How to excite a rogue wave

We propose initial conditions that could facilitate the excitation of rogue waves. Understanding the initial conditions that foster rogue waves could be useful both in attempts to avoid them by seafarers and in generating highly energetic pulses in optical fibers.

Figure 1

(Color online) (a) Collision of three ABs with eigenvalues: $l_1=0.2+i0.95$, $l_2=+i0.92$, and $l_3=-0.2+i0.95$. Coordinate and phase offsets are set to zero. All three ABs collide at one point with the resulting highest maximum amplitude being around 7. (b) Collision of three ABs with eigenvalues: $l_1=0.5+i0.95$, $l_2=+i0.92$, and $l_3=-0.5+i0.95$. Coordinate offsets: $t_0=12$ and $x_0=-12$. There are three points of collision with potential maximum amplitude around 5 in each case provided that the phase offsets at those points are 0. This is not the case here.

Figure 2

(Color online) Two types of ABs with nonzero velocity. (a) Galilean transformation is applied to AB with zero velocity. Here $V=1$ and ${\nu }_1=0.7$. (b) Direct calculation of AB with an eigenvalue $l_1=-0.08+i0.9$.

Figure 3

(Color online) Velocity of a single AB as a function of $l_r$. Here we have $l_i=0$ (blue curve passing through origin), $l_i=\pm 1$ (red curves giving velocity $\pm 1$ for small real part), and $l_i=\pm 2$ (green).

Figure 4

(Color online) Excitation of a single AB with nonzero velocity with a single sideband in the spectrum. (a) Periodic evolution of the peak amplitude. (b) Evolution of the solution from $x=0$ to $x=60$ [blue shaded area in (a)].

Figure 5

(Color online) Single AB with eigenvalue $l=0.2+i$. The AB crosses the input line, giving a complicated wave envelope at $x=-5$. This example shows that the initial conditions are not as trivial as those considered in Fig. 4.

Figure 6

Evolution of the peak amplitude when the initial cw has two harmonics of modulation. The high peak at the end of simulation is due to the in-phase collision of two ABs.

Figure 7

(Color online) (a) Evolution of the maximum of the wave envelope when the initial cw is perturbed by two harmonics. The initial condition is written at the top of part of this figure. The inset shows the location of the two frequencies within the MI gain curve. The three curves are calculated for the set of amplitudes $A_i$ written inside the plot. Only one of these sets produces a rogue wave with a high peak (red solid curve). Its evolution in the $\left(x,t\right)$ plane is shown in (b).

Figure 8

(Color online) (a) Maxima of the wave envelope vs $x$ for the initial condition written on top of the figure. The location of the frequency of the initial perturbation and its harmonics within the MI gain band is shown in the inset. (b) Contour plot of the wave amplitudes in the $\left(x,t\right)$ plane. The point in the middle of the dashed circle corresponds to the high amplitude rogue wave with the highest amplitude in (a).