On a (2+1)-dimensional generalization of the Ablowitz-Ladik lattice and a discrete Davey-Stewartson system

TL;DR

This paper introduces a (2+1)-dimensional integrable generalization of the Ablowitz-Ladik lattice, serving as a space discretization of the Davey-Stewartson system, with exact solutions constructed via Hirota's method.

Contribution

It presents a novel integrable space discretization of the Davey-Stewartson system derived from a (2+1)-dimensional Ablowitz-Ladik lattice, including symmetry properties and exact solutions.

Findings

The discretized system has a Lax pair and preserves complex conjugation reduction.

The system is symmetric under space reflections.

Exact multidromion solutions are constructed.

Abstract

We propose a natural (2+1)-dimensional generalization of the Ablowitz-Ladik lattice that is an integrable space discretization of the cubic nonlinear Schroedinger (NLS) system in 1+1 dimensions. By further requiring rotational symmetry of order 2 in the two-dimensional lattice, we identify an appropriate change of dependent variables, which translates the (2+1)-dimensional Ablowitz-Ladik lattice into a suitable space discretization of the Davey-Stewartson system. The space-discrete Davey-Stewartson system has a Lax pair and allows the complex conjugation reduction between two dependent variables as in the continuous case. Moreover, it is ideally symmetric with respect to space reflections. Using the Hirota bilinear method, we construct some exact solutions such as multidromion solutions.