Camassa-Holm型方程的相关研究

The project is mainly focused on reciprocal transformations and bi-Hamiltonian structures as well as peakon solutions of Camassa-Holm(CH) type equations, and construction of some new 2+1，3+1 dimensional CH type equations. It contains getting bi-Hamiltonian structures of some CH type equations which spectral problems are known; constructing reciprocal transformations for some CH type equations to connect the equations with some classical integrable systems admitting physically significant; obtaining some new high-dimensional CH type equations and their spectral problems by tri-Hamiltonian duality method, solving the multi-peakon solutions of some CH type equations.

本项目主要研究Camassa-Holm（CH）型方程的反向变换、双Hamilton结构、尖峰解的求解问题以及新的2+1、3+1维CH型方程的构造。 即研究若干谱问题已知的CH型方程的双哈密顿结构；利用reciprocal变换建立若干CH型方程与已知具有物理意义的经典系统之间的联系；利用三哈密顿对偶方法构造一些新的高维CH型方程并得到其谱问题；求解若干CH型方程多尖峰解。

该项目研究了CH型方程的互反变换、Hamilton结构以及多孤子解。首先找了一种系统的办法将CH型方程通过互反变换与约束KP族负一流或其修正系统联系起来（这些约束KP族负一流或修正系统往往具有一定的物理意义），从而间接证明CH型方程的Painlevé性质，同时我们还考虑了Hamilton结构在互反变换下的变化情况；其次利用CH型方程谱问题的特点，我们构造了一个新的3*3谱问题，导出一个新的3-DP方程并研究了其可积性和短波极限；最后我们找到了一个利用互反变换和达布变换的统一方法构造CH型方程的多孤子解。