  Based on the Peregrine solution (PS) of the nonlinear Schr\"odinger (NLS)
equation, the evolution of rational fraction pulses surrounded by zero
background is investigated. These pulses display the behavior of a
breather-like solitons. We study the generation and evolution of such solitons
extracted, by means of the spectral-filtering method, from the PS in the model
of the optical fiber with realistic values of coefficients accounting for the
anomalous dispersion, Kerr nonlinearity, and higher-order effects. The results
demonstrate that the breathing solitons stably propagate in the fibers. Their
robustness against small random perturbations applied to the initial background
is demonstrated too.
