多分量可积系统的Reciprocal变换和Backlund变换

In this project, we shall study the integrability and the explicit solutions for the multi-component integrable systems based on the Reciprocal and B?cklund transformations. Firstly, we study the two-component quasilinear higher-order systems of evolution equations of the most general form which are transformed into the two-component semilinear systems by utilizing a class of Reciprocal transformations. All systems admittting such transformations are completely classified. We establish the connection between the two-component quasilinear higher-order systems and some known semilinear integrable systems, and illustrate the relationship between the two-component integrable systems by combinations of a series of transformations including the Reciprocal transformations. Next, we give a complete classification for the two-component C-integrable quasilinear systems of evolution equations based on the Hodograph-type transformations. In addition, we construct the explicit solutions for the Cauchy problem of the two-component quasilinear systems. Thirdly, we exploit the complete classfication for the two-component third-order nonlinear systems of evolution equations based on the B?cklund transformations. Finally, we demonstrate how to find new multi-component bi-Hamiltonian integrable systems in terms of the B?cklund transformations. Topics will be investigation for the new bi-Hamiltonian integrable systems including their auto-B?cklund transformations, the exsistence of the soliton solutions and the analysis of the associated scattering problems which are based on their bi-Hamiltonian structure, hereditary symmetries and the Lax pairs and so on.

本项目主要针对多分量可积系统，利用Reciprocal 变换和Backlund 变换，研究其可积性质和精确解。首先，基于非局部Reciprocal 变换给出二分量高阶拟线性发展方程组到半线性发展方程组的完全分类；建立二分量高阶拟线性发展方程组与已知半线性可积系统之间的联系；通过以Reciprocal 变换为核心的系列变换，建立二分量可积系统之间的联系。其次，基于点变换形式Hodograph变换，给出C-可积的二阶拟线性发展方程组的完全分类；构造二分量拟线性发展方程组柯西问题的精确解。第三，研究允许Backlund 变换的二分量三阶非线性发展方程组的完全分类。最后，由已知的多分量双哈密顿系统，利用Backlund 变换寻找新的多分量可积模型；研究其自Backlund 变换；分析其孤立子解存在性和形成机理；探讨其双哈密顿结构、遗传对称、Lax对等可积性质，以及在此基础上的散射和反散射问题。

本项目的研究领域是数学物理，研究方向是可积系统及其应用，主要研究内容为可积方程族的Liouville相关性；基于结合Bäcklund变换和tri-Hamiltonian对偶理论的方法，多分量可积系统的构造及其行波解的分类；不同几何中若干可积曲线流的Bäcklund变换的几何结构；几种多分量CH类系统可积性质和精确解。已取得的重要结果和科学意义体现在以下四个方面。首先，系统地研究了几类具有典型非线性特征和非局部效应的可积方程族的Liouville相关性问题，分别证明了mCH和mKdV可积方程族、Novikov和SK可积方程族、DP和KK可积方程族、short-pulse和sine-Gordon可积方程族之间的Liouville相关性，发现了mCH和CH方程、DP和Novikov方程之间的非平凡内在联系。其次，提出了结合Bäcklund变换和tri-Hamiltonian对偶理论的方法，该方法为构造支持tri-Hamiltonian对偶结构的非线性可积系统提供了可行方案。研究了具有典型物理背景的两分量和三分量可积的色散水波系统和修正的色散水波系统的对偶可积结构，得到了多个新的对偶可积系统，并给出了行波解的分类，发现了几种新的具有间断点的Cuspon型非光滑孤立波。第三，系统研究了不同几何中若干可积曲线流的Bäcklund变换的几何结构，得到了联系两组可积曲线流方程的Bäcklund变换，并利用某些显式的Bäcklund变换，构造相应可积方程的精确解。最后，研究了非局域µCH系统的非局部对称和守恒律；利用不变子空间结合分离变量以及扰动的方法研究了两分量b族CH类系统自相似解的结构、解的全局存在性以及有限时间爆破的性质。