  We study the non-leptonic decays of $B\to PP$ in the modified perturbative
QCD approach, where $P$ stands for pseudoscalar mesons. Transverse momenta of
partons and Sudakov factor are included, which help to suppress the
contributions of soft interactions. The wave function of $B$ meson obtained
from the relativistic potential model is used, then the contributions in the
infrared region can not be suppressed completely. So a soft cutoff scale and
soft form factors are introduced. The contributions with the scale higher than
the soft cutoff scale are calculated with perturbative QCD, while the
contributions lower than the cutoff scale are replaced by the soft form
factors. To explain experimental data, we find that contributions of
color-octet operators for the quark-antiquarks in the mesons in the final state
need to be considered. The contributions of the color-octet operators are
parameterized by a few parameters with the help of SU(3) flavor symmetry and
symmetry breaking. These parameters for color-octet contributions are universal
for all the non-leptonic decay modes of $B$ meson, where the mesons in the
final state belong to the same flavor SU(3) nonet. Both the branching ratios
and $CP$ violations are studied. We find that the theoretical calculation can
well explain the experimental data of $B$ factories.
