MCH方程能量守恒的N-peakon解

孤立波在不同领域，如等离子体、声学、液晶物理学等，有着极其重要的应用。本项目的研究对象是modified Camassa-Holm(mCH)方程，该方程是具有特殊孤立波弱解的可积系统，其孤立波解被称为“尖峰孤子解”（即peakon解）。本项目主要研究mCH方程能量守恒N-peakon解的相关性质，具体研究内容如下：（1）拟采用黏性粒子方法研究能量守恒N-peakon解的全局存在性；（2）利用磨光特征方程的方法给出能量守恒N-peakon解唯一性的选择原理，研究此类弱解的长时间行为，并证明其渐近稳定性；（3）研究基于能量守恒N-peakon解的粒子方法的收敛性问题；最后把得到的相关结果，推广到广义mCH方程上。

Solitary waves have a lot of applications in different areas such as plasma, acoustics, liquid crystal, etc.. In this project, we are going to study an integrable system with special solitary waves as weak solutions, which is referred to as the modified Camassa-Holm (mCH) equation. The solitary waves of the mCH equation are known as peakon solutions. We are going to study the properties of the conservative N-peakon solutions to the mCH equation. Our main targets are as follows: (1) We plan to use a sticky particle method to show the global existence of the conservative N-peakon solutions. (2) A selection principle for the uniqueness of conservative N-peakon solutions will be provided via a mollification of the characteristics equation. The long time behavior of these solutions will be investigated, and we plan to obtain the asymptotic stability of N-peakon solutions. (3) Based on the conservative N-peakon solutions, we are going to prove the convergence of the particle method. At last, all the results obtained for the mCH equation will be studied for the generalized mCH equation.