研究集会のご案内

11月26日(月)～30日(金) に開催されるRIMS研究集会「パンルヴェ方程式を
めぐる諸相」の "satellite seminar" として、その前後の週に、上記
研究集会の参加者の中で特にパンルヴェ方程式の漸近解析の専門家の方々に、
京都大学数理解析研究所において下記のような講演をお願い致しました。
興味をお持ちの皆さんにお集まり頂ければ幸いです。

どうぞよろしくお願い申し上げます。

京都大学数理解析研究所
竹井 義次
(tel) 075-753-7249
(fax) 075-753-7272
(e-mail) takei@kurims.kyoto-u.ac.jp

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``Seminar on Singular Perturbations''

〈第１回〉

Date    :  November 19 (Mon), 16:00 -- 17:30
Place   :  Room No. 206 of RIMS, Kyoto Univ.
Speaker :  Dr. Davide Guzzetti (SISSA-ISAS, Trieste, Italy)
Title   :  A review of the sixth Painlev\a'e equation

〈第２回〉

Date    :  December 3 (Mon), 13:30 -- 17:30
Place   :  Room No. 206 of RIMS, Kyoto Univ.

13:30 -- 14:30 : Victor Novokshenov (Ufa, Russia)
     Special functions and isomonodromic deformations

15:00 -- 16:00 : Andrei Kapaev (SISSA, Trieste, Italy)
     On the Riemann-Hilbert approach to the Malgrange divisor: $P_I^2$ case

16:30 -- 17:30 : Shingo Kamimoto (RIMS, Kyoto, Japan)
     On the decomposition of WKB solutions to monomially summable series

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以下は、各講演のアブストラクトです。

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〈第１回〉

◆ D. Guzzetti : "A review of the sixth Painlev\'e equation"

The isomonodromy deformation method provides a unitary description of the
critical behaviors of the solutions of the Painlev\'e 6 equation, their
connection formulae and the asymptotic distribution of the poles close
to a critical point. I will discuss the results known on the subject,
including those which I have obtained during my stay in RIMS as a COE
fellow (2004-8), and in KIAS in Seoul (2010-11). The purpose of the talk
is to introduce a table of Painleve 6 transcendents, which will be
presented at the workshop "Various aspects of the Painlev\'e equations",
RIMS, Nov. 26-30.
References: arXiv:1210.0311, arXiv:1108.3401

〈第２回〉

◆ V. Novokshenov : "Special functions and isomonodromic deformations"

A generic scheme based on modern soliton theory is proposed for description
of classical special functions of mathematical physics. To this end,
the special function is considered as isomonodromic deformation of some
linear ODE with rational coefficients. This ODE plays a role of one of two
equations of the Lax pair. Thus the ODE, or difference equation satisfied
by the special function is integrable, i.e., it has the commuting integrals
of motions, the invariant submanifolds and the corresponding action-angle
variables. Moreover, an integration scheme for the Lax pair involves a
matrix Riemann-Hilbert problem which provides an integral representation of
the special function. The examples of relevant Riemann-Hilbert problems are
given for hypergeometric functions and orthogonal polynomials. We discuss
the way how to get the non-Abelian generalizations of the Riemann-Hilbert
problems, leading to new examples of special functions, such as Painlev\'e
transcendents.

◆ A. Kapaev : 
"On the Riemann-Hilbert approach to the Malgrange divisor: $P_I^2$ case"

Equation $P_I^2$ is the second member in the hierarchy of ODEs associated
with the classical Painlev\'e first equation $P_I$. $P_I^2$, which is a
nonlinear ODE in $x$ depending on a parameter $t$, together with the KdV
equation, which is an evolutionary PDE in $t$ and $x$, governs the
isomonodromic deformations of a particular $2\times 2$ matrix linear ODE
with polynomial coefficients and can be studied using the Riemann-Hilbert
(RH) problem approach. Given the relevant monodromy data, the set of points
$(x,t)$, where the above mentioned RH problem is not solvable, is called
the Malgrange divisor. The function $x=f(t)$, which parametrizes locally
the Malgrange divisor, satisfies a nonlinear ODE which admits a Lax pair
representation and can be studied using the RH problem. We discuss the
relations between these two kinds of the RH problems and the properties of
their $t$-large genus 1 asymptotic solutions.

◆ S. Kamimoto :
"On the decomposition of WKB solutions to monomially summable series"

We discuss the exact WKB analysis of singularly perturbed ordinary
differential equations. The principal aim of this talk is to study
asymptotic properties of WKB solutions at an irregular singular point
via the decomposition of them to monomially summable series. The notion
of monomial summability was introduced by M. Canalis-Durand, J. Mozo-
Fern\'andez and R. Sch\"afke in their study of doubly singular equations.
As an application of their theory, we give a sufficient condition that
guarantees 1-summability of WKB solutions in a perturbation parameter.
Stability of the Newton polygon of the total symbol of the equation plays
an important role in our discussion.