Internal modes of solitons and near-integrable highly-dispersive nonlinear systems
Abstract:
The transition from integrable to non-integrable highly-dispersive
nonlinear models is investigated. The sine-Gordon and $\varphi^4$-equations
with the additional fourth-order spatial and spatio-temporal derivatives,
describing the higher-order 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 are 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 $\varphi^4$-equations. The
appearance of the soliton internal modes is believed to be a criterion
of the transition between integrable and non-integrable equations and it
is considered as the sufficient condition of non-trivial (inelastic) interactions
of solitons in the systems.