Integrable symplectic maps associated with the ZS-AKNS spectral problem

Abstract

The Darboux transformations (DTs) associated with the well-known ZS-AKNS spectral problem are regarded as discrete spectral problems. Two kinds of integrable symplectic maps are derived from them through nonlinearization procedures. As an application, they are used to calculate the finite genus solutions of some new integrable lattice models constructed from these DTs.

Soliton Solutions for a Nonisospectral Semi-Discrete Ablowitz–Kaup–Newell–Segur Equation

“…Without loss of generality, we only focus on the dynamics of u n , since v n behaves similarly. We first pay attention to u n in (16). For the sake of brevity, we denote ω 1 = sgn(α 1 )e ρ 1 and σ 1 = sgn(β 1 )e 1 , i.e.,…”

“…Generally speaking, a continuous equation could have several discrete versions upon the discretisation procedure. Until now, many results for semi-discrete Ablowitz-Kaup-Newell-Segur (AKNS) type equations have been derived [16][17][18][19][20][21]. In [22] (see also [23]), Ablowitz and Ladik propose a semi-discrete spectral problem, called the Ablowitz-Ladik spectral problem,…”

Symmetry Constraint of the Differential-difference KP Hierarchy and a Second Discretization of the ZS-AKNS System

“…can be used as a Lax pair to generate fully discrete integrable systems. As examples one can refer to [5,6,16,24]. It is natural that the semidiscrete equations generated from a Darboux transformation Co-published by Atlantis Press and Taylor & Francis Copyright: the authors (as a discrete spectral problem) are related to the original continuous spectral problem via suitable continuum limits.…”

Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

“…In contrasts to the usual cases treated in Refs. 16,41,42, where potentials themselves satisfy the corresponding discrete models, the solutions here are expressed in terms of the derivative of a special theta function with respect to the auxiliary Darboux variable.…”

“…In this example, we actually have a different type of situation from the one of the previous examples 16,41,42 since the relation between the discrete potential and the continuous potentials , is implicit. However, based on the Lax structure of the Kaup-Newell equation, (6), the system can still be nonlinearized, in the sense mentioned above, as an integrable symplectic map.…”