From rogue wave solution to solitons

Using a generalized nonlinear Schrödinger equation, we investigate the transformation of a fundamental rogue wave solution to a collection of solitons. Taking the third-order dispersion, self-steepening, and Raman-induced self-frequency shift as the generalizing effects, we systematically observe how a fundamental rogue wave has an impact on its surrounding continuous wave background and reshapes its own characteristics while a group of solitons are created. Applying a local inverse scattering technique based on the periodization of an isolated structure, we show that the third-order dispersion and Raman-induced self-frequency shift generates a group of solitons in the neighborhood where the rogue wave solution emerges. Using a volume interpretation, we show that the self-steepening effect stretches the rogue wave solution by reducing its volume. Also, we find that with the Raman-induced self-frequency shift, a decelerating rogue wave generates a red-shifted Raman radiation while the rogue wave itself turns into a slow-moving soliton. We show that when third-order dispersion, self-steepening, and Raman-induced self-frequency shift act together on the rogue wave solution, each of these effects favor the rogue wave to generate a group of solitons near where it first emerges while the rogue wave itself also becomes one of these solitons.

Figure 1

(a) Temporal amplitude evolution of the emergence of a first-order rogue wave at $z=30$ with ${\epsilon }_3=-0.03$ (b) Corresponding spectral intensity evolution. It shows the emission of dispersive wave shown by the white arrow as observed in Ref. [39]. Note that in the analytic solution, the rogue wave appears at $z=0$ where as in simulation it is at $z=30$.

Figure 2

Top panel: Temporal amplitude evolution of a rogue wave under the influence of TOD. The rogue wave appears at $z=30$, and a skewed background wave dynamics is visible with a combination of different types of wave entities. Midpanel: The wave envelop is an instance extracted at $z=40$ after the emergence of the rogue wave. Bottom panel: Local-IST spectra of various types of wave profiles.

Figure 3

(a) Amplitude and (b) phase profiles of the rogue wave from Eq. (5), with $s=0.2$. It has a maximum amplitude of 3 with a distorted phase shift of $\pi$ across the peak.

Figure 4

(a) Rogue wave volume $V$ (blue) and the rate of volume change $dV/ds$ (red) versus $s$. (b) Onset of a rogue wave departing from its original state to become a soliton when $s=2$.

Figure 5

(a) Temporal phase and (b) spectral intensity evolutions of the rogue wave solution, Eq. (5) with $s=0$, 0.2, and 1. Note a spectral discontinuity is arise in the spectrum indicated by the red arrow.

Figure 6

(a) Extended amplitude evolution of a rogue wave with the SS effect when $s=0.2$. The recurring emergence of breathers is also achieved with the same amount of translations as the primary rogue wave. The rogue wave has appeared at a translated distance $t\approx 13$.

Figure 7

(a) Temporal and (b) spectral intensity evolutions of a fundamental rogue wave under the influence of RIFS effect when ${\tau }_R=0.012$. The rogue wave appearing at $z=30$ is decelerating along the propagation. A frequency down-shifting is observed in the spectral domain.

Figure 8

Trajectories of the center of mass of the rogue wave solution in the temporal (blue) and spectral (red) domains under the influence of RIFS effect.

Figure 9

Top panel: Temporal amplitude evolution of a rogue wave solution under the influence of the RIFS effect. A skewed background wave is visible with a various types of wave entities appearing atop. Midpanel: An extracted wave envelop at $z=40$ after the emergence of rogue wave. Bottom panel: Local-IST spectra of various localized wave profiles.

Figure 10

(a)–(c) are three different window sizes of the structure (iii) in the midpanel from Fig. 9. Corresponding periodization are presented in (d)–(f). Insets (g)–(i) are the respective local-IST spectrums of (d)–(f).

Figure 11

(a) Temporal and (b) spectral intensity evolutions of the fundamental rogue wave when ${\epsilon }_3=-0.03$, $s=0.2$, and ${\tau }_R=0.012$.

Figure 12

Decelerating rogue wave under the influence of TOD (${\epsilon }_3=0.03$), SS ($s=0.1$), and RIFS (${\tau }_R=0.008$). It transforms into a soliton.

Figure 13

(a) Temporal and (b) spectral intensity evolutions showing disintegration of $N=200$ soliton under the influence of TOD (${\epsilon }_3=0.03$), SS ($s=0.1$), and RIFS (${\tau }_R=0.001$).