Synthetic Research on nonlinear complete integrable systems and combinatorics

从1997年到2000年，本项目得到了科学研究补助金的支持。我们追求的目标是双重的：从组合学的观点研究非线性完全可积系统，以及用完全可积系统的各种方法来研究组合学理论。具体地说，我们的主要研究内容如下：(A)非线性完全可积系统(简称CIS)的理论研究，(B)该理论在数学科学中的各个领域的应用，(C)完全可积系统的辛性，Painleve方程无疑是非线性可积系统最重要的例子之一。该项目的首席研究员发表了一篇关于Painleve方程的文章，在本报告的参考文献的顶部被引用；本文的前半部分致力于对Painleve方程的结果进行综述，而关于Garnier系统的最新结果则在Hal…中给出与上述每一个主题相对应，我们总结了在本项目推广过程中所取得的结果：(A)对Painleve方程首席研究员所构造的初始条件空间进行了几何解释。利用这一观点，建立了连续型和离散型可积系统的几何刻画和分类。(B)发现了Garnier系统的一种新的族结构，即把Painleve方程推广到多个变量的情形。(C)从数学上建立了Painleve方程的另一族结构；在这种情况下，推广的方程具有与Painleve方程相同的一组对称性。我们可以毫不犹豫地认为，我们目前对非线性完全可积系统的研究在理论和应用方面都取得了丰硕的成果，我们相信这一领域的研究将会取得进展。本研究项目的研究者们继续他们对可积系统的研究，并在不同的情况下公布了他们在1997-2000四年间的研究结果。他们已经在期刊上发表了一些成果。较少

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