On an isomonodromy deformation equation without the Painlev\'e property

TL;DR

This paper investigates a fourth order nonlinear ODE linked to pole dynamics in solutions of a specific integrable PDE, revealing its role in isomonodromy deformations and its lack of the Painlevé property, with asymptotic analysis of related Riemann-Hilbert problems.

Contribution

It demonstrates that this nonlinear ODE governs isomonodromy deformations without having the Painlevé property, expanding understanding of integrable systems and their deformation equations.

Findings

The ODE controls isomonodromy deformations of a matrix linear ODE.

It does not possess the Painlevé property.

Large t asymptotics of the associated Riemann-Hilbert problem are obtained.

Abstract

We show that the fourth order nonlinear ODE which controls the pole dynamics in the general solution of equation $P_I^2$ compatible with the KdV equation exhibits two remarkable properties: 1) it governs the isomonodromy deformations of a $2\times2$ matrix linear ODE with polynomial coefficients, and 2) it does not possesses the Painlev\'e property. We also study the properties of the Riemann--Hilbert problem associated to this ODE and find its large $t$ asymptotic solution for the physically interesting initial data.