  The problem of stability and spectrum of linear excitations of a soliton
(kink) of the dispersive sine-Gordon and $\varphi^4$ - equations is solved
exactly. It is shown that the total spectrum consists of a discrete set of
frequencies of internal modes and a single band spectrum of continuum waves. It
is indicated by numerical simulations that a translation motion of a single
soliton in the highly dispersive systems is accompanied by the arising of its
internal dynamics and, in some cases, creation of breathers, and always by
generation of the backward radiation. It is shown numerically that a fast
motion of two topological solitons leads to a formation of the bound soliton
complex in the dispersive sine-Gordon system.
