A novel hierarchy of integrable lattices

Abstract

In the framework of the reduction technique for Poisson-Nijenhuis structures, we derive a new hierarchy of integrable lattice, whose continuum limit is the AKNS hierarchy.In contrast with other differential-difference versions of the AKNS system, our hierarchy is endowed with a canonical Poisson structure and, moreover, it admits a vector generalisation.We also solve the associated spectral problem and explicity contruct action-angle variables through the r-matrix approach.

Selected Topics in Classical Integrability

“…It is worth noting that the latter expressions are valid in the quantum case as well (see e.g. [30,31,35]). …”

A Family of Integrable Different-Difference Equations, Its Hamiltonian Structure, and Darboux-Bäcklund Transformation

“…They play the important roles in mathematical physics, lattice soliton theory, cellular automata, and so on. Many nonlinear integrable differentdifference equations have been proposed and discussed, such as Ablowitz-Ladik lattice [1], Toda lattice [2], relativistic Toda lattices in polynomial form and rational form [3,4], modified Toda lattice [5,6], Volterra (or Langmuir) lattice [7], deformed reduced semi-discrete Kaup-Newell lattice [8], Merola-Ragnisco-Tu lattice [9], and so forth [10][11][12][13][14][15][16][17][18]. As we know, searching for novel integrable nonlinear differentdifference equations is still an important and very difficult research topic.…”

Solving an integrable coupling system of Merola–Ragnisco–Tu lattice equation by Darboux transformation of Lax pair

“…Many integrable differential-difference equations have been presented. Their integrable properties have been studied from different points of view, such as the symmetries and master symmetries [12,13], continuum limit [14], inverse scattering transformation [15], Hamiltonian structure [9,10,16,17], r-matrix structure [14], the symmetry constraints [18,19], the conservation laws [20], the Darboux transformations [21][22][23][24], and so forth. Among them, the Darboux transformation is a powerful tool to find explicit solutions of the integrable differential-difference equations.…”