Generation of higher-order rogue waves from multibreathers by double degeneracy in an optical fiber

In this paper, we construct a special kind of breather solution of the nonlinear Schrödinger (NLS) equation, the so-called breather-positon (b-positon for short), which can be obtained by taking the limit ${\lambda }_j\to {\lambda }_1$ of the Lax pair eigenvalues in the order-$n$ periodic solution, which is generated by the $n$-fold Darboux transformation from a special “seed” solution–plane wave. Further, an order-$n$ b-positon gives an order-$n$ rogue wave under a limit ${\lambda }_1\to {\lambda }_0$. Here, ${\lambda }_0$ is a special eigenvalue in a breather of the NLS equation such that its period goes to infinity. Several analytical plots of order-2 breather confirm visually this double degeneration. The last limit in this double degeneration can be realized approximately in an optical fiber governed by the NLS equation, in which an injected initial ideal pulse is created by a frequency comb system and a programable optical filter (wave shaper) according to the profile of an analytical form of the b-positon at a certain position $z_0$. We also suggest a new way to observe higher-order rogue waves generation in an optical fiber, namely, measure the patterns at the central region of the higher-order b-positon generated by above ideal initial pulses when ${\lambda }_1$ is very close to the ${\lambda }_0$. The excellent agreement between the numerical solutions generated from initial ideal inputs with a low signal-to-noise ratio and analytical solutions of order-2 b-positon supports strongly this way in a realistic optical fiber system. Our results also show the validity of the generating mechanism of a higher-order rogue waves from a multibreathers through the double degeneration.

Figure 1

A sketchy demonstration of the limit ${\lambda }_3\to {\lambda }_1$ in an order-2 breather $|q^{\left[2\right]}{|}^2$ (density plot) with $a=0,c=\frac{1}{2},{\lambda }_1=\frac{2}{5}i,s_1=0$. The central region of the left (right) column is a fundamental (triangular) pattern. From top to bottom, the ratio of two breathers are 4:5, 8:9, 15:16, and there are 3, 7, 14 peaks in one period of time. The other parameters of the breathers, respectively, are (a) ${\lambda }_3=\frac{\sqrt{7}}{8}i,s_0=0$, (b) ${\lambda }_3=\frac{\sqrt{7}}{8}i,s_0=1$, (c) ${\lambda }_3=\frac{\sqrt{871}}{80}i,s_0=0$, (d) ${\lambda }_3=\frac{\sqrt{871}}{80}i,s_0=1$, (e) ${\lambda }_3=\frac{3\sqrt{41}}{50}i,s_0=0$, (f) ${\lambda }_3=\frac{3\sqrt{41}}{50}i,s_0=1$.

Figure 2

A sketchy demonstration of the limit ${\lambda }_1\to {\lambda }_0$ in an order-2 b-positon $|q_{\mathrm{b}-\mathrm{positon}}^{\left[2\right]}{|}^2$ (density plot) with ${\lambda }_0=-\frac{a}{2}+ic,a=0,c=\frac{1}{2},s_0=0$. The central region of the left (right) column is a fundamental (triangular) pattern. Note that the left column is a continuous limit of the left column in Fig. 1, but the right column is not because values of $s_0$ in two figures are different. The other parameters of the b-positons, respectively, are (a) ${\lambda }_1=\frac{2}{5}i,s_1=0$, (b) ${\lambda }_1=\frac{2}{5}i,s_1=100$, (c) ${\lambda }_1=\frac{12}{25}i,s_1=0$, (d) ${\lambda }_1=\frac{12}{25}i,s_1=100$, (e) ${\lambda }_1=\frac{72}{145}i,s_1=0$, (f) ${\lambda }_1=\frac{72}{145}i,s_1=100$.

Figure 3

The comparison of an order-2 b-positon $|q_{\mathrm{b}-\mathrm{positon}}^{\left[2\right]}{|}^2$ (red dash line) with an order-2 fundamental rogue wave $|q_{\mathrm{rw}}^{\left[2\right]}{|}^2$ (blue solid line) at $z=0$. The order-2 b-positons are generated through the same parameters as Figs. 2, 2, and 2, respectively. The order-2 fundamental rogue wave is generated by Eq. (A2) with parameters $a=0,c=\frac{1}{2},s_0=s_1=0$. The parameter ${\lambda }_1$ of the breathers is $\frac{2}{5}i$ in (a), $\frac{12}{25}i$ in (b), and $\frac{72}{145}i$ in (c).

