The non-linear Schr\"odinger equation and the conformal properties of non-relativistic space-time

TL;DR

This paper explores the conformal properties of non-relativistic space-time to explain the integrability of the non-linear Schrödinger equation with specific time-dependent coefficients, revealing connections to conformal transformations and integrability conditions.

Contribution

It demonstrates how conformal transformations relate to the integrability of the non-linear Schrödinger equation with time-dependent nonlinear coefficients.

Findings

The NLS passes the Painlevé test only for specific time-dependent coefficients.

Transformations relate the time-dependent NLS to constant-coefficient NLS.

Conformal properties explain the integrability in certain force fields.

Abstract

The cubic non-linear Schr\"odinger equation where the coefficient of the nonlinear term is a function $F(t,x)$ only passes the Painlev\'e test of Weiss, Tabor, and Carnevale only for $F=(a+bt)^{-1}$, where $a$ and $b$ are constants. This is explained by transforming the time-dependent system into the constant-coefficient NLS by means of a time-dependent non-linear transformation, related to the conformal properties of non-relativistic space-time. A similar argument explains the integrability of the NLS in a uniform force field or in an oscillator background.