Asymptotic and numerical description of the kink/antikink interaction

We consider a class of semi-linear wave equations with a small parameter and nonlinearities which provide the equations having exact kink-type solutions. We declare sufficient conditions for the nonlinearities under which the kink-kink and kink-antikink collisions occur, in the asymptotic sense, without changing the shape of the waves and with only some shifts of the solitary wave trajectories. Furthermore, we create an absolutely stable finite differences scheme to simulate the solution of the Cauchy problem and obtain some numerical results for two-wave interaction. We present also some unexpected results about three-wave interaction.