非线性可积系统的对偶系统包括Camassa-Holm方程，修正的Camassa-Holm方程，对偶薛定谔方程和两份量的Camassa-Holm方程等。由于这些方程（或方程组）是可积的，但具有非光滑的孤立波解如尖峰孤子解和cuspon解，近年来引起了数学物理和偏微分方程领域学者的广泛关注。本项目首先将发展Olver和Fuchssteiner等人的三重 Hamilton对偶方法和Lie-Backlund对称方法来分类具有非光滑孤子解的非线性色散可积系统；研究这些新的可积系统的对称、几何可积性和物理背景, 深入探讨它们和某些几何中不变几何流的密切联系。研究这些可积系统的非光滑孤子解的存在性、结构和稳定性，探讨非光滑孤子解形成的机理。讨论其初值问题的适定性问题，包括逆散射方法，强解的局部和整体适定性，弱解的存在性等。研究这些方程解的奇性，寻找解爆破的机制，给出解爆破和波破缺的充分或必要条件。

Dual integrable systems include the well-known Camassa-Holm equation, the modified Camassa-Holm equation, a novel Schrodinger equation and the two-component Camassa-Holm system,etc. These equations have attracted much attention among the experts in the areas of mathmeatical physics and partial differential equations. In this project, we first develop the tri-Hamilton duality approach due to Olver， Rosenau and Fuchssteiner and the Lie-Backlund symmetry method to classify dual integrable systems with non-smooth solitary wave solutions, so that some new dual integrable systems will be obtained. We also study the local and nonlocal symmetries, physical background and geometric integrability of these integrable systems. Furthermore, we study the relationship between these systems and invariant geometric flows in certain geometries. Second, we study the existence of non-smooth solitary waves, their structure and dynamical stability. It is of our interest to explore the formation mechanism of non-smooth solitary waves. Third, we study the well-posedness of the Cauchy problem for these systems, which include the inverse scattering approach, the existence of local and global strong solutions, and the existence of weak solutions of these systems. Finally, the singularities of the solutions to the new dual integrable systems will be investigated. Blow up criterion and wave breaking mechanism for certain initial profiles will be given.