  We have previously formulated a simple criterion for deducing the intervals
of oscillations in the solutions of second-order linear homogeneous
differential equations. In this work, we extend analytically the same criterion
to the cubic nonlinear Schr\"odinger equations that describe Bose-Einstein
condensates. With this criterion guiding the search for solutions, we classify
all types of solutions and we find new stationary solutions in the
free-particle cases that were not noticed previously because of limited
coverage in the adopted boundary conditions. The new solutions are produced by
the nonlinear terms of the differential equations and they continue to exist
when various external potentials are also incorporated. Surprisingly, these
solutions appear when the nonlinearities are small.