Figure 4

The density plots of three patterns in central region of an order-3 b-positon $|q_{\mathrm{b}-\mathrm{positon}}^{\left[3\right]}{|}^2$ with $s_0=0,a=0,c=\frac{1}{2}$. (a) The fundamental pattern with $s_1=0,s_2=0,{\lambda }_1=\frac{6}{13}i$, (b) the triangular pattern with $s_1=50,s_2=0,{\lambda }_1=\frac{6}{13}i$, (c) the circular pattern with $s_1=0,s_2=500,{\lambda }_1=\frac{6}{13}i$, (d) the circular pattern with different directions with $s_1=0,s_2=500,{\lambda }_1=\frac{13}{24}i$.

Figure 5

The density plots of four patterns in central region of an order-4 b-positon $|q_{\mathrm{b}-\mathrm{positon}}^{\left[4\right]}{|}^2$ with $a=0,c=\frac{1}{2},s_0=s_2=0,{\lambda }_1=\frac{6}{13}i$. (a) The fundamental pattern with $s_1=0,s_3=0$, (b) the triangular pattern with $s_1=50,s_3=0$, (c) the circular pattern with an inner fundamental pattern when $s_1=0,s_3=5\times {10}^3$, (d) the circular pattern with an inner triangular pattern when $s_1=20,s_3=5\times {10}^4$.

Figure 6

The density plots of six patterns in central region of an order-5 b-positon $|q_{\mathrm{b}-\mathrm{positon}}^{\left[5\right]}{|}^2$ with $a=0,c=\frac{1}{2},s_0=0,{\lambda }_1=\frac{6}{13}i$. (a) The fundamental pattern with $s_1=0,s_2=0,s_3=0,s_4=0$, (b) the triangular pattern with $s_1=50,s_2=0,s_3=0,s_4=0$, (c) the circular pattern with an inner fundamental pattern when $s_1=0,s_2=0,s_3=0,s_4=5\times {10}^5$, (d) the circular pattern with an inner triangular pattern when $s_1=20,s_2=0,s_3=0,s_4=2\times {10}^6$, (e) the circular pattern with inner decomposed peaks when $s_1=0,s_2=500,s_3=0,s_4=5\times {10}^5$, (f) two-ring pattern with $s_1=0,s_2=0,s_3=5\times {10}^4,s_4=0$.

Figure 7

Pulses of the fundamental pattern of an order-2 b-positon with $c=\frac{1}{2},{\lambda }_1=\frac{2}{5}i,a=s_0=s_1=0$. (a) Output pulse (theoretically predicted) at $z_1=0$, the amplitude is 4.41 at $t=0$. (b) Ideal initial input pulse at $z_0=-6.80$. Note that the amplitude is associated with the main peak in Figs. 15 and 15.

Figure 8

Pulses of the triangular pattern of an order-2 b-positon with $c=\frac{1}{2},{\lambda }_1=\frac{2}{5}i,a=s_0=0,s_1=50$. (a) Output pulse-2 (theoretically predicted) at $z_2=4.42$, the amplitude is 1.54 at $t=\pm 4.47$. (b) Output pulse-1 (theoretically predicted) at $z_1=-4.81$, the amplitude is 1.86 at $t=0$. (c) Ideal initial input pulse at $z_0=-7.43$. Note that three amplitudes are associated with the triplets in Figs. 15 and 15 around the coordinate origin.

Figure 9

Pulses of the fundamental pattern of an order-3 b-positon with $c=\frac{1}{2},{\lambda }_1=\frac{2}{5}i,a=s_0=s_1=s_2=0$. (a) Output pulse (theoretically predicted) at $z_1=0$, the amplitude is 8.41 at $t=0$. (b) Ideal initial input pulse at $z_0=-12.4$. Note that the amplitude is associated with the main peak in Figs. 16 and 16.

