几类奇异非线性波方程的特殊解及其分支

Nonlinear wave phenomena are of great importance and exist in the physical world, and have been for a long time a challenging topic in depth research. Based on the structural features of the singular nonlinear wave systems, by using the qualitative theory of differential equations, the bifurcation method of dynamical systems and the soliton theory of mathematical physics, this project is aimed at the special solutions and their bifurcations of several singular nonlinear wave equations. Specially, the existence of Peakon solution, Compacton solution, Cuspon solution, periodic Peakon solution, periodic Cuspon solution and other non-smooth solutions will be demonstrated. At the same time, the exact expressions of the special solutions and their bifurcations are provided under the given parameter settings. The bifurcation phenomenon of these special solutions in the equations will be revealed by analyzing the bifurcation portraits in detail and combining the results of numerical simulation. The mechanism will be mentioned to solve the problems that how to understand the dynamics of the special solutions strictly and what is the reason of the smoothness change of the travelling wave solutions.

非线性波现象在现实世界中是普遍存在的，对其深入研究一直是非常重要的且具有一定挑战性的课题。本项目基于奇异行波系统的特有结构的性质，结合微分方程定性理论、动力系统分支方法以及数学物理孤立子理论等工具来研究几类奇异非线性波方程的特殊解及其分支。特别地，本项目拟证明奇异非线性波方程的Peakon 解，Compacton 解，Cuspon 解，周期Peakon 解，周期Cuspon 解以及其它非光滑特殊解的存在性，并给出在一定参数条件下这些特殊解的精确表达式。进一步综合分支相图的详尽分析和数值分析，本项目拟揭示这些方程的若干特殊解所蕴含的分支现象，从而期望构建如何理解这些特殊解的动力学行为以及行波解失去光滑性的机理。