Uniform semiclassical approximations of the nonlinear Schr�dinger equation by a Painlev� mapping (original) (raw)
A useful semiclassical method to calculate eigenfunctions of the Schrödinger equation is the mapping to a well-known ordinary differential equation, such as for example Airy's equation. In this paper, we generalize the mapping procedure to the nonlinear Schrödinger equation or Gross-Pitaevskii equation describing the macroscopic wavefunction of a Bose-Einstein condensate. The nonlinear Schrödinger equation is mapped to the second Painlevé equation (PII), which is one of the best-known differential equations with a cubic nonlinearity. A quantization condition is derived from the connection formulae of these functions. Comparison with numerically exact results for a harmonic trap demonstrates the benefit of the mapping method. Finally we discuss the influence of a shallow periodic potential on bright soliton solutions by a mapping to a constant potential.
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