Peregrine solitons and resonant radiation in cubic and quadratic media

Abstract

We present the fascinating phenomena of resonant radiation emitted by transient rogue waves in cubic and quadratic nonlinear media, particularly those shed from Peregrine solitons, one of the main wavepackets used today to model real-world rogue waves. In cubic media, it turns out that the emission of radiation from a Peregrine soliton can be attributed to the presence of higher-order dispersion, but is affected by the intrinsic local longitudinal variation of the soliton wavenumber. In quadratic media, we reveal that a two-color Peregrine rogue wave can resonantly radiate dispersive waves even in the absence of higher-order dispersion, subjected to a phase-matching mechanism that involves the second harmonic wave, and to a concomitant difference-frequency generation process. In both cubic and quadratic media, we provide simple analytic criteria for calculating the radiated frequencies in terms of material parameters, showing excellent agreement with numerical simulations.

Figures (8)

• Figure 1: Propagation of a perturbed radiating Peregrine soliton in the cubic case with $\beta_1=-1$ and $\sigma_1=-0.1$: (a) pseudo-color plot of spatio-temporal evolution of the intensity $|\psi|^2$, the dashed white line and dotted red line mark the predicted and numerical Peregrine soliton velocity, respectively; (b) evolution of the Fourier spectrum (in log scale); (c) output spectrum (in log scale, solid red) at $\xi=2$ superposed to the input spectrum (solid blue). In (b-c) the dashed black line shows the predicted RR frequency from Eq. \ref{['eq:CubicPM']}.
• Figure 2: Same as Fig. \ref{['fig:cubic1']} for stronger TOD $\sigma_1=-0.5$. In (c) the three dashed black lines mark the three predicted RR frequencies, while the dashed green line marks the first-order approximation $\omega_{RR} \simeq \textcolor{black}{3/\sigma_1}$.
• Figure 3: Impact of the third-order dispersion in the cubic case: frequencies $\omega_{RR1}$, $\omega_{RR2}$ and $\omega_{RR3}$ against the normalized coefficient $\sigma_1$; blue dots: numerical data; black dashed lines: theoretical approximation from Eq. \ref{['eq:CubicPM']}; red dashed line: first-order approximation $\tilde{\omega}_{RR}=\textcolor{black}{3/\sigma_1}$.
• Figure 4: Pseudo-color plot of the spatio-temporal evolution of intensities (a) at FF $|u_1|^2$ and (b) at SH $|u_2|^2$ components of a typical walking Peregrine soliton in the $(\tau,\xi)$ plane. (c,d): corresponding evolution of the FF (c) and SH (d) Fourier spectra (in log scale); the dashed black line marks the predicted resonant frequencies $\omega_{1,FC}^+$ in (c) and $\omega_{2,RR}^+$ in (d). Here $\beta_1=\beta_2=1$, $v=7.5$, $\delta k=-15$.
• Figure 5: RR frequencies $\omega_{2,RR}=\omega_{2,RR}^\pm$ versus GVD at SH $\beta_2$ at $v=5$, comparing theoretical predictions (dashed black curves) and numerical simulations (blue dots). Here $\beta_1=-1$, $\delta k=20$.