Integrable discrete systems on R and related dispersionless systems

TL;DR

This paper develops a comprehensive framework for integrable discrete systems on the real line, introducing regular grain structures, constructing hierarchies with bi-Hamiltonian structures, and exploring their continuous limits and deformation quantization.

Contribution

It introduces the concept of regular grain structures on R and constructs new integrable hierarchies with bi-Hamiltonian structures, advancing the understanding of discrete integrable systems.

Findings

Construction of two integrable hierarchies of discrete chains

Development of bi-Hamiltonian structures for these hierarchies

Analysis of continuous limits and deformation quantization schemes

Abstract

The general framework for integrable discrete systems on R in particular containing lattice soliton systems and their q-deformed analogues is presented. The concept of regular grain structures on R, generated by discrete one-parameter groups of diffeomorphisms, through which one can define algebras of shift operators is introduced. Two integrable hierarchies of discrete chains together with bi-Hamiltonian structures are constructed. Their continuous limit and the inverse problem based on the deformation quantization scheme are considered.