Soliton excitations on a continuous-wave background in the modulational instability regime with fourth-order effects

We study the correspondence between modulational instability and types of fundamental nonlinear excitation in a nonlinear fiber with both third-order and fourth-order effects. Some soliton excitations are obtained in the modulational instability regime which have not been found in nonlinear fibers with second-order effects and third-order effects. Explicit analysis suggests that the existence of solitons is related to the modulation stability circle in the modulation instability regime, and they just exist in the modulational instability regime outside of the modulational stability circle. It should be emphasized that the solitons exist only with two special profiles on a continuous-wave background at a certain frequency. The evolution stability of the solitons is tested numerically by adding some noise to initial states, which indicates that they are robust against perturbations even in the modulation instability regime. Further analysis indicates that solitons in the modulational instability regime are caused by fourth-order effects.

Figure 1

Modulation instability (MI) gain $G$ distribution on the background frequency $\omega$ and perturbation frequency $\mathrm{\Omega }$ plane. MS, modulation stability. There is an MS circle in the MI band, shown by the dashed white line. The MI gain value is not 0 for the regime outside and inside of the MS circle, shown in the lower frame. Parameters are $A=1$, $\beta =\frac{1}{6}$, and $\gamma =-\frac{5+\sqrt{15}}{48}$.

Figure 2

(a) Phase diagram for different types of nonlinear waves on the modulation instability gain spectrum plane corresponding to Fig. 1. AB, Akhmediev breather; RW, rogue wave; K-M, Kuznetsov-Ma breather; PW, periodic wave; WST, $\mathsf{W}$-shaped soliton train; ${\mathrm{WS}}_r$, rational $\mathsf{W}$-shaped soliton; AD, antidark soliton; ${\mathrm{WS}}_{\mathrm{nr}}$, $\mathsf{W}$-shaped soliton. (b) Resonant line in the MI regime, on which AD and ${\mathrm{WS}}_{\mathrm{nr}}$ solitons can exist. Parameters are the same as in Fig. 1.

Figure 3

Optical amplitude distributions $|{\psi }_s|$ of (a) a nonrational $\mathsf{W}$-shaped soliton and (b) an antidark soliton on a continuous-wave background. Parameters are $A=1$, $\beta =\frac{1}{12}$, $\gamma =-\frac{1}{36}$, $\omega =0$, $b=2$.

Figure 4

(a) Numerical evolution from the initial conditions ${\psi }_p(t,-5)={\psi }_{\mathrm{AD}}[1+0.01\mathrm{random}\left(t\right)]$. (b) Numerical evolution of $\left|\psi \right(t,z\left)\right|$ from the AD with the addition of weak Gaussian pulse perturbations in the form ${\psi }_{\mathrm{AD}}[1+0.1e^{-{(t+1)}^2/4}]$. Parameters are $A=1$, $\beta =\frac{1}{12}$, $\gamma =-\frac{1}{36}$, $\omega =0$, $b=2$.

Figure 5

Positions of the AD and ${\mathrm{WS}}_{\mathrm{nr}}$ with different energies on the modulational instability gain spectrum plane. Green, aqua, and purple points correspond to the AD and ${\mathrm{WS}}_{\mathrm{nr}}$ with different energies ${\varepsilon }_s$; blue points are modulation stability points on the resonance line. It shows that the position approaches the MS point as the perturbation energy decreases (the soliton amplitude becomes lower). Parameters are $A=1$, $\beta =\frac{1}{6}$, $\gamma =-\frac{5+\sqrt{15}}{48}$.