An integrable time-dependent non-linear Schr\"odinger equation

TL;DR

This paper investigates a time-dependent non-linear Schrödinger equation, demonstrating it is integrable only when the non-linear coefficient follows a specific inverse linear time dependence, through a transformation linked to conformal symmetry.

Contribution

It identifies the precise form of the non-linear coefficient for integrability and connects it to conformal transformations, extending understanding of integrable non-linear Schrödinger equations.

Findings

The equation passes the Painlevé test only for F=(a+bt)^{-1}.

A transformation relates the time-dependent system to the standard NLS.

The transformation is connected to conformal properties of space-time.

Abstract

The cubic non-linear Schr\"odinger equation (NLS), where the coefficient of the non-linear term can be a function $F(t,x)$, is shown to pass the Painlev\'e test of Weiss, Tabor, and Carnevale only for $F=(a+bt)^{-1}$, where $a$ and $b$ constants. This is explained by transforming the time-dependent system into the ordinary NLS (with $F=\const$.) by means of a time-dependent on-linear transformation, related to the conformal properties of non-relativistic space-time.