TheR-matrix approach to integrable systems on time scales

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 δdifferential operators in terms of which one can construct an infinite hierarchy of commuting vector fields. The theory is illustrated by two infinite-field integrable hierarchies on time scales which are-differential counterparts of KP and mKP. The difference counterparts of AKNS and Kaup-Broer soliton systems are constructed as related finite-field restrictions.

Bi-Hamiltonian structures for integrable systems on regular time scales

“…According to the classical R-matrices (20) we have two Lax hierarchies of commuting evolution equations [15] …”

“…for details see [15]. Hence, there arises natural constraint between dynamical fields from (23) given by…”

“…The above approach was initiated in [14] where the Gelfand-Dickey construction was extended. The theory was further developed in [15], where systematic construction of (1 + 1)-dimensional integrable systems on regular time scales was presented, and a very effective tool, classical R-matrix formalism, was utilized. The R-matrix formalism provides a construction of infinite hierarchies of mutually commuting vector fields.…”

“…The R-matrix formalism provides a construction of infinite hierarchies of mutually commuting vector fields. In [15], we examined the general classes of admissible Lax operators, presented examples of integrable systems on time scales that can be written in an explicit form. We also explained the source of constraints first observed in [14].…”

“…In Section 6, in order to find the bi-Hamiltonian structures, we introduce a trace functional on the algebra of δ-pseudo-differential operators in terms of which we construct the linear Poisson tensors and the related Hamiltonians. The quadratic Poisson tensors are reconstructed in the frame of the recursion operators [15] of the Lax hierarchies. Finally, in Section 7, the theory is illustrated by bi-Hamiltonian formulation of finite-field integrable hierarchies on regular time scales which are ∆-differential counterparts of AblowitzKaup-Newell-Segur (AKNS) and Kaup-Broer hierarchies.…”