一类高阶非线性波方程的行波解研究

发布时间：2018-02-28 07:55

本文关键词： 行波解非线性波方程动力系统分支理论不变流形　出处：《浙江理工大学》2017年硕士论文　论文类型：学位论文

【摘要】：行波解是一种广泛存在于各类非线性方程中的一种相似解,其典型的特征是这类解在空间传播中能够保持平移不变。许多在实验下观察到的物理、化学和生物现象都可以用方程的行波解来描述,而这些方程往往都是非线性方程,比如用来描述我们在日常生活中经常会遇到的各种各样的水波、声波、电磁波等非线性波方程的行波解描述了这些波在媒介中的传播过程。认识和发现这些非线性方程所蕴含的各种波的内在机理成为当今非线性波研究领域中理论研究与数值分析的重要课题。在非线性科学研究中,动力系统理论和方法由于其理论的深刻性与应用的广泛性已成为相关非线性科学领域中非常活跃的前沿方向之一,因此把动力系统理论和方法应用于非线性波方程行波解的研究具有广阔前景。本文首先研究了一类Ito五阶mKdV方程,其对应的行波方程只能约化为含有参数的四阶常微分方程,对于所对应的四阶行波系统,借助计算机符号计算,在某些参数条件下我们得到了这个含参四维系统的由平面动力系统确定的二维不变流形,通过利用平面动力定性分析和分支理论研究确定了这个二维不变流形的二维系统在各类参数条件下的分支和精确解,从而得到了这类高阶非线性波方程的各类光滑的有界行波解,其中包括孤波解、扭波解和不同振幅的周期波解。其次,研究了一类复mKdV方程,利用平移、旋转和尺度对称将方程转化为在一定参数条件下实的参数方程,该方程是一个二阶常微分方程,我们通过利用动力系统定性和分支理论分析了该二阶常微分方程所对应平面系统的相图和分支,得到了该方程在各种参数条件下的有界解,从而得到了该复mKdV方程的各类行波解,其中包含了振幅为周期函数的包络解。
[Abstract]:Traveling wave solutions are similar solutions widely found in all kinds of nonlinear equations. The typical characteristic of traveling wave solutions is that they can keep their translation invariant in space propagation. Both chemical and biological phenomena can be described by traveling wave solutions to equations that are often nonlinear, such as the various water waves, sound waves that we often encounter in our daily lives. The traveling wave solutions of nonlinear wave equations such as electromagnetic waves describe the propagation process of these waves in media. Understanding and discovering the intrinsic mechanism of various waves contained in these nonlinear equations has become a theoretical study in the field of nonlinear wave research today. Research and numerical analysis. In nonlinear scientific research, The theory and method of dynamic systems have become one of the most active frontier directions in the field of nonlinear science because of its deep theory and extensive application. Therefore, the application of dynamic system theory and method to the study of traveling wave solutions of nonlinear wave equations has broad prospects. In this paper, we first study a class of Ito fifth order mKdV equations, the corresponding traveling wave equations can only be reduced to fourth order ordinary differential equations with parameters. For the corresponding fourth-order traveling wave system, with the aid of the computer symbolic calculation, we obtain the two-dimensional invariant manifold of the four-dimensional system with parameters determined by the plane dynamic system. By using the qualitative analysis of plane dynamics and bifurcation theory, the bifurcation and exact solutions of this two-dimensional invariant manifold system under various parameter conditions are determined. Thus, all kinds of smooth and bounded traveling wave solutions of this kind of high order nonlinear wave equations are obtained, including solitary wave solutions, torsional wave solutions and periodic wave solutions with different amplitudes. Secondly, we study a class of complex mKdV equations, using translation, Rotation and scale symmetry transform the equation into a real parametric equation under certain parameter conditions, which is a second-order ordinary differential equation. By using the qualitative and bifurcation theory of dynamical system, we analyze the phase diagram and bifurcation of the plane system corresponding to the second order ordinary differential equation, and obtain the bounded solution of the equation under various parameter conditions. The traveling wave solutions of the complex mKdV equation are obtained, including the envelope solution of the periodic function of the amplitude.
【学位授予单位】：浙江理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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