Recursion and Hamiltonian operators for integrable nonabelian difference equations

TL;DR

This paper develops a Hamiltonian and recursion operator framework for integrable nonabelian difference equations, introducing new nonabelian integrable systems and analyzing their symmetries and operators.

Contribution

It adapts Hamiltonian formalism and recursion operators to nonabelian difference equations, and introduces the nonabelian Narita-Itoh-Bogoyavlensky lattice.

Findings

Constructed recursion and Hamiltonian operators for the nonabelian Narita-Itoh-Bogoyavlensky lattice.

Proved the locality of infinitely many symmetries generated by nonlocal recursion operators.

Extended the nonabelian framework to systems like the relativistic Toda chain and Ablowitz-Ladik lattice.

Abstract

In this paper, we carry out the algebraic study of integrable differential-difference equations whose field variables take values in an associative (but not commutative) algebra. We adapt the Hamiltonian formalism to nonabelian difference Laurent polynomials and describe how to obtain a recursion operator from the Lax representation of an integrable nonabelian differential-difference system. As an application, we propose a novel family of integrable equations: the nonabelian Narita-Itoh-Bogoyavlensky lattice, for which we construct their recursion operators and Hamiltonian operators and prove the locality of infinitely many commuting symmetries generated from their highly nonlocal recursion operators. Finally, we discuss the nonabelian version of several integrable difference systems, including the relativistic Toda chain and Ablowitz-Ladik lattice.