Probing triple Higgs couplings of the two Higgs doublet model at a linear collider

We study double Higgs production at the future linear collider in the framework of the two Higgs doublet models through the following channels: $e^{+}{e}^{-}\to {\Phi }_i{\Phi }_{j}Z$, ${\Phi }_i=h^0$, $H^0$, $A^0$, $H^{\pm }$. All these processes are sensitive to triple Higgs couplings. Hence observations of them provide information on the triple Higgs couplings that help reconstructing the scalar potential. We also discuss the double Higgs-Strahlung $e^{+}{e}^{-}\to h^0{h}^{0}Z$ in the decoupling limit where $h^0$ mimics the standard model Higgs boson. The processes $e^{+}{e}^{-}\to h^0{h}^{0}Z$ and $e^{+}{e}^{-}\to h^0{H}^{0}Z$ are also discussed in the fermiophobic limit where distinctive signatures such as $4\gamma +X$, $2\gamma +X$, and $6\gamma +X$ are expected in the Type-I two Higgs doublet model.