  In this work we compare the results of the Gross-Pitaevskii and modified
Gross-Pitaevskii equations with ab initio variational Monte Carlo calculations
for Bose-Einstein condensates of atoms in axially symmetric traps. We examine
both the ground state and excited states having a vortex line along the z-axis
at high values of the gas parameter and demonstrate an excellent agreement
between the modified Gross-Pitaevskii and ab initio Monte Carlo methods, both
for the ground and vortex states.
