具有peakon解的可积模型，蕴含丰富的数学与物理结构，近年来的研究十分活跃。本课题将研究具有peakon、kink以及kink-peakon相互作用解的可积系统的构造和精确求解。主要研究内容包括：1）新的具有peakon、kink以及kink-peakon相互作用解的可积系统的构造； 2）两peakon系统、kink-peakon相互作用系统的求解与碰撞研究；3）N-peakon系统以及kink-peakon相互作用系统的可积性；4）该类系统的光滑孤立子解、达布变换、Lax对非线性化与代数几何解。

In recent years, the integrable systems with peakon solutions have attracted much attention in soliton theory. In this project, we will study integrable systems with peakon, kink, and kink-peakon interactional solutions. The studies mainly focus on: 1) constructing new integrable systems with peakon, kink, and kink-peakon interactional solutions; 2) the solutions of two-peakon and kink-peakon interactional dynamic systems, and the collisions between two-peakon and kink-peakon; 3) integrability of N-peakon and kink-peakon interactional systems; 4) smooth soliton solutions, Darboux transformations and algebra-geometric solutions of these systems.