Rogue waves in nonlocal media

The generation of rogue waves is investigated in a class of nonlocal nonlinear Schrödinger (NLS) equations. In this system, modulation instability is suppressed as the effect of nonlocality increases. Despite this fact, there is a parameter regime where the number and amplitude of the rogue events increase as compared to the standard NLS equation, which is a limit of the system when nonlocality vanishes. Furthermore, the nature of these waves is investigated; while no analytical solutions are known to model these events, it is shown, numerically, that these rogue events differ significantly from the rational soliton (Peregrine) solution of the limiting NLS equation. The universal structure of the associated rogue waves is discussed and a local description is presented. These results can help in the experimental realization of rogue waves in these media.

Figure 1

Top: Growth rates for different values of the nonlocal parameter $\nu$. The far right curve corresponds to $\nu =0$ and the curves become smaller in amplitude and width until the value $\nu =200$, the far left curve. Bottom: The change of critical values $\mathrm{Im}\left\{{\omega }_{\max }\right\}$ and $k_c$ with the nonlocality $\nu$.

Figure 2

Probability density functions of the maximum value or $max\left\{\tilde{u}\right\}$ for different values of the nonlocal parameter $\nu$.

Figure 3

Top: The mean value of the PDFs and the mean value of the max 10% events with the nonlocal parameter $\nu$. The horizontal dashed lines indicate the relative values for $\nu =0$ and the vertical dashed lines the values of $\nu$ for which these values surpass the NLS system. Bottom: A zoom in around $\nu =0$.

Figure 4

Snapshots of typical evolutions around the rogue event for different values of the nonlocal parameter. The white circle depicts the rogue event.

Figure 5

Comparisons of a (randomly chosen) rogue event of the nonlocal equation with the known soliton and rational solutions.

Figure 6

A zoom in around a rogue event for the different values of the nonlocal parameter $\nu$. A fourth order rational solution has been fitted (red line) in all cases.

Figure 7

Left: Amplitude of rogue events for the different values of the nonlocal parameter $\nu$; right: their corresponding phases.

Figure 8

Left: Amplitude of rogue events for the different values of the nonlocal parameter $\nu$; right: their corresponding phases.

Figure 9

The local description of rogue waves for different values of the nonlocal parameter $\nu$.