Wave functions and k-point functions for the AKNS hierarchy

TL;DR

This paper introduces wave functions for the AKNS hierarchy, providing a new formula for k-point correlation functions and demonstrating that its tau-functions are also KP tau-functions.

Contribution

It develops a wave function approach to express matrix resolvents and derive k-point correlation functions for the AKNS hierarchy, linking it to KP tau-functions.

Findings

Derived a new formula for k-point correlation functions

Expressed matrix resolvent in terms of wave functions

Showed AKNS tau-functions are KP tau-functions

Abstract

For an arbitrary solution to the AKNS hierarchy, the logarithmic derivatives of the tau-function of the solution can be computed by the matrix-resolvent method [14,21]. In this paper, we introduce a pair of wave functions of the solution and we use them to express the corresponding matrix resolvent. Based on this, we derive a new formula for the k-point correlation function of the AKNS hierarchy expressed in terms of wave functions. As an application, we show that the tau-function of an arbitrary solution to the AKNS hierarchy is a KP tau-function.