  We review recent works on modeling of dynamics of kinks in 1+1 dimensional
$\phi^4$ theory and other related models, like sine-Gordon model or $\phi^6$
theory. We discuss how the spectral structure of small perturbations can affect
the dynamics of non-perturbative states, such as kinks or oscillons. We
describe different mechanisms, which may lead to the occurrence of the resonant
structure in the kink-antikink collisions. We explain the origin of the
radiation pressure mechanism, in particular, the appearance of the negative
radiation pressure in the $\phi^4$ and $\phi^6$ models. We also show that the
process of production of the kink-antikink pairs, induced by radiation is
chaotic.
