arXiv subjects

Hiroshi Kawakami

Publications and source records attributed to Hiroshi Kawakami.

9 recordsLinked to original sources

arXiv2023

A $q$-analogue of the matrix fifth Painlevé system

We consider a degeneration of the $q$-matrix sixth Painlevé system. As a result, we obtain a system of non-linear $q$-difference equations, which describes a deformation of a certain non-Fuchsian linear $q$-difference system. We define the spectral type for non-Fuchsian $q$-difference systems and characterize the associated linear problem in terms of the spectral type. We also consider a continuous limit of the non-linear $q$-difference system and show that the resulting system of non-linear differential equations coincides with the matrix fifth Painlevé system.

nlin.SI↗

arXiv2021

Four-dimensional Painlevé-type difference equations

We focus on Fuchsian equations with four accessory parameters and three singular points. We see that the Fuchsian equations admit a "degeneration scheme" in some sense, which is expected to give rise to a degeneration scheme of discrete isomodromic deformation equations with four-dimensional phase space. We compute an example of discrete isomonodromic deformation equations of a certain Fuchsian equation.

math.CA↗

arXiv2020

A $q$-analogue of the matrix sixth Painlevé system

We derive a $q$-analogue of the matrix sixth Painlevé system via a connection-preserving deformation of a certain Fuchsian linear $q$-difference system. In specifying the linear $q$-difference system, we utilize the correspondence between linear differential systems and linear $q$-difference systems from the viewpoint of the spectral type. The system of non-linear $q$-difference equations thus obtained can also be regarded as a non-abelian analogue of Jimbo-Sakai's $q$-$P_{\mathrm{VI}}$.

math.CA↗

arXiv2019

Emergence of long-term rhythmicity within a frustrated triangle oscillator-network

This study tries to simulate a brain cell network using an electric circuit oscillator called electronic firefly. Multiple stability was observed in the electric circuit oscillator which is expressed by simple mathematical models.

nlin.AO↗

arXiv2018

Regular flat structure and generalized Okubo system

We study a relationship between regular flat structures and generalized Okubo systems. We show that the space of variables of isomonodromic deformations of a regular generalized Okubo system can be equipped with a flat structure. As its consequence, we introduce flat structures on the spaces of independent variables of generic solutions to (classical) Painlevé equations (except for PI). In our framework, the Painlevé equations PVI-PII can be treated uniformly as just one system of differential equations called the four-dimensional extended WDVV equation. Then the well-known coalescence cascade of the Painlevé equations corresponds to the degeneration scheme of the Jordan normal forms of a square matrix of rank four.

math.CA↗

arXiv2017

Four-Dimensional Painlevé-Type Equations Associated with Ramified Linear Equations III: Garnier Systems and Fuji-Suzuki Systems

This is the last part of a series of three papers entitled "Four-dimensional Painlevé-type equations associated with ramified linear equations". In this series of papers we aim to construct the complete degeneration scheme of four-dimensional Painlevé-type equations. In the present paper, we consider the degeneration of the Garnier system in two variables and the Fuji-Suzuki system.

math.CA↗

arXiv2017

Four-dimensional Painlevé-type equations associated with ramified linear equations I: Matrix Painlevé systems

As a sequel to Kawakami-Nakamura-Sakai (arXiv:1209.3836), this series of papers constructs the complete degeneration scheme of four-dimensional Painlevé-type equations which includes the Painlevé-type equations associated with linear systems of ramified type. In the present paper, we consider the degeneration of Painlevé-type equations which we call the matrix Painlevé systems.

math.CA↗

arXiv2017

Four-dimensional Painlevé-type equations associated with ramified linear equations II: Sasano systems

This is a continuation of the paper "Four-dimensional Painlevé-type equations associated with ramified linear equations I: Matrix Painlevé systems" (arXiv:1608.03927). In this series of three papers we aim to construct the complete degeneration scheme of four-dimensional Painlevé-type equations. In the present paper, we construct the degeneration scheme of what we call the Sasano system.

math.CA↗

arXiv2016

Degeneration scheme of 4-dimensional Painlevé-type equations

Four 4-dimensional Painlevé-type equations are obtained by isomonodromic deformation of Fuchsian equations: they are the Garnier system in two variables, the Fuji-Suzuki system, the Sasano system, and the sixth matrix Painlevé system. Degenerating these four source equations, we systematically obtained other 4-dimensional Painlevé-type equations. If we only consider Painlevé-type equations whose associated linear equations are of unramified type, there are 22 types of 4-dimensional Painlevé-type equations: 9 of them are partial differential equations, 13 of them are ordinary differential equations. Some well-known equations such as Noumi-Yamada systems are included in this list. They are written as Hamiltonian systems, and their Hamiltonians are neatly written using Hamiltonians of the classical Painlevé equations.

math.CA↗