Breatherlike solitons extracted from the Peregrine rogue wave

Based on the Peregrine solution (PS) of the nonlinear Schrödinger (NLS) equation, the evolution of rational fraction pulses surrounded by zero background is investigated. These pulses display the behavior of a breatherlike 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.

Figure 1

The contour plots of (a) the Peregrine rogue wave given by Eq. (2) and (b) the rogue wave with zero CW background given by Eq. (4). The growing intensity distributions at propagation distances $z=3.5$, $z=4$, $z=4.5$, and $z=5$, respectively, are shown in (c) for the Peregrine rogue wave, and in (d) for the rogue wave with zero CW background. The parameters are $A=1$, $z_0=5$, and $t_0=0$.

Figure 2

The evolution of (a) the peak power, (b) FWHM, and (c) energy of the rogue wave with zero background. The parameters are the same as in Fig. 1.

Figure 3

The evolution of the rational fraction pulses, initiated by the configurations extracted from Eq. (4) at the corresponding positions: (a) $z_0=3.5$; (b) $z_0=4$; (c) $z_0=4.5$; (d) $z_0=5$ (see further details in the text). (e–h) The evolution of the corresponding pulse intensity and FWHM. The parameters are the same as in Fig. 1.

Figure 4

The evolution of (a) the Peregrine rogue wave excited by initial condition (10), and (b) the corresponding evolution of its spectral intensity. (c) The growing temporal-domain pulse shapes and (d) the corresponding spectral shapes at $\xi =1.45$ km, $1.50$ km, and $1.60$ km, respectively. The parameters are $P_0=0.3$ W, $\epsilon =0.0765$, and $T_1=3.7722$ ps.

Figure 5

(a) The evolution of the fraction pulse in the case when the background was removed from the Peregrine rogue wave. (b) and (c) The corresponding temporal and spectral shapes of the pulse at $\xi =1.45$ km, $1.50$ km, and $1.60$ km, respectively. (d), (e), and (f) The evolution of the peak power, FWHM, and soliton number $N$, respectively. The parameters are the same as in Fig. 4.

Figure 6

(a), (c), and (e) The evolution of the initial state (10), corresponding to the fraction pulse, with the background extracted from the PS at (a) $\xi =1.45$ km, (c) $\xi =1.50$ km, and (e) $\xi =1.60$ km, respectively. (b), (d), and (f) The corresponding evolution of the peak power. Parameters of initial condition (10) are the same as in Fig. 4.

Figure 7

The same as in Fig. 6, but with the Raman effect, with $T_R=3$ fs, taken into regard. The other parameters are the same as in Fig. 4.

Figure 8

The evolution of the breathing soliton extracted from the PS at $\xi =1.45$ km, under the the action of small perturbations added to initial condition (10). (a) The initial profile (10), perturbed by $0.1\%$ random noise. (b) The long-distance evolution of the pulse. (c) The corresponding evolution of the peak power. The parameters are the same as in Fig. 4.