  At the Large Hadron Collider (LHC), both the ATLAS and CMS Collaborations
have been searching for light charged Higgs bosons via top (anti)quark
production and decays channels, like $pp\to t \bar{t}$ with one top (anti)quark
decaying into a charged Higgs boson and a $b$ (anti)quark, when the decay is
kinematically open (i.e., when $m_{H^\pm}\lesssim m_t$). In this paper, we
propose new searches at the LHC involving light charged Higgs bosons via their
pair production channels like $pp\to H^\pm h/A$ and $pp\to H^+ H^-$ in the
2-Higgs Doublet Model (2HDM) Type-I and -X scenarios. By focusing on the case
where the heavy $H$ state plays the role of the Standard Model (SM)-like Higgs
boson with a mass near 125 GeV, we study the aforementioned Higgs boson pair
production channels and investigate their bosonic decays, such as $H^\pm \to
W^{\pm } h$ and/or $H^\pm \to W^{\pm } A$. We demonstrate that for a light
charged Higgs boson state, with $m_{H^\pm}\lesssim m_t$, at the LHC, such
di-Higgs production and decay channels can give rise to signatures with event
rates much larger than those emerging from $pp\to t\bar{t}\to t\bar{b} H^-$ +
c.c.. We specifically study $h/A\to b\bar b$ and $\tau^+\tau^-$ decays. We,
therefore, claim that the discussed combination of new production and decay
modes can result in an alternative discovery channel for charged Higgs bosons
lighter than the top (anti)quark at the LHC within the above two 2HDM Types.
Finally, in order to motivate experimentalists in ATLAS and CMS to search for
such signatures, we propose 16 Benchmark Points (BPs) which are compatible with
both theoretical and experimental constraints.
