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Generalized discrete integrable operator and integrable hierarchy

Huan Liu

TL;DR

The paper extends the discrete IIKS integrable-operator framework by introducing two new operator classes: $2\times 2$ matrix-kernel operators and generalized differential-kernel operators, establishing their RH problem characterizations and resolvent formulas. It demonstrates that these generalized discrete operators encode higher-order pole data and generate complete integrable hierarchies (notably the $n$-component focusing NLS, KP, and non-commutative KP) via a unified RH formalism, with explicit constructions of higher-order pole solutions. Through connections to discrete orthogonal polynomials and explicit examples (including second-order poles in NLS, mKdV, and KP-I), the work broadens the toolkit for integrable systems on discrete spaces and links to KPZ fluctuations. The results pave the way for applications to discrete kernels, spectral statistics, and potentially the KPZ universality class, while offering a versatile framework for matrix and differential-integrable operators.

Abstract

We introduce and systematically develop two classes of discrete integrable operators: those with $2\times 2$ matrix kernels and those possessing general differential kernels, thereby generalizing the discrete analogue previously studied. A central finding is their inherent connection to higher-order pole solutions of integrable hierarchies, contrasting sharply with standard operators linked to simple poles. This work not only provides explicit resolvent formulas for matrix kernels and differential operator analogues but also offers discrete integrable structures that encode higher-order behaviour.

Generalized discrete integrable operator and integrable hierarchy

TL;DR

The paper extends the discrete IIKS integrable-operator framework by introducing two new operator classes: matrix-kernel operators and generalized differential-kernel operators, establishing their RH problem characterizations and resolvent formulas. It demonstrates that these generalized discrete operators encode higher-order pole data and generate complete integrable hierarchies (notably the -component focusing NLS, KP, and non-commutative KP) via a unified RH formalism, with explicit constructions of higher-order pole solutions. Through connections to discrete orthogonal polynomials and explicit examples (including second-order poles in NLS, mKdV, and KP-I), the work broadens the toolkit for integrable systems on discrete spaces and links to KPZ fluctuations. The results pave the way for applications to discrete kernels, spectral statistics, and potentially the KPZ universality class, while offering a versatile framework for matrix and differential-integrable operators.

Abstract

We introduce and systematically develop two classes of discrete integrable operators: those with matrix kernels and those possessing general differential kernels, thereby generalizing the discrete analogue previously studied. A central finding is their inherent connection to higher-order pole solutions of integrable hierarchies, contrasting sharply with standard operators linked to simple poles. This work not only provides explicit resolvent formulas for matrix kernels and differential operator analogues but also offers discrete integrable structures that encode higher-order behaviour.

Paper Structure

This paper contains 9 sections, 10 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. A brief view of integrable operators
  3. Integrable matrix-type kernel
  4. Connection to discrete orthogonal polynomials
  5. Generalized discrete integrable operator
  6. Connection to the higher-order pole solutions of integrable hierarchy
  7. Integrable hierarhcy
  8. Higher-order pole solution
  9. Concluding remarks

Key Result

Theorem 2.2

(Its–Izergin–Korepin–Slavnov IIKS1990IIKS1993$\&$ Borodin Borodin2000) If $\mathbbm{1}-\mathsf{K}$ is invertible, the resolvent operator $\mathsf{R}:L^2(\Omega, \mu)\rightarrow L^2(\Omega, \mu)$ has the kernel where

Theorems & Definitions (27)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.4
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 17 more