R-matrix approach to integrable systems on time scales

TL;DR

This paper introduces an R-matrix framework for integrable soliton systems on time scales, unifying continuous and discrete cases, and constructs hierarchies like KP and mKP as well as related soliton systems.

Contribution

It develops a unifying R-matrix approach for integrable systems on time scales, extending classical soliton hierarchies to a broader unified setting.

Findings

Constructed infinite hierarchies of integrable systems on time scales.

Derived difference counterparts of KP and mKP hierarchies.

Built finite-field restrictions for AKNS and Kaup-Broer systems.

Abstract

A general unifying framework for integrable soliton-like systems on time scales is introduced. The $R$-matrix formalism is applied to the algebra of $\delta$-differential operators in terms of which one can construct infinite hierarchy of commuting vector fields. The theory is illustrated by two infinite-field integrable hierarchies on time scales which are difference counterparts of KP and mKP. The difference counterparts of AKNS and Kaup-Broer soliton systems are constructed as related finite-field restrictions.