Integrable discretization of recursion operators and unified bilinear forms to soliton hierarchies

TL;DR

This paper introduces a method for discretizing recursion operators in integrable hierarchies using unified bilinear forms, demonstrated on AKNS and KdV hierarchies, with discrete operators converging to continuous ones.

Contribution

It presents a novel procedure for discretizing recursion operators via unified bilinear forms, bridging continuous and discrete integrable hierarchies.

Findings

Discrete recursion operators converge to continuous forms

Unified bilinear forms derived for AKNS and KdV hierarchies

Discretization preserves integrability properties

Abstract

In this paper, we give a procedure for discretizing recursion operators by utilizing unified bilinear forms within integrable hierarchies. To illustrate this approach, we present unified bilinear forms for both the AKNS hierarchy and the KdV hierarchy, derived from their respective recursion operators. Leveraging the inherent connection between soliton equations and their auto-B\"acklund transformations, we discretize the bilinear integrable hierarchies and derive discrete recursion operators. These discrete recursion operators exhibit convergence towards the original continuous forms when subjected to a standard limiting process.