Tronquée solutions of the painlevé II equation

Abstract

We study special solutions of the Painleve II (PII) equation called tronquee solutions, i.e., those having no poles along one or more critical rays in the complex plane. They are parameterized by special monodromy data of the Lax pair equations. The manifold of the monodromy data for a general solution is a twodimensional complex manifold with one- and zero-dimensional singularities, which arise because there is no global parameterization of the manifold. We show that these and only these singularities (together with zeros of the parameterization) are related to the tronquee solutions of the PII equation. As an illustration, we consider the known Hastings-McLeod and Ablowitz-Segur solutions and some other solutions to show that they belong to the class of tronquee solutions and correspond to one or another type of singularity of the monodromy data.