Numerical simulation of the interactions of breather solutions of $(2+1)$-dimensional $O(3)$ nonlinear sigma model

By methods of numerical simulation the processes of interaction of breather solutions in the phase space of the $(2+1)$-dimensional supersymmetric $O(3)$ nonlinear sigma model are investigated. Frontal collision models are obtained, where, depending on the dynamic parameters of the system, processes of combining the breathers, the formation of bound states (doubled breathers), collision and reflection, the passage of breathers through each other, as well as their destruction are observed. It is shown that the breathers of the $O(3)$ nonlinear sigma model in the interaction are more stable with respect to similar solutions of the sine-Gordon equation. In the presence of rotational isospin dynamics, the system of breather fields after the collision emits a certain part of the energy preserves structural stability with a characteristic periodic oscillation. Properties of longitudinal-transverse oscillations of doubled breathers and a sudden increase in the speed of breathers, reflected from each other after interaction, are revealed. Numerical models are constructed on the basis of methods of the theory of finite difference schemes, by using the properties of stereographic projection, taking into account the group-theoretical features of constructions of the $O(N)$ class of nonlinear sigma-models of field theory. A complex program module has been developed that implements the numerical calculation algorithm.