An important task at future colliders is the investigation of the Higgs-boson sector. Here the measurement of the triple Higgs coupling(s) plays a special role. Based on previous analyses, within the framework of Two Higgs Doublet Models (2HDM) type~I and~II, we define and analyze several two-dimensional benchmark planes, that are over large parts in agreement with all theoretical and experimental constraints. For these planes we evaluate di-Higgs production cross sections at future high-energy $e^+e^-$ colliders, such as ILC or CLIC. We consider two different channels for the neutral di-Higgs pairs $h_i h_j=hh,hH,HH,AA$: $e^+e^- to h_i h_j Z$ and $e^+e^- to h_i h_j u bar u$. In both channels the various triple Higgs-boson couplings contribute substantially. We find regions with a strong enhancement of the production channel of two SM-like light Higgs bosons and/or with very large production cross sections involving one light and one heavy or two heavy 2HDM Higgs bosons, offering interesting prospects for the ILC or CLIC. The mechanisms leading to these enhanced production cross sections are analyzed in detail. We propose the use of cross section distributions with the invariant mass of the two final Higgs bosons where the contributions from intermediate resonant and non-resonant BSM Higgs bosons play a crucial role. We outline which process at which center-of-mass energy would be best suited to probe the corresponding triple Higgs-boson couplings.
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