  The transition from integrable to non-integrable highly-dispersive nonlinear
models is investigated. The sine-Gordon and $\phi^4$-equations with the
additional fourth-order spatial and spatio-temporal derivatives, describing the
higher dispersion, and with the terms originated from nonlinear interactions
are studied. The exact static and moving topological kinks and soliton-complex
solutions are obtained for a special choice of the equation parameters in the
dispersive systems. The problem of spectra of linear excitations of the static
kinks is solved completely for the case of the regularized equations with the
spatio-temporal derivatives. The frequencies of the internal modes of the kink
oscillations are found explicitly for the regularized sine-Gordon and
$\phi^4$-equations. The appearance of the first internal soliton mode is
believed to be a criterion of the transition between integrable and
non-integrable equations and it is considered as the sufficient condition for
the non-trivial (inelastic) interactions of solitons in the systems.
