  The formation and evolution of stationary and moving breather solutions in
(2+1)-dimensional O(3) nonlinear $\sigma$-model are investigated. The
analytical form of oscillating solutions for (2+1)-dimensional sine-Gordon
equation, which evolve to periodic (breather) radially symmetric solutions is
determined. On the basis of the found solutions by adding the rotations to the
A3-field vector in isotopic space of S^2, the solutions for the O(3) nonlinear
$\sigma$-model are obtained. By numerical study of the solutions dynamics their
stability in a stationary and a moving state for quite a long time (45000
cycles), although in the presence of weak radiation is shown.
