Bi-Hamiltonian structures for integrable systems on regular time scales

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.