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The non-linear Schrödinger equation and the conformal properties of non-relativistic space-time

P. A. Horvathy·J. -C. Yera
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Abstract

The cubic non-linear Schrödinger equation where the coefficient of the nonlinear term is a function $F(t,x)$ only passes 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.

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P. A. Horvathy, J. -C. Yera. 2009-07-28. The non-linear Schrödinger equation and the conformal properties of non-relativistic space-time. https://doi.org/10.1007/s10773-009-0113-z

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