  Based on the fitting results of the LHCb collaboration on the contributions
of various intermediate resonances to the $B^{+}\to\pi^{+}\pi^{+}\pi^{-}$
decay, we make systematically calculate the branching fractions and localized
$CP$ asymmetries of the quasi-two-body $B^{+}\to \pi^{+} \left( \rho(770),
\omega(782), \rho(1450), f_{2}\left(1270\right)\to\right)\pi^{+}\pi^{-}$ decays
within the perturbative QCD (PQCD) approach. Our theoretical predictions for
the branching fractions agree well with the data within the errors. In order to
further test the framework of the three-body $B$ decays in PQCD and the wave
functions of $\pi\pi$ pair, we also calculate the branching fractions of the
corresponding two-body $B^{+}\to\pi^{+}\rho(770)$, $B^{+}\to\pi^{+}\omega(782)$
and $B^{+}\to\pi^{+}f_{2}\left(1270\right)$ decays under the
narrow-width-approximation, which are in consistence with the experimental
data. The direct $CP$ asymmetries of $B^{+}\to\pi^{+}(\rho(770/1450)\to)
\pi^{+}\pi^{-}$ decays are found to be very small because these decay modes are
tree-dominated. However, due to the large penguin contributions from the chiral
enhanced annihilation diagrams, $B^{+}\to\pi^{+} (f_{2}\left(1270\right)\to
)\pi^{+}\pi^{-}$ decay has a large direct $CP$ asymmetry, which is also
discovered by LHCb collaboration. A relatively large $CP$ asymmetry is also
expected in the $B^+\to\pi^+ (\omega\to)\pi^+\pi^-$ decay occurring via
$\rho-\omega$ mixing, which would be measured LHCb and Belle-II experiments in
future.
