We studied the kink-antikink collision process for the “double sine-Gordon” (DSG) equation in 1+1 dimensions at different values of the potential parameter $R>\mathrm{}0.\mathrm{}$ For small values of R we discuss the problem of resonance frequencies. We give qualitative explanation of the frequency shift in comparison with the frequency of the discrete level in the potential well of isolated kink. We show that in this region of the parameter R the effective long-range interaction between kink and antikink takes place.

We study excitation spectra of BPS-saturated topological solutions — the kinks — of the φ⁸ 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 φ⁸ model, and find the critical collision velocity v_cr that separates the different collision regimes: inelastic bounce of the kinks at v_in ≥ v_cr, and capture at v_in< v_cr. We also observe escape windows at some values of v_in< v_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.

We study kink scattering processes in the $(1+1)$-dimensional ${$\varphi${}}^{6}$ model in the framework of the collective coordinate approximation. We find critical values of the initial velocities of the colliding kinks. These critical velocities distinguish different regimes of collisions. The exact equation of motion for the ${$\varphi${}}^{6}$ model is also solved numerically with the same initial conditions. We discuss advantages and disadvantages of the collective coordinate approximation, and also outline its applicability limits. Resonance phenomena and the so-called escape windows are also observed in the kink collisions.

We study the scattering of kink and antikink of the double sine-Gordon model. There is a critical value of the initial velocity [equation]of the colliding kinks, which separates different regimes of...