Statistics of vector Manakov rogue waves

We present a statistical analysis based on the height and return-time probabilities of high-amplitude wave events in both focusing and defocusing Manakov systems. We find that analytical rational or semirational solutions, associated with extreme, rogue wave (RW) structures, are the leading high-amplitude events in this system. We define the thresholds for classifying an extreme wave event as a RW. Our results indicate that there is a strong relationship between the type of RW and the mechanism which is responsible for its creation. Initially, high-amplitude events originate from modulation instability. Upon subsequent evolution, the interaction among these events prevails as the mechanism for RW creation. We suggest a strategy for confirming the basic properties of different extreme events. This involves the definition of proper statistical measures at each stage of the RW dynamics. Our results point to the need for redefining criteria for identifying RW events.

Figure 1

Localized patterns in the focusing Manakov system for $a_1\left(0\right)=a_2\left(0\right)=1$ and (a) $q_1=q_2=0$ and $q=0$ $\left[\right|u^{\left(1\right)}(t,z)\left|\right]$ and (b) and (c) $q_2=-q_1=1$ and $q=1$ [plots of two components $|u^{\left(1\right)}(t,z)|$ and $|u^{\left(2\right)}(t,z)|$ are shown separately]. The absolute values of the corresponding amplitudes are shown in the color bar in (c) and it is mutual for all plots in the figure. Initial small uniform random and periodic perturbations are added to the plane-wave background, which is amplitude modulated by a super Gaussian.

Figure 2

Localized patterns [$|u^{\left(1\right)}(t,z)|$ and $|u^{\left(2\right)}(t,z)|$] in the defocusing Manakov system for (a) and (b) $a_1\left(0\right)=a_2\left(0\right)=0.8$, $q_2=-q_1=1$, and $q=1$ (region of existence of rational solitons). (c) The case outside the region of the rational RW existence reported in [20] is presented, with $a_1=a_2=1$ and $q=0$ $\left[\right|u^{\left(1\right)}(t,z)\left|\right]$. The absolute values of the corresponding amplitudes are shown in the color bar in (c) and it is mutual for all plots in the figure. Initially small cosine and uniform random perturbations are added to the plane-wave background, amplitude modulated by a super Gaussian.

Figure 3

Schematic illustration of determination of the wave height. The quantity $\eta \left(z\right)=\left|u\right(z\left)\right|-\left|u\right(0\left)\right|$ is the wave amplitude elevation [7].

Figure 4

Amplitude evolution plots for the focusing Manakov system with initial plane-wave parameters (a) $a_1=a_2=1$ and $q=0$ ($q_1=q_2=0$) and (b) $a_1=a_2=1$ and $q=1$ ($q_2=-q_1=1$). Both sets of parameters belong to the MI development region. Two regions in these plots can be distinguished with respect to the presence of isolated localized patterns: the region up to $z\approx 30$ and above $z\approx 50$ (indicated by dashed lines). Since the purpose of this figure is to provide a general picture of the dynamics of the system, we only plot the amplitudes of the first component of the vector fields, for the sake of simplicity. The maximum wave amplitude is (a) 4.5 and (b) 4.

Figure 5

The $P_h$ vs $h$ in (a) and (b) linear and (c) and (d) semilogarithmic scales for initial plane-wave parameters (a) and (c) $a_1=a_2=1$ and $q=0$ and (b) and (d) $a_1=a_2=1$ and $q=1$ ($q_2=-q_1=1$). Different curves correspond to the height distributions of events belonging to different $z$ ranges: black squares, $z=10-40$; blue circles, $z=10-80$; red triangle, $z=40-80$. In (c) and (d), solid lines present GGD fits of the corresponding $P_h$ curves.

Figure 6

The $P_r$ vs $r/R$ for (a) and (b) $a_1=a_2=1$ and $q=0$ and (c) and (d) $a_1=a_2=1$ and $q=1$ ($q_2=-q_1=1$), with respect to different threshold amplitudes $h_r$. Three curves are obtained for (a) and (c) $h_r=0.7h_s$ and (b) and (d) $h_r=2.2h_s$. Different curves correspond to different calculation windows, as in Fig. 5. The amplitude threshold values are summarized in Table 2 for completeness.

Figure 7

Amplitude evolution plots in the defocusing Manakov system with initial plane-wave parameters (a) $a_1=a_2=1$ and $q=0$ and (b) $a_1=a_2=0.8$ and $q=1$ ($q_2=-q_1=1$). The first set of parameters is outside the baseband MI region, while the second is inside of it. The maximum amplitude is (a) 4 and (b) 3.

Figure 8

The $P_h$ vs $h$ in (a) and (b) linear and (c) and (d) semilogarithmic scales for initial plane-wave parameters (a) and (c) $a_1=a_2=1$ and $q=0$ and (b) and (d) $a_1=a_2=0.8$ and $q=1$ ($q_2=-q_1=1$). Different curves corresponding to the height distributions of events belonging to regions of $z$ are explicitly reported in the plots. In (c) and (d) solid lines represent GGD fits of the corresponding $P_h$ curves.

Figure 9

The $P_r$ vs $r/R$ for (a) and (b) $a_1=a_2=1$ and $q=0$ and (c) and (d) $a_1=a_2=0.8$ and $q=1$ ($q_2=-q_1=1$), with respect to different threshold amplitudes $h_r$. Three curves are obtained for (a) and (c) $h_r=0.7h_s$ and (b) and (d) $h_r=2.2h_s$. Different curves correspond to different calculation windows, as in Fig. 8.