高阶非线性方程和奇波方程的精确解分支及其相关研究

With the help of bifurcation theory and computer software such as maple and mathematica, the following subjects will be investigated. (1) For higher-order nonlinear equations, choosing proper transformations and transforming higher-order nonlinear equations into lower-order ones. The existence of homoclinic orbits, heteroclinic orbits and periodic orbits of corresponding lower-order equations will be studied to discuss solutions of higher-order nonlinear equations and derive the parametric expressions of solutions. (2) For the equation with fractional derivatives, based on the properties of fractional derivatives, the proper traveling wave transformation is introduced to transform the fractional equation into integer equation. The exact solutions of the fractional equation will be obtained, and the ideas and formulas for solving the exact solutions of the fractional equation will be established. (3) Using the dynamic system method, the traveling wave solutions and their dynamic properties of Lakshmanan-Porsezian-Daniel model and complex Ginzburg-Landau equation under various non-linear conditions will be discussed. The explicit parametric expressions of smooth and non-smooth traveling wave solutions are obtained by analyzing the phase diagrams of corresponding traveling wave systems under different parameters.

本项目对高阶非线性方程和几类波动方程，借助分支理论，结合maple和mathematica等计算机软件开展如下研究：(1)对于高阶非线性方程，选择恰当的变换，将高阶非线性方程转化为较低阶的形式，利用相关的理论和方法研究相应低阶方程的同宿轨道、异宿轨道和周期轨道等的存在性，进一步探讨高阶非线性方程的解，并求出解的参数表达式；(2)对于具有分数阶导数的方程，基于分数阶导数的性质，引入恰当的行波变换，将分数阶方程转化为整数阶方程进行研究，得到分数阶方程的精确解，建立求解分数阶方程精确解的思路与方法；(3)运用动力系统方法，讨论Lakshmanan-Porsezian-Daniel模型和复Ginzburg-Landau方程在各种不同非线性条件下的行波解及其动力学性质，通过分析相应行波系统在不同参数条件下的相图，获得光滑行波解和非光滑行波解的显式参数表达式。

本项目研究了复Ginzburg-Landau方程、复Kundu-Eckhaus方程、具有抛物型非线性项Lakshman-Porcezian-Daniel模型、Gerdjikov-Ivanov方程、广义Burgers-αβ方程的分支问题和精确解；利用平面动力系统的分支理论，分别得到了相应行波系统的相图，相图随着参数的变化而变化；对应于一些特殊的曲线，给出了不同参数区域下所有可能行波解的参数表达式，这些行波解包括光滑和非光滑行波解。基于平面动力系统的分支理论，研究了求解非线性分数阶和整数阶偏微分方程精确行波解的统一方法，并应用该方法对整数阶BM方程和分数阶BM方程的精确行波解进行了比较。结合动力学系统的分支理论和几何奇异摄动方法，给出了扰动KdV方程存在一个周期解、一个孤立波解以及孤立波解与无穷多个周期解共存的参数条件和波速条件。利用切比雪夫准则分析阿贝尔积分的比值，证明了波速的单调性，得到了波速的上下界。本项目的结果丰富了非线性微分方程精确解及其动力学性质的研究。