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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