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| 担当者 | 大島 利雄, 坂井 秀隆 |
|---|
2012年12月05日(水)
16:00-17:30 数理科学研究科棟(駒場) 270号室
Andrei Kapaev 氏 (SISSA, Trieste, Italy)
On the Riemann-Hilbert approach to the Malgrange divisor: $P_I^2$ case (ENGLISH)
Andrei Kapaev 氏 (SISSA, Trieste, Italy)
On the Riemann-Hilbert approach to the Malgrange divisor: $P_I^2$ case (ENGLISH)
[ 講演概要 ]
Equation $P_I^2$ is the second member in the hierarchy of ODEs associated with the classical Painlev\\’e first equation $P_I$ and can be solved via the Riemann-Hilbert (RH) problem approach. It is known also that solutions of equation $P_I^2$ as the functions of $x$ depending on the parameter $t$ can be used to construct a 4-parameter family of isomonodromic solutions to the KdV equation. Given the monodromy data, the set of points $(x,t)$, where the above mentioned RH problem is not solvable, is called the Malgrange divisor. The function $x=a(t)$, which parametrizes locally the Malgrange divisor, satisfies a nonlinear ODE which admits a Lax pair representation and can be also studied using an 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.
Equation $P_I^2$ is the second member in the hierarchy of ODEs associated with the classical Painlev\\’e first equation $P_I$ and can be solved via the Riemann-Hilbert (RH) problem approach. It is known also that solutions of equation $P_I^2$ as the functions of $x$ depending on the parameter $t$ can be used to construct a 4-parameter family of isomonodromic solutions to the KdV equation. Given the monodromy data, the set of points $(x,t)$, where the above mentioned RH problem is not solvable, is called the Malgrange divisor. The function $x=a(t)$, which parametrizes locally the Malgrange divisor, satisfies a nonlinear ODE which admits a Lax pair representation and can be also studied using an 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.
