Symmetries, conserved quantities, and hierarchies for some lattice systems with soliton structure

Abstract

Basic invariants, such as conserved quantities, symmetries, mastersymmetries, and recursion operators are explicitly constructed for the following nonlinear lattice systems: The modified Korteweg–de Vries lattice, the Ablowitz–Ladik lattice, the Brusci–Ragnisco lattice, the Ragnisco–Tu lattice and some cases of the class of integrable systems introduced by Bogoyavlensky. The algorithmic basis for obtaining these quantities is described and the interrelation between the underlying mastersymmetry approach and the Lax pair analysis is discussed. By explicit presentation of the higher-order members of the corresponding hierarchies new completely integrable lattice flows are found. For all systems, multi-Hamiltonian formulations are given.

A novel hierarchy of integrable lattices

“…In fact, we present and discuss an integrable differential-difference version of socalled AKNS hierarchy [9], already mentioned in [10]. In comparison with other discretisations [11] [3], it has certain advantages and some drawbacks.…”

New hierarchies of integrable positive and negative lattice models and Darboux transformation

“…We believe that the present studies can be useful for other applications asnd can be extended to more complicated spectral problems in higher order. Furthermore, we would emphasize that the mathematical and physical background as well as the deeper properties, such as the symmetries, infinitely many conservation laws, nonlinearization, and so forth [20][21][22][23] of these nice rational equations from positive hierarchy (12) would be exploited gradually in another occasion.…”

A Lattice Hierarchy with a Free Function and Its Reductions to the Ablowitz-Ladik and Volterra Hierarchies

“…Recently, the spectral problem (1.4) was used to study the integrable discretization of the derivative Schrödinger equation and the N -soliton solution of a discrete system by Tsuchida, Ito and Kakuhata [24,25]. Zhang and Tu ever investigated the symmetries and conservation of a hierarchy related to the following spectral problem [5] ψ n+1 =Ũ nψn = λ +s nqñ r n 1 ψ n .…”