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SIGMA 2 (2006), 047, 12 pages nlin.PS/0604076
https://doi.org/10.3842/SIGMA.2006.047
Internal Modes of Solitons and Near-Integrable Highly-Dispersive Nonlinear Systems
Oksana V. Charkina and Mikhail M. Bogdan
B. Verkin Institute for Low Temperature Physics and Engineering of the NAS of Ukraine,
47 Lenin Ave., Kharkiv, 61103 Ukraine
Received November 30, 2005, in final form April 11, 2006; Published online April 28, 2006
Abstract
The transition from integrable to non-integrable
highly-dispersive nonlinear models is investigated. The
sine-Gordon and φ4-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 φ4-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.
Key words:
solitons; integrable and non-integrable equations; internal modes; dispersion.
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