  Very recently, the LHCb Collaboration reported the observation of several
enhancements in the invariant mass spectrum of a $J/\psi$ pair between 6.2 and
7.4 GeV. In this work, we propose the dynamical mechanism to mimic the
experimental data of a di-$J/\psi$ mass spectrum given by LHCb, which is based
on the reactions, where all the possible combinations of a double charmonium
directly produced by a proton-proton collision are transferred into a final
state $J/\psi J/\psi$. We find that the LHCb experimental data can be well
reproduced. We further extend our framework to study a di-$\Upsilon(1S)$
system, and give the line shape of a differential cross section of a partner
process in a $b\bar{b}$ system on the invariant mass of
$\Upsilon(1S)\Upsilon(1S)$, which shows that there should exist possible
enhancements near $m_{\Upsilon(1S)\Upsilon(1S)}=$19.0, 19.3, 19.7 GeV in the
$\Upsilon(1S)$-pair invariant mass spectrum. These predictions can be tested in
LHCb and CMS, which can be as a potential research issue in near future.
