尖峰孤子相关的若干等谱及非等谱问题

“Peakon” is one of the most important discoveries within the field of integrable systems in the past two decades. The search for new integrable dynamic models with peakon solutions (peakons) and constructing explicit formulae for peakons are fundamental subjects in this topic. This program is devoted to the exploration on these two aspects and related subjects. Noticing that there exist some equations, whose peakons are still unknown, and almost all of the integrable systems with peakons are of isospectral evolution in the literature, we plan to focus on three parts: (i) solving the peakons for some known integrable systems explicitly by use of inverse spectral method and analyzing the dynamics of the peakons with the help of the obtained explicit formulae; (ii) investigating the orthogonal polynomials and approximation problems appearing in the inverse problems of solving peakons, and related integrable systems; (iii) searching for new nonisospectral integrable systems with peakons. This program has important theoretical significance. It is expected to enrich the contents on peakons and will be helpful for better understandings on peakons.

尖峰孤子是近二十年可积系统领域重大发现之一。在相关的研究中，寻找新的具有尖峰孤子解的可积动力模型以及给出尖峰孤子解析解一直是基本且重要的课题，本项目将围绕这两方面及相关问题进行探索。本项目主要分为三部分：（一）、运用反谱方法研究一些可积动力系统的尖峰孤子解的精确构造，分析其尖峰孤子的动力性质；（二）、由于反谱问题中经常遇到某种正交多项式、逼近问题，本项目将深入探索新问题中的正交多项式、逼近问题以及相关的可积系统；（三）、文献中几乎所有的具有尖峰孤子解的方程都是等谱演化方程，本项目将寻找新的具有尖峰孤子解的非等谱可积系统。 本项目的开展将丰富尖峰孤子方面的结果，有助于对尖峰孤子更进一步的理解，具有重要的理论意义。

本项目主要围绕几个尖峰孤子相关的等谱及非等谱问题进行了探索，在修正Camassa-Holm方程尖峰孤子以及两分量修正Camassa-Holm方程的overlapping尖峰孤子相关的等谱问题及非等谱问题、Degasperis-Procesi方程和Novikov方程尖峰孤子相关的等谱问题等方面取得了若干重要进展，主要结果发表在知名学术刊物上。本项目的实施极大的丰富了尖峰孤子方面的结果，必然有助于对尖峰孤子更进一步的理解，具有重要的理论意义。