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Wave functions and k-point functions for the AKNS hierarchy

Ang Fu

TL;DR

This work extends the matrix-resolvent MR framework to the AKNS hierarchy by constructing a pair of wave functions that encode the Lax structure and yield the matrix resolvent. It provides explicit generating-series formulas for the AKNS k-point functions in terms of the wave-functions (via a function D) and an alternative B-form, and proves that the AKNS tau-function is a KP-tau-function, connecting AKNS to the KP hierarchy in the big cell. The results generalize earlier MR-based analyses for KdV and Toda, offering new tools for computing multivariate derivatives of AKNS tau-functions and for identifying KP structure in AKNS solutions.

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.

Wave functions and k-point functions for the AKNS hierarchy

TL;DR

This work extends the matrix-resolvent MR framework to the AKNS hierarchy by constructing a pair of wave functions that encode the Lax structure and yield the matrix resolvent. It provides explicit generating-series formulas for the AKNS k-point functions in terms of the wave-functions (via a function D) and an alternative B-form, and proves that the AKNS tau-function is a KP-tau-function, connecting AKNS to the KP hierarchy in the big cell. The results generalize earlier MR-based analyses for KdV and Toda, offering new tools for computing multivariate derivatives of AKNS tau-functions and for identifying KP structure in AKNS solutions.

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.
Paper Structure (5 sections, 8 theorems, 59 equations)

This paper contains 5 sections, 8 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Review of the MR method to tau-functions for the AKNS hierarchy
  3. A pair of wave functions for the AKNS hierarchy
  4. Explicit formulas for the $k$-point functions
  5. Acknowledgements

Key Result

Theorem 1.1

Let $k\geq 2$ be an integer. The generating series of $k$-point correlation functions of the solution $(q({\bf t};\epsilon),r({\bf t};\epsilon))$ is given by where $\Omega_{i_{1},i_{2},\dots,i_{k}}({\bf t};\epsilon)$ are the $k$-point correlation functions (defined in Section sec2) of the solution $(q({\bf t};\epsilon),r({\bf t};\epsilon))$ for the AKNS hierarchy.

Theorems & Definitions (19)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Theorem 1.2
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 9 more