Camassa-Holm型方程及其短波极限方程的可积离散

The project mainly focuses on integrable discretizations of the Camassa-Holm type equations and their short-wave limits. The models that we will study mainly include the Camassa-Holm equation, the Degasperis-Procesi equation, the Hunter-Saxton equation, the Vakhnenko equation, some multi-component Camassa-Holm type equations and so on. We will apply the following methods to construct their integrable discretizations: firstly, based on the Lax pairs of the continuous Camassa-Holm type equations and their short-wave limits, we seek the integrable discretizations with equal step length using the algebraization of the difference operator; secondly, based on the integrable discretizations of the classical solition equations related to the continuous Camassa-Holm type equations and their short-wave limits through the reciprocal transformations, we present the integrable discretizations of them by the discrete reciprocal transformations; thirdly, based on the integrable discretizations of the classical solition equations which are the dual versions of the Camassa-Holm type equations,we propose the integrable discretizations of them through the discrete generalization of the tri-Hamiltonian duality method.

本项目主要研究Camassa-Holm型方程及其短波极限方程的可积离散。研究的模型主要包括Camassa-Holm方程、Degasperis-Procesi方程、Hunter-Saxton方程、Vakhnenko方程及多分量Camassa-Holm型方程等。我们主要采用以下手段可积离散：一是从连续Camassa-Holm型方程及其短波极限方程的Lax对出发，采用差分算子代数化离散，寻找它们的等步长的可积离散格式；二是从与连续Camassa-Holm型方程及其短波极限方程通过reciprocal变换联系的经典孤子方程的可积离散格式出发，利用半离散reciprocal变换建立它们的可积离散格式；三是从连续Camassa-Holm型方程的对偶方程的可积离散格式出发，经三哈密顿对偶方法的离散推广推导它们的可积离散格式。

本项目主要研究了Camassa-Holm型方程及其短波极限方程的可积性质和孤子解。一方面通过对已有的Camassa-Holm型方程的Lax对空间部分做矩阵延拓得到新的多分量Lax对及对应的多分量Camassa-Holm型方程，进而研究这些多分量方程的可积性质、约化和对偶方程。另一方面通过reciprocal transformation和Darboux transformation提出了求某些 Camassa-Holm型方程及其短波极限方程孤子解的简单方法，并通过reciprocal transformation寻找这些Camassa-Holm型方程的可积离散。