R-matrix approach to lattice integrable systems

Abstract

An r-matrix formalism is applied to the construction of the integrable lattice systems and their bi-Hamiltonian structure. Miura-like gauge transformations between the hierarchies are also investigated. In the end the ladder of linear maps between generated hierarchies is established and described.

Generation of discrete integrable systems and some algebro-geometric properties of related discrete lattice equations

“…Therefore, the Poisson tensor presents P (L) = 0 2q n (1 − E −1 ) 2(E − 1)q n 0 , which is consistent to that presented in [1]. The Hamiltonian structure of (4.12) is given by q n p n t 2 = P (L) δH δu .…”

“…According to the Riemann theorem [6], there exist constant vectors M (1) and M (2) such that the theta function θ (A(p(λ)…”

“…In what follows, we want to seek the Hamiltonian structure of the discrete integrable system (4.12) by applying new approach called variational formulas for discrete vector fields which is different from the way presented in [1][2][3]. The gradient of the functional H : g → R presents that…”

“…In addition, we reduce the isospectral problem introduced in the paper to two shift operators. Then, by applying the R-matrix formalism [1][2][3], we give rise to a new form of the Toda lattice hierarchy which can be reduced to the Toda lattice equation and its series of the generalized discrete integrable systems. Again, utilizing the Casimir functions of the Lie-Poisson bracket presented in the paper and the Novikov equation, we obtain the representation of the Lax pair of the GTL system derived from the Toda lattice hierarchy.…”

Bi-Hamiltonian structures for integrable systems on regular time scales

“…The approach of time scales allows the unification of such classes of nonlinear evolution equations like field soliton systems [1,2,3,4,5], lattice soliton systems [6,7,8,9], q-discrete soliton systems [10,11,12,13] and others. The above approach was initiated in [14] where the Gelfand-Dickey construction was extended.…”