带有三次非线性项和尖峰解的Camassa-Holm类型的方程和系统的若干问题研究

In this project we mainly study several problems of Camassa-Holm type equations and systems with cubic nonlinearity and peakons. Concerning the Camassa-Holm equation and the Degasperis-Procesi equation, we mainly study some open problems on them. As for the Novikov equation with cubic nonlinearity, the modified Camassa-Holm equation with cubic nonlinearity, the generalized CH equation and two-component CH system with quadratic and cubic nonlinearity, the three-component CH system with cubic nonlinearity, the four-component CH system with cubic nonlinearity, the two-component Novikov system with cubic nonlinearity, the four-component Novikov system with cubic nonlinearity, we mainly investigate the Cauchy problem of these equations and systems, such as, local well-posedness, global existence and blow-up phenomena of strong solutions, the existence and uniqueness of global weak solutions, the existence and semigroup property of conservative weak solutions and disspiatice weak solutions, and the orbital stabilities of peakons and cuspons. The Camassa-Holm equation and the Degasperis-Procesi equation are two important shallow water wave equations which are completely integrable and can exhibit both phenomena of soliton and wave breaking. In recent years, they have been studied extensively. Our research in this project on the above mentioned problems of Camassa-Holm type equations and systems with cubic nonlinearity and peakons will help us to characterize deeply two important phenomena of soliton and wave breaking from different mathematical points of view. Therefore, our research of this project will be very important and useful in theories and applications.

本项目主要研究带有三次非线性项和尖峰解的Camassa-Holm类型的方程和系统的柯西问题. 关于CH方程和DP方程,主要研究与之有关的未解决的问题.关于带有三次非线性项的Novikov方程,修正的CH方程,带有二次和三次非线性项的广义CH方程和系统，三个分量的CH系统，四个分量的CH系统,两个分量的Novikov系统,四个分量的Novikov系统, 主要研究这些方程和系统的柯西问题的局部适定性,强解的爆破和整体存在性,整体弱解的存在唯一性,守恒弱解和耗散弱解的存在性和半群性质，尖峰解和扭结解的轨道稳定性. CH方程和DP方程是能描述孤立子和波破裂现象的重要可积浅水波方程,近年来得到了广泛的关注和研究.本项目拟对带有三次非线性项和尖峰解的CH类型的方程和系统的上述问题研究，有助于我们从不同的数学角度对孤立子和波破裂现象加以描述和刻画，因而有重要理论和实际意义.

本项目主要研究带有三次非线性项和尖峰解的Camassa-Holm类型的方程和系统的柯西问题. 关于CH方程和DP方程,主要研究与之有关的未解决的问题.关于带有三次非线性项的Novikov方程,修正的CH方程,带有二次和三次非线性项的广义CH方程和系统，三个分量的CH系统，四个分量的CH系统,两个分量的Novikov系统,四个分量的Novikov系统, 主要研究这些方程和系统的柯西问题和初边值问题的的局部适定性和不适定性，强解的爆破和整体存在性，整体弱解的存在性和唯一性，全局守恒弱解的存在性和唯一性，解的Gevrey正则性和衰减性，解析解的存在性，尖峰孤立子解的存在性,以及尖峰孤立子解的轨道稳定性等问题进行了细致深入的研究，推广和改进了原有的理论，并在所研究的各个方面都取得了比较大的突破和进展。所得的研究成果从数学角度上对尖峰孤立子和波破裂现象这两个重要的浅水波领域的物理现象有细致的刻画和深入的理解。