Bi-Hamiltonian structures for integrable systems on regular time scales

TL;DR

This paper develops bi-Hamiltonian structures for integrable systems on regular time scales, introducing trace functionals and Poisson tensors, and illustrates the theory with specific differential hierarchies.

Contribution

It presents a novel construction of bi-Hamiltonian structures on regular time scales, including trace functionals and Poisson tensors applicable to various integrable systems.

Findings

Constructed trace functional on algebra of $ ext{delta}$-pseudo-differential operators.

Derived linear and quadratic Poisson tensors using recursion operators.

Applied theory to $ ext{Delta}$-differential versions of known hierarchies.

Abstract

A construction of the bi-Hamiltonian structures for integrable systems on regular time scales is presented. The trace functional on an algebra of $\delta$-pseudo-differential operators, valid on an arbitrary regular time scale, is introduced. The linear Poisson tensors and the related Hamiltonians are derived. The quadratic Poisson tensors is given by the use of the recursion operators of the Lax hierarchies. The theory is illustrated by $\Delta$-differential counterparts of Ablowitz-Kaup-Newell-Segur and Kaup-Broer hierarchies.