 应用平面动力系统理论在三参数空间内作定性分析 ,首先获得三次多项式系统的完整拓扑分类 ,再将相平面分析的结果返回到非线性波解u(ξ) · 考虑到解关于变量 ξ =x-ct在“奇线”近旁的不连续性 ,可得到各种光滑与非光滑行波的存在条件·

The travelling wave solutions (TWS) in a class of P.D.E. is studied. The travelling wave equation of this P.D.E. is a planar cubic polynomial system in three-parameter space. The study for TWS became the topological classifications of bifurcations of phase portraits defined by the planar system. By using the theory of planar dynamical systems to do qualitative analysis, all topological classifications of the cubic polynomial system can be obtained. Returning the results of the phase plane analysis to TWS, u(ξ), and considering discontinuity of the right side of the equation of TWS when ξ=x-ct is varied along a phase orbit and passing through a singular curve, all conditions of existence of smooth and nonsmooth travelling waves are given.