Synthetic Research on nonlinear complete integrable systems and combinatorics

本项目在1997年至2000年期间得到科学研究补助金的支持。我们的目标是双重的：从组合学的角度研究非线性完全可积系统，以及从完全可积系统的角度研究组合学理论的各种途径。本课题的主要研究内容包括：（a）非线性完全可积系统（CIS）的理论研究;（B）非线性完全可积系统在数学科学各个领域的应用;（c）完全可积系统的对称性。该项目的主要研究者发表了一篇关于Painleve方程的文章，引用在本报告的参考文献的顶部;本文的前半部分致力于Painleve方程结果的综述，后半部分给出了Garnier系统的最新结果。 ...更多信息 F.针对上述每一个课题，我们总结了本项目在推进过程中所取得的成果。(a)给出了初始条件空间的几何解释，该空间是由Painleve方程的首席研究员构造的。利用这一观点，不仅对连续型可积系统，而且对离散型可积系统，都建立了一个几何特征和分类。(b)对Garnier系统发现了一种新的层次结构，它被称为Painleve方程到多变量情形的推广。我们相信，前者承认后者的一般解决方案，作为特殊的孤子。(c)Painleve方程的另一种层次结构已经在数学上建立起来;在这种情况下，扩展方程具有与Painleve方程相同的对称群。我们可以毫不犹豫地坚持认为，我们目前对非线性完全可积系统的研究正在为该学科的理论和应用带来丰硕的成果，我们确信这一领域的研究正在取得进展。本研究项目的研究人员继续他们对可积系统的研究，并在不同的场合宣布了他们在1997-2000年四年期间所获得的结果。他们已经在期刊上发表了一些结果。少

The present project has been supported by Grant-in-Aid for Scientific Research from 1997 to 2000. The aim of our pursuite is double: studies on nonlinear completely integrable systems from viewpoints of combinatorics, and various approaches to theory of combinatorics in terms of completely integrable systems. In particular, our main subjects of this research project are listed as follows:(a) theoretical investigation on nonlinear completely integrable systems (CIS, in short) ,(b) application of the theory to various domains in mathematical sciences,(c) symmmeries of completely integrable systems,The Painleve equations are surely one of the most important examples of nonlinear inteegrable systems. The head investigator of the project has published an article on the Painleve equations, cited at the top of references of this reprt; the former half of this paper is devoted to an survey of results on the Painleve equations and recent results on the Garnier sytems are given in the latter hal … More f. Corresponding to each of the subjects mentioned above, we make a summary of results, obtained during promotion of the present project.(a) A geometrical interpretation is given to the space of initial conditions, which had been constructed by head investigator for the Painleve equations. By the use of this viewpoint, a geometrical characterization and classification are established for integrable systems, not only of continuous type but also of discrete one.(b) A new structure of hierarchy is discovered for the Garnier systems, which are known as extension of the Painleve equations to several variable cases. We persuade that the former admits the general solution of the latter, as special solitions.(c) Another structure of hierarchy for the Painleve equations has been established mathematically; in this case, extendes equations have the same group of symmetries as that of the Painleve equations. We can insist without hesitation that our present investigation on nonlinear completely integrable systems is giving fruitful results to theories and applications of the subjects, and we convince ourselves of development of researches on this domain.The investigators of this research project have continued their studies on integrable systems and announced their own results obtained during four years, 1997-2000, in various occasins. They have published some of results in journals. Less