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

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.

Thoroughly revised version, in the light of new interest in non-relativistic conformal tranformation, with a new reference list. 8 pages, LaTex, no figures. To be published in Int. J. Theor. Phys

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Cited by in corpus (3)

• Schrödinger Symmetry in Gravitational Mini-Superspaces
• Free Particle Wave Function and Niederer's Transformation
• Weyl Anomaly in NonRelativistic CFTs