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 $bbar{b}$ 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.
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