伪球面上的可积簇及其非局部对称和几何性质研究

Based on the inextensional curvilinear (toroidal) motion in centre geometry, we reveal, from the positive and negative aspects, the relationship between 1-component integrable hierarchy and pseudospherical surface, and further research the analogues in the situation of N-component integrable hierarchy (N>1), based on which, the methods of differential geometry are extended to the area of N-component integrable system (N>1). By means of local equivalent transformation, we propose an effective method to extend the quadric potential related with geodesic in pseudospherical surface, and construct some nonlocally related PDE(s) systems and tree structures of the integrable hierarchy. Compared with nonlocal symmetries obtained by classic methods, then we get some new ones. Furthermore，for the given integrable hierarchy, we theoretically present an effectiveness rule, which can be used to determine whether nonlocally related PDE(s) systems can generate some useful nonlocal symmetries (conservation laws). Under the rule, we further propose an effective method for obtaining nonlocally symmetries(conseration laws), which can improve the efficiency of the original algorithm. According to the properties of nonlocal symmetries (conservation laws), one can present the analysis of invertible transformation, linearizable transformation, analytical solution algorithm and geometrical integrabilities for some kinds of integrable hierarchies. Finally, through improving and perfecting the applications of nonlocal symmetries in integrable system, the nonlocal methods can be further extended to the area of applications in pseudospherical surface.

基于中心几何上的非伸缩曲线(曲面)运动，本课题致力于从正、反两方面揭示1-分量可积簇与伪球面之间的紧密刻画关系，并将研究结果推广到N-分量情形(N>1)，从而扩展微分几何在N-分量可积系统中的应用范围(N>1)。 通过局部等价变换，提出一种能够推广伪球面上与测地线相关联的二次势的有效方法，用于构造可积簇的非局部PDE(s)系统及其树形结构，从而可以获得之前不能得到的非局部对称。 此外，对给定的可积簇，从理论上给出其非局部PDE(s)系统对获取非局部对称(守恒律)有效性的判定法则，进而给出获取非局部对称(守恒律)的有效算法，从而提高原算法的运算效率。 根据非局部对称(守恒律)的性质，给出几类可积簇及其之间的可逆性变化分析、可线性化分析、非局部解析算法和几何可积性质等。最后，通过完善和改进非局部对称在可积系统中的应用，进一步推广非局部方法在伪球面上的应用范围。

单分量可积簇在可积系统系统研究领域被国际数学物理学家广泛研究，但是多分量可积簇的研究远远不足。我们按照课题的原计划进行研究，成功地将单分量的结果推广到多分量可积簇中去。在多分量可积簇的几何性质、非局部PDE(s)系统及其树形结构、非局部PDE(s)系统有效性的判定法则、及非局部对称的各种应用等几方面开展了深入的研究，获得了一些新的研究成果。此外，我们进一步研究了多分量可积簇的保对称离散格式、分数阶李对称及相关初边值问题。在项目组成员的共同努力下，发表相关SCI学术性论文18篇，撰写20余篇。课题方向上培养硕士研究生11名。同时，出访从事合作研究或参加国内外学术会议近10次。这些研究成果将对进一步研究可积系统及相关几何不变量等理论具有一定的促进作用。

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