  In this paper we continue our study of the Kelvin-Helmholtz (KH) instability
in relativistic planar jets following the long-term evolution of the numerical
simulations which were introduced in Paper I. The models have been classified
into four classes (I to IV) with regard to their evolution in the nonlinear
phase, characterized by the process of jet/ambient mixing and momentum
transfer. Models undergoing qualitatively different non-linear evolution are
clearly grouped in well-separated regions in a jet Lorentz
factor/jet-to-ambient enthalpy diagram. Jets with a low Lorentz factor and
small enthalpy ratio are disrupted by a strong shock after saturation. Those
with a large Lorentz factor and enthalpy ratio are unstable although the
process of mixing and momentum exchange proceeds to a longer time scale due to
a steady conversion of kinetic to internal energy in the jet. In these cases,
the high value of the initial Lorentz seems to prevent transversal velocity
from growing far enough to generate the strong shock that breaks the slower
jets. Finally, jets with either high Lorentz factors and small enthalpy ratios
or low Lorentz factors and large enthalpy ratios appear as the most stable.
