Virtual turning points and bifurcation of Stokes curves for higher order ordinary differential equations

TL;DR

This paper investigates the bifurcation of Stokes curves in higher order linear differential equations, emphasizing virtual turning points, and applies this understanding to resolve a paradox in the Noumi-Yamada system related to the fourth Painleve equation.

Contribution

It introduces the concept of virtual turning points to explain Stokes curve bifurcations in higher order ODEs, advancing the theoretical understanding of their complex behavior.

Findings

Stokes curve bifurcation occurs when hitting another turning point.

Virtual turning points are crucial for understanding bifurcations.

Application to the Noumi-Yamada system resolves a recent paradox.

Abstract

For a higher order linear ordinary differential operator P, its Stokes curve bifurcates in general when it hits another turning point of P. This phenomenon is most neatly understandable by taking into account Stokes curves emanating from virtual turning points, together with those from ordinary turning points. This understanding of the bifurcation of a Stokes curve plays an important role in resolving a paradox recently found in the Noumi-Yamada system, a system of linear differential equations associated with the fourth Painleve equation.