Tau-functions for the Ablowitz--Ladik hierarchy: the matrix-resolvent method

TL;DR

This paper extends the matrix-resolvent method to the Ablowitz--Ladik hierarchy, providing formulas for tau-function derivatives and applications to unitary group integrals.

Contribution

It introduces a novel extension of the matrix-resolvent method specifically for the Ablowitz--Ladik hierarchy, enabling new computations of tau-function derivatives.

Findings

Derived a formula for the generating series of logarithmic derivatives

Provided a method to compute integrals over the unitary group

Extended the matrix-resolvent approach to a new integrable hierarchy

Abstract

We extend the matrix-resolvent method for computing logarithmic derivatives of tau-functions to the Ablowitz--Ladik hierarchy. In particular, we derive a formula for the generating series of the logarithmic derivatives of an arbitrary tau-function in terms of matrix resolvents. As an application, we provide a way of computing certain integrals over the unitary group.