基于定性和分岔理论分析的高阶非线性波方程动力学研究

The dynamical system approach has been widely applied to study the existence, bifurcation and the exact traveling wave solutions of the nonlinear differential equations which can be reduced to planar Hamiltonian system or integrable systems. It has also been used to show the real reason why some smooth systems possess non-smooth traveling wave solutions. However, many problems in the real-world are modeled by high-order nonlinear wave equations which correspond to high-dimensional dynamical systems or non-integrable systems. In this project, based on the qualitative and bifurcation theorems of dynamical systems, the dynamical behaviors of high-order nonlinear wave equations, with the help of computer algebraic systems and symbolic computation, are investigated by using symmetry analysis and asymptotic analysis. At first, the symmetries of certain high-order nonlinear wave equations are studied and then these are used to transform the nonlinear wave equations to nonlinear ordinary differential equations (ODEs). The nonlinear wave equations can be classified by the invariant manifold of the reduced equations which can be constructed by the symmetries or other methods. Finally, after investigating the flow on the invariant manifold via applying the qualitative and bifurcation analysis, varieties of exact bounded traveling wave solutions, self-similar solutions, other invariant solutions and even some multi-wave solutions of the high-order nonlinear wave equations can be obtained. The dynamical behaviors such as bifurcation and chaos of the high-order nonlinear wave equations will be considered in the same way. We aim to develop methods to study the dynamical behaviors of high-order nonlinear wave equations on their invariant manifolds via qualitative and bifurcation theorems. The results of this project will build up theoretical foundation and provide scientific support to explore the transportation of the nonlinear waves, the very common physical phenomena in our daily life and in many scientific fields.

动力系统理论方法已被有效地研究可约化为平面哈密顿系统或可积系统的一些非线性偏微分方程行波解的存在性、分岔和精确行波解，完美解释了光滑非线性波方程出现非光滑奇异行波解的内在原因，然而实际应用中大量高阶非线性波模型方程对应于高维动力系统或非可积系统。本项目基于动力系统定性和分岔理论分析，综合运用对称性分析和渐近分析方法，借助计算机符号计算研究高阶非线性波方程的动力学行为。通过研究经典高阶非线性波方程的对称，利用对称约化方程并构造其不变流形，根据方程蕴含的不变流形将方程进行分类研究，并在得到的不变流形上利用定性和分岔理论研究非线性波方程的各类精确有界行波解、相似不变解等群不变解和多孤子解的精确表达式及其分岔和混沌等动力学行为，进而发展在不变流形上利用定性和分岔理论研究高维非线性动力系统动力学行为的方法。本项目的研究结果将为探索广泛存在于日常生活和科学应用领域中的非线性波传播机理提供严格理论支持。

动力系统理论方法已被有效地应用于研究可约化为平面哈密顿系统或可积系统的某些非线性偏微分方程行波解的存在性、分岔和精确行波解，并完美解释了光滑非线性波方程出现非光滑奇异行波解的内在原因，然而实际应用中大量高阶非线性波模型方程对应于高维动力系统或非可积系统。本项目基于动力系统定性和分岔理论分析，综合运用对称性分析和渐近分析方法，借助计算机符号计算研究高阶非线性波方程的动力学行为。通过研究经典高阶非线性波方程的对称，利用对称约化方程并构造其不变流形，根据方程蕴含的不变流形将方程进行分类研究，并在得到的不变流形上利用定性和分岔理论研究非线性波方程的各类精确有界行波解、相似不变解等群不变解和多孤子解的精确表达式及其分岔和混沌等动力学行为，进而发展在不变流形上利用定性和分岔理论研究高维非线性动力系统动力学行为的方法。.本项目研究在不变流形上利用定性和分岔理论以及对称性分析研究高维非线性动力系统动力学方面取得了预期的研究成果，研究了五阶Ito方程等高阶非线性波方程的行波解，KdV- Sawada-Kotera-Ramani方程的精确拟周期行波解和二波解，研究了(2+1)-维BKK方程的行波解、双 Wronskian解及其守恒律；通过研究一类广义的Camassa-Holm-Novikov方程和一类非线性色散水波方程分析了奇异线对高阶非线性波方程行波解的影响，得到了具有奇异线的非线性系统的奇异行波解存在的充要条件以及其非唯一性；本项目的一个重要结果是通过研究具有奇异扰动的mKdV方程发现了非线性波在奇异扰动后会出现一类既有波峰又有波谷的新孤立波解，该结果极大刷新了对高阶非线性波的认识。此外在利用几何奇异慑动研究通过离子通道的离子流动力学研究方面做了一些尝试，初步取得了可喜的研究成果。本项目的研究结果将为探索广泛存在于日常生活和科学应用领域中的非线性波传播机理提供严格理论支持。