Abstract. The transition from integrable to non-integrable highly-dispersive nonlinear models is investigated. The sine-Gordon and ϕ 4 -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 spatiotemporal derivatives. The frequencies of the internal modes of the kink oscillations are found explicitly for the regularized sine-Gordon and ϕ 4 -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 nontrivial (inelastic) interactions of solitons in the systems. We dedicate this paper to the memory of A.M. Kosevich.
The nonstationary dynamics of topological solitons (dislocations, domain walls, fluxons) and their bound states in one-dimensional systems with high dispersion are investigated. Dynamical features of a moving kink emitting radiation and breathers are studied analytically. Conditions of the breather excitation and its dynamical properties are specified. Processes of soliton complex formation are studied analytically and numerically in relation to the strength of the dispersion, soliton velocity, and distance between solitons. It is shown that moving bound soliton complexes with internal structure can be stabilized by an external force in a dissipative medium then their velocities depend in a step-like manner on a driving strength.
Equations for the antiferromagnetism vector are used to study the spectrum and scattering of spin waves on a domain wall with precessing spins in an easy-axis antiferromagnet with a constant magnetic field directed along the easy axis. It is shown that this kind of magnetic field can be completely eliminated from the equations of motion, so that they can be reduced to a Lorentz invariant form. The spectral problem for weak excitation of a precessing domain wall is solved and exact solutions are found for the linearized equations describing the propagation of spin waves in antiferromagnets with this kind of domain wall. An explicit expression is found for the reflection coefficient of spin waves from a domain wall as a function of the wave vectors of the incident and transmitted waves, along with its dependence on the spin wave frequency. The range of frequencies within which the spin waves are fully reflected is found and it is shown that the reflection coefficient falls off sharply above the upper limit of this range. These results can be generalized to the case of a moving domain wall in a three-dimensional crystal.
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