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

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

1999-02-27

arxiv.org/abs/math-ph/9903001v4

Int.J.Theor.Phys.48:3139-3146,2009

Physics

Mathematical Physics

Thoroughly revised version, in the light of new interest in non-relativistic conformal tranformation, with a new reference lis

Scientific paper

10.1007/s10773-009-0113-z

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.

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