2 research outputs found
Internal Modes of Solitons and Near-Integrable Highly-Dispersive Nonlinear Systems
The transition from integrable to non-integrable highly-dispersive nonlinear
models is investigated. The sine-Gordon and -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
-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.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Vibration spectra of benzene-like models with Hooke’s law interactions
The harmonic oscillations of a spring-ball model of benzene-like nanosystems with Hooke’s law interactions between nearest, second, and third neighbors are explored. We show that in the cylindrical coordinates the dynamics of this cyclic hexagonal system is described by the Lagrange equations similar to those of the one-dimensional two-component crystal model. We expose that the vibration frequencies of the hexagonal model lie on the branches of the dispersion law of the associated lattice model, and their positions are determined by the cyclic Born-Von Karman condition. The hexagonal model is generalized to one describing the benzene molecule and the fully deuterated and halogenated benzenes. The effect of hybridization of vibration modes and pushing apart of spectral branches in the crossover situation is revealed. All the discrete frequency spectrum and normal modes of oscillations and their explicit dependencies on all the constants of elastic interactions are exactly found