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

The cubic non-linear Schrödinger equation where the coefficient of the non-linear term can be a function F (t, x), is shown to pass the Painlevé 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.