Singularity analysis of a new discrete nonlinear Schrodinger equation

TL;DR

This paper applies the Painleve test to a new discrete nonlinear Schrödinger equation, concluding its nonintegrability due to logarithmic singularities, and proposes a related potentially integrable discrete equation.

Contribution

It introduces a novel singularity analysis for the Leon-Manna equation and suggests a new discrete equation that may be integrable.

Findings

The Leon-Manna equation is nonintegrable.

Logarithmic terms in solutions indicate nonintegrability.

A new discrete equation potentially admits integrability.

Abstract

We apply the Painleve test for integrability to a new discrete (differential-difference) nonlinear Schrodinger equation introduced by Leon and Manna. Since the singular expansions of solutions of this equation turn out to contain nondominant logarithmic terms, we conclude that the studied equation is nonintegrable. This result supports the observation of Levi and Yamilov that the Leon-Manna equation does not admit high-order generalized symmetries. As a byproduct of the singularity analysis carried out, we obtain a new discrete equation which should be integrable according to a conjecture of Weiss.