可积系统的代数与几何结构

项目名称： 可积系统的代数与几何结构

项目编号： No.11331008

项目类型： 重点项目

立项/批准年度： 2014

项目学科： 数理科学和化学

项目作者： 耿献国

作者单位： 郑州大学

项目金额： 240万元

中文摘要： 研究与3×3矩阵谱问题相联系的Lax矩阵特征多项式产生的非超椭圆曲线及紧化给出的三叶Riemann面，引入BA函数和带有因子数据的代数函数并探讨它们的性质。建立Abel坐标与连续型和离散型孤子方程族解在原坐标下的关系，导出与3×3矩阵谱问题相联系的连续型和离散型孤子方程族的代数几何解。借助Hankel行列式和Pfaffian技巧研究离散系统的可积性质和代数结构。利用designants技巧和Clifford代数为工具推导新的非交换外推算法，并研究这些算法的奇性规则和cross-rules。用拟行列式技巧寻找对应非交换可积系统的孤子解。构造有限域上新的可积系统并研究它们的代数和几何性质。构造新超可积和超对称系统及其对称、Hamiltonian结构和守恒律，系统地发展构造新超可积方程的方法。构造超对称可积系统的反向变换并研究它们的应用。发展有效方法构造超对称可积系统的显式解.

中文关键词： 可积系统;代数结构;几何结构;;

英文摘要： The main aim of this project is to study algebraic and geometric structures of integrable systems. Based on the characteristic polynomial of Lax matrix for the stationary soliton equation associated with a given 3×3 matrix spectral problem, we shall derive the corresponding non-hyperelliptic curve and a three-sheeted Riemann surface by its compactification. We shall introduce Baker-Akhiezer functions and algebraic functions carrying the data of the divisor and study their properties. We shall develop the methods of constructing three kinds of holomorphic differentials on the Riemann and study the asymptotic behavior at infinities and zeros. The relationship between Abel-Jacobi coordinates and solutions of soliton equations under original coordinates will be established, from which algebro-geometric solutions of certain continuous and discrete soliton equations associated with 3×3 matrix spectral problems may be constructed.The integrability and algebraic structures of explicit solutions for discrete systems will be studied with the help of Hankel determinants and Pfaffian technique. We shall derive new non-commutative extrapolation algorithms by using designants and Clifford algebras and study singular rules and cross rules for the resulting algorithms and further seek soliton solutions for the corresponding non-commutative integrable systems via quasideterminant technique. We also plan to construct new integrable systems over finite fields and study their algebraic and geometric properties. The higher-dimensional integrable equations will be decomposed by using the bilinear method. Based on the theory of integrable systems, a systematic approach will be developed to search for new super integrable systems and supersymmetric ones. We shall construct infinite symmtries, Hamiltonian structures, conserved quantities for these new super integrable systems and supersymmetric ones. The reciprocal transformations for supersymmetric integrable systems may be constructed and employed to study their properies. The effective methods will be developed to obtain their explicit solutions of supersymmetric integrable systems, including soliton solitions and algebro-geometric solutions and others.

英文关键词： integrable systems;algebraic structures;geometric structures