  Explicit solutions are obtained for a class of semilinear radial Schrodinger
equations with power nonlinearities in multi-dimensions. These solutions
include new similarity solutions and other new group-invariant solutions, as
well as new solutions that are not invariant under any symmetries of this class
of equations. Many of the solutions have interesting analytical behavior
connected with blow-up and dispersion. Several interesting nonlinearity powers
arise in these solutions, including the case of the critical (pseudo-conformal)
power. In contrast, standard symmetry reduction methods lead to nonlinear ODEs
for which few if any explicit solutions can be derived by standard integration
methods.
