Dynamics and stability of solitons in the sine-Gordon models with a strong dispersion

Influence of the strong dispersion on dynamic properties of solitons (kinks) is investigated in one-dimensional near-discrete systems. Typical examples of kinks are dislocations in the Frenkel-Kontorova model, fluxons in the long Josephson junctions and domain walls in anisotropic magnets. In the case of the strong dispersion the solitons exhibit an intrinsic structure with internal degrees of freedom. Analytical description of novel effects caused by the dispersion is performed in the framework of the sine-Gordon and the double sine-Gordon equations with a fourth-order spatio-temporal derivative. This approach allows to find exactly a total spectrum of linear excitations of the static kink. It consists of a discrete set of frequencies of internal modes and a bounded spectrum of continuum waves. It is shown analytically and numerically that a translational motion of a single soliton in the dispersive systems is accompanied by exciting its internal modes leading to creation of the breather state, and by generation of the regular or chaotic radiation. It is demonstrated that interaction of two fast solitons can lead to a formation of the bound soliton complex which moves radiationlessly in the highly dispersive medium. Conditions of stability and dissociation of the soliton complexes are studied numerically in the dispersive sine-Gordon and double sine-Gordon systems.