The matrix-resolvent method to tau-functions for the nonlinear Schr\"odinger hierarchy

TL;DR

This paper extends the matrix-resolvent method to the nonlinear Schrödinger hierarchy, providing new proofs and applications, including improved algorithms for correlator computations in hermitian matrix models.

Contribution

It introduces an extension of the matrix-resolvent method to the NLS hierarchy and offers a detailed proof of a key theorem relating Toda and NLS hierarchies.

Findings

Extended the matrix-resolvent method to NLS hierarchy

Provided a detailed proof of the Carlet-Dubrovin-Zhang theorem

Improved algorithms for computing correlators in hermitian matrix models

Abstract

We extend the matrix-resolvent method of computing logarithmic derivatives of tau-functions to the nonlinear Schr\"odinger (NLS) hierarchy. Based on this method we give a detailed proof of a theorem of Carlet, Dubrovin and Zhang regarding the relationship between the Toda lattice hierarchy and the NLS hierarchy. As an application, we give an improvement of an algorithm of computing correlators in hermitian matrix models.