  We study small-amplitude, nonlinear pulsations of uniformly and
differentially rotating neutron stars employing a two-dimensional evolution
code for general-relativistic hydrodynamics. Using Fourier transforms at
several points inside the star, both the eigenfrequencies and two-dimensional
eigenfunctions of pulsations are extracted. The centrifugal forces and the
degree of differential rotation have significant effects on the
mode-eigenfunction. We find that near the mass-shedding limit, the pulsations
are damped due to shocks forming at the surface of the star. This new damping
mechanism may set a small saturation amplitude for modes that are unstable to
the emission of gravitational waves. After correcting for the assumption of the
Cowling approximation (used in our numerical code), we construct empirical
relations that predict the range of gravitational-wave frequencies from
quasi-periodic post-bounce oscillations in the core collapse of massive stars.
We also find that the fundamental quasi-radial mode is split, at least in the
Cowling approximation and mainly in differentially rotating stars, into two
different sequences.
