  We introduce a nonlinear Schroedinger equation to describe the dynamics of a
superfluid Bose gas in the crossover from the weak-coupling regime, where $a
n^{1/3}\ll 1$ with $a$ the inter-atomic s-wave scattering length and $n$ the
bosonic density, to the unitarity limit, where $a\to +\infty$. We call this
equation the {unitarity Schroedinger equation} (USE). The zero-temperature bulk
equation of state of this USE is parametrized by the Lee-Yang-Huang low-density
expansion and Jastrow calculations at unitarity. With the help of the USE we
study the profiles of quantized vortices and vortex-core radius in a uniform
Bose gas. We also consider quantized vortices in a Bose gas under
cylindrically-symmetric harmonic confinement and study their profile and
chemical potential using the USE and compare the results with those obtained
from the Gross-Pitaevskii-type equations valid in the weak-coupling limit.
Finally, the USE is applied to calculate the breathing modes of the confined
Bose gas as a function of the scattering length.
