Mathematical Studies on the Painleve equations

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The aim of present project is to study in synthetic manner the Painleve equations and the Garnier systems from many different points of view ; the latter is a generalization in the case of several variables of the former. In other words, we pursue studies on the Painleve equations with analytic, geometric and algebraic method and magnify, based on results obtained concerning Painleve equations, our research to investigation on general integrable systems, such as the Ganier system, discrete and q-difference Painleve equations. As for analytic behavior of solutions of Painleve equations, we have obtained new results by using Nevanlinna theory on meromorphic functions. The space of initial conditions of Painleve equations was constructed by K. OKAMOTO ; recently an analogous object has been considered for the Garnier systems and this gives us a geometric aspect on studies of the Garnier systems. From an algebraic view point of nonlinear integrable systems, we have studied birational canonical transformations of the Garnier systems by exteoding results on those of Painleve equations.At the beginning of pursuit of present project, we studied algebraic transformation of Painleve equations, called folding transformations of the space of initial conditions, we have arrived at the final result, which are published in the first articles cited in the list of publications. By means of geometrical researches on Painleve equations, done by H.SAKAI, the third Painleve equations has two special degenerate types and it is natural to consider the eight types of Painleve equations from a geometric point of view. We have completed studies on Painleve equations by considering the Hamiltonian structure of these two types of equations ; the space of initial conditions, the group of birational canonical transformations and special solutions are given in the second article of the list of the next page.