  A useful semiclassical method to calculate eigenfunctions of the Schroedinger
equation is the mapping to a well-known ordinary differential equation, as for
example Airy's equation. In this paper we generalize the mapping procedure to
the nonlinear Schroedinger equation or Gross-Pitaevskii equation describing the
macroscopic wave function of a Bose-Einstein condensate. The nonlinear
Schroedinger equation is mapped to the second Painleve equation, 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.
