Finite-amplitude longitudinal collective excitations in spin and isospin saturated zero-temperature nuclear matter are investigated in the semiclassical limit, using a self-consistent density-dependent interaction of the Skyrme type. There are no solitons and no shocks, only zero-sound modes. Due to trapping of particles in the troughs of the self-consistent potential, there is a relatively small maximum amplitude that a zero-sound mode of a given wavelength can have. The theory of slowly modulated wave-trains of finite-amplitude zero sound is also developed. This theory shows that zero sound is stable, and that a modulation of finite extent ultimately breaks into two separated disturbances. In order to test the adequacy of nuclear hydrodynamics in the regime of large amplitudes, all calculations are performed twice, once exactly using the Vlasov equation, and once using a hydrodynamic approach. In the small-amplitude limit, hydrodynamics is semi-quantitatively correct, while in the large amplitude region it is qualitatively incorrect. It incorrectly predicts the existence of zero-sound solitons, and fails to take into account some of the most interesting new features which can arise, such as particle trapping, discontinuity of Fermi surface, and splitting of the Fermi sea into two disconnected parts.