Figure 10

Pulses of the triangular pattern of an order-3 b-positon with $c=\frac{1}{2},{\lambda }_1=\frac{2}{5}i,a=s_0=s_2=0,s_1=50$. (a) Output pulse-3 (theoretically predicted) at $z_3=8$, the amplitudes are 1.43 at $t=\pm 8.65$ and 1.46 at $t=0$. (b) Output pulse-2 (theoretically predicted) at $z_2=-1.28$, the amplitudes are 1.65 at $t=\pm 4.21$. (c) Output pulse-1 (theoretically predicted) at ($z_1=-8$), the amplitude 2.05 at $t=0$. (d) Ideal initial input pulse at $z_0=-12.8$. Note that the six amplitudes are associated with the six peaks in triangular Figs. 16 and 16.

Figure 11

Pulses of the circular pattern of an order-3 b-positon with $c=\frac{1}{2},{\lambda }_1=\frac{2}{5}i,a=s_0=s_1=0,s_2=500$. (a) Output pulse-3 (theoretically predicted) at $z_3=7$, the amplitudes are 1.16 at $t=-7.03$ and 1.78 at $t=2.22$). (a) Output pulse-2 (theoretically predicted) at $z_2=0$, the amplitudes are 1.69 at $t=0$ and 1.28 at $t=8.06$. (c) Output pulse-1 (theoretically predicted) at $z_1=-7$ is the same as the result of $z_3$. (d) Ideal initial input pulse at $z_0=-12.35$. Note that the six amplitudes are associated with six peaks in circle of Figs. 16 and 16.

Figure 12

The evolution of an order-2 b-positon $|q_{\mathrm{b}-\mathrm{positon}}^{\left[2\right]}{|}^2$ (density plot) on $z-t$ plane with $c=\frac{1}{2},s_1=0,{\lambda }_1=\frac{2}{5}i$. (a) The fundamental pattern with $s_0=0,a=0$; (b) the titled propagation with a minor rotation when $s_0=0,a=\frac{1}{2}$; (c) the fundamental pattern with $s_0=5,a=0$, which has been shifted along negative direction of $t$ axis about $\mathrm{Re}\left(s_0\right)$ unit by comparing with (a); (d) the fundamental pattern with $s_0=i,a=0$, which has been shifted along negative direction of $z$ axis with deformation of the profile in central region about $\mathrm{Im}\left(s_0\right)$ unit by comparing with (a). These shifts are originated from contributions of $\mathrm{Re}\left(s_0\right)$ or $\mathrm{Im}\left(s_0\right)$.

Figure 13

The evolution of an order-2 b-positon $|q_{\mathrm{b}-\mathrm{positon}}^{\left[2\right]}{|}^2$ (density plot) on $z-t$ plane with $a=0,c=\frac{1}{2},{\lambda }_1=\frac{12}{25}i$. (a) The fundamental pattern with $s_0=10,s_1=0$, (b) the triangular pattern with $s_0=2i,s_1=0$. By comparing the two pictures in the last row of Fig. 12, there exists a larger shift of the central profile and also a significantly increase of the distance of two peaks along the time axis. This effects reflect the contribution of the nonzero value of $s_0$.

Figure 14

The tilted propagation of an order-3 b-positon $|q_{\mathrm{b}-\mathrm{positon}}^{\left[3\right]}{|}^2$ on $z-t$ plane with $a=1,c=\frac{1}{2},s_0=s_1=s_2=0,{\lambda }_1=\frac{6}{13}i$. (a) The density plot, (b) the central profile of (a), (c) three dimensional profile of $|q_{\mathrm{b}-\mathrm{positon}}^{\left[3\right]}{|}^2$, (d) the central profile of (c).

Figure 15

Two patterns of an order-2 b-positon $|q_{\mathrm{b}-\mathrm{positon}}^{\left[2\right]}{|}^2$ with $c=\frac{1}{2},{\lambda }_1=\frac{2}{5}i,a=s_0=0$. (a) The fundamental pattern with $s_1=0$ (density plot), (b) the fundamental pattern with $s_1=0$ (three dimensional profile), (c) the triangular pattern with $s_1=50$ (density plot), (d) the triangular pattern with $s_1=50$ (three dimensional profile). The ideal initial input and output(theoretