多分量可积系统的Reciprocal变换和Backlund变换

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。

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.