Synthetic Research on nonlinear complete integrable systems and combinatorics

点击翻译按钮获取中文摘要

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