The double scaling limit method in the Toda hierarchy

TL;DR

This paper introduces a double scaling limit method for regularizing critical points in the dispersionful Toda hierarchy, analyzing their properties and extending the concept of critical points from matrix models.

Contribution

It formulates a new double scaling limit approach for the Toda hierarchy and proves the doubling property of equations at critical points.

Findings

Identifies and analyzes critical points in the dispersionful Toda hierarchy.

Proves the doubling of equations property at critical points.

Introduces a broad family of critical points and discusses their double scaling limits.

Abstract

Critical points of semiclassical expansions of solutions to the dispersionful Toda hierarchy are considered and a double scaling limit method of regularization is formulated. The analogues of the critical points characterized by the strong conditions in the Hermitian matrix model are analyzed and the property of doubling of equations is proved. A wide family of sets of critical points is introduced and the corresponding double scaling limit expansions are discussed.