On an integrable discretisation of the Ablowitz-Ladik hierarchy

TL;DR

This paper explores an integrable discretisation of the Ablowitz-Ladik hierarchy, focusing on Bäcklund transformations, boundary conditions, Hamiltonian properties, and providing explicit transformations along with analytical and numerical examples.

Contribution

It introduces explicit integrable discretisations of the Ablowitz-Ladik hierarchy based on Bäcklund transformations, with detailed analysis of their properties and conditions.

Findings

Explicit transformations for the Ablowitz-Ladik hierarchy are derived.

Hamiltonian properties of the discrete maps are analyzed.

Numerical examples demonstrate the effectiveness of the discretisation.

Abstract

Following the general results on the relationships about Backlund transformations (BTs) and exact discretisation given in a previous work [12], we consider the Ablowitz-Ladik hierarchy and a corresponding family of BTs. After discussing the boundary conditions, we show how to get explicit transformations. The Hamiltonian properties of the maps and of the discrete flows are examined. The conditions on the parameters of the map giving exact discretisations are discussed. Finally, analytical and numerical examples are given.