Generation mechanisms of fundamental rogue wave spatial-temporal structure

We discuss the generation mechanism of fundamental rogue wave structures in $N$-component coupled systems, based on analytical solutions of the nonlinear Schrödinger equation and modulational instability analysis. Our analysis discloses that the pattern of a fundamental rogue wave is determined by the evolution energy and growth rate of the resonant perturbation that is responsible for forming the rogue wave. This finding allows one to predict the rogue wave pattern without the need to solve the $N$-component coupled nonlinear Schrödinger equation. Furthermore, our results show that $N$-component coupled nonlinear Schrödinger systems may possess $N$ different fundamental rogue wave patterns at most. These results can be extended to evaluate the type and number of fundamental rogue wave structure in other coupled nonlinear systems.

Figure 1

The rogue wave patterns in the third component for (a) five-component case and (b) six-component coupled case. It is seen that there are two anti-eye-shaped rogue waves, two four-petaled ones, and one eye-shaped one in (a). There are four anti-eye-shaped ones, one eye-shaped one, and one four-petaled one in (b). The anti-eye-shaped rogue waves are a bit different from each other. The results agree precisely with the patterns predicted by modulational instability analysis. Parameters: (a) $a_i=1,b_1=-2,b_2=-1,b_3=0,b_4=1,b_5=2,x_1=-20,x_2=-10,x_3=0,x_4=10,x_5=20,t_i=0,(i=1,2,3,4,5)$; (b) $a_i=1,b_1=-3,b_2=-2,b_3=-1,b_4=1,b_5=2,b_6=3,x_1=-37.5,x_2=-22.5,x_3=-7.5,x_4=7.5,x_5=22.5,x_6=37.5,t_i=0(i=1,2,3,4,5,6)$.

Figure 2

The numerical stability test of the rogue wave against parameter deviations in a three-component case. The numerical evolution from the ideal initial condition given by the exact solution at $t=-3$ is shown in the left columns. The results with parameter deviation are shown in the right columns. The figures in the $i\mathrm{th}$ row represent the density plot of $|q_i|(i=1,2,3)$, (a), (b) $|q_1|$; (c), (d) $|q_2|$; (e), (f) $|q_3|$. It is shown that rogue waves are stable against parameter deviations.

Figure 3

The numerical evolution from the ideal initial condition with Gaussian white noises. The three figures represent the density plot for the three different components $\left(\right|q_1|,|q_2|,|q_3\left|\right)$ for the three-component NLSE. It is shown that the rogue wave signals are robust against the noises.