New integrable semi-discretizations of the coupled nonlinear Schrodinger equations

TL;DR

This paper introduces new integrable semi-discretizations of coupled nonlinear Schrödinger equations, derived through an algorithmic search involving Lax pairs, revealing novel systems with specific continuum limits.

Contribution

The authors discover a new integrable coupled nonlinear Schrödinger system combining elements of known lattices, expanding the class of integrable semi-discrete models.

Findings

New integrable semi-discretizations of coupled NLS equations

Continuum limit yields uncoupled complex mKdV and NLS equations

Method involves compatibility conditions for discrete Lax operators

Abstract

We have undertaken an algorithmic search for new integrable semi-discretizations of physically relevant nonlinear partial differential equations. The search is performed by using a compatibility condition for the discrete Lax operators and symbolic computations. We have discovered a new integrable system of coupled nonlinear Schrodinger equations which combines elements of the Ablowitz-Ladik lattice and the triangular-lattice ribbon studied by Vakhnenko. We show that the continuum limit of the new integrable system is given by uncoupled complex modified Korteweg-de Vries equations and uncoupled nonlinear Schrodinger equations.