Scaling Self-Similar Formulation of the String Equations of the Hermitian Matrix Model

TL;DR

This paper reformulates the string equations of the Hermitian matrix model in terms of self-similar conditions across multiple integrable hierarchies, revealing deep connections between different models.

Contribution

It introduces a unified framework for understanding string equations as self-similar conditions in various integrable hierarchies, extending previous formulations.

Findings

Reformulation of string equations as self-similar conditions in the Ur--KdV hierarchy

Identification of the complexified NLS hierarchy with a string equation via non-scaling limit analysis

Establishment of equivalences between the string equations and hierarchies like AKNS and Heisenberg ferromagnet

Abstract

The string equation appearing in the double scaling limit of the Hermitian one--matrix model, which corresponds to a Galilean self--similar condition for the KdV hierarchy, is reformulated as a scaling self--similar condition for the Ur--KdV hierarchy. A non--scaling limit analysis of the one--matrix model has led to the complexified NLS hierarchy and a string equation. We show that this corresponds to the Galilean self--similarity condition for the AKNS hierarchy and also its equivalence to a scaling self--similar condition for the Heisenberg ferromagnet hierarchy.