The non-linear Schrödinger equation and the conformal properties of non-relativistic space-time · arXivDesk
math-ph/9903001Feb 27, 1999Thoroughly 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
The non-linear Schrödinger equation and the conformal properties of non-relativistic space-time
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
Nearby in the stack
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.