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The non-linear Schrödinger equation and the conformal properties of non-relativistic space-time. (English) Zbl 1184.81047

Summary: 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.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
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