Decay of shrinking solitons in multidimensional sine-Gordon equation

The dynamics of contracting ring-like solitons (kinks and breathers) described by rotationally or spherically symmetric sine-Gordon equation is considered by means of the perturbation theory, the small parameter being the ratio of the soliton's width to its radius, i.e. solitons being assumed quasi-one-dimensional. For a shrinking kink the rate of the energy emission is calculated in the logarithmic approximation, assuming the kink's initial energy be large. In the three-dimensional case the radiatively damped kink remains quasi-one-dimensional during all the collapse time, while for a two-dimensional kink this condition is broken at the exponentially small terminal stage of collapsing. For a shrinking breather the process of its decay into kink-antikink pair is investigated. The decay time is calculated, and initial conditions generating anomalously long-lived breathers are revealed. Dynamics of the escaping pair is investigated too. Then the collapse of a breathers in a dissipative medium is considered. It is demonstrated that in the presence of a diffusion type dissipation the breather does not decay, but acquires an asymptotically constant value of the amplitude. The collapse ofa nonlinear Schrödinger equation soliton in a dissipative medium is also investigated. Some results obtained by means of the perturbation theory are directly compared with results of numerical experiments.