Kink excitation spectra in the (1+1)-dimensional $\varphi^8$ model

We study excitation spectra of BPS-saturated topological solutions -- the kinks -- of the $\varphi^8$ scalar field model in $(1+1)$ dimensions, for three different choices of the model parameters. We demonstrate that some of these kinks have a vibrational mode, apart from the trivial zero (translational) excitation. One of the considered kinks is shown to have three vibrational modes. We perform a numerical calculation of the kink-kink scattering in one of the considered variants of the $\varphi^8$ model, and find the critical collision velocity $v_{\scriptsize \mbox{cr}}$ that separates the different collision regimes: inelastic bounce of the kinks at $v_{\scriptsize \mbox{in}}\ge v_{\scriptsize \mbox{cr}}$, and capture at $v_{\scriptsize \mbox{in}}<v_{\scriptsize \mbox{cr}}$. We also observe escape windows at some values of $v_{\scriptsize \mbox{in}}<v_{\scriptsize \mbox{cr}}$ where the kinks escape to infinity after bouncing off each other two or more times. We analyse the features of these windows and discuss their relation to the resonant energy exchange between the translational and the vibrational excitations of the colliding kinks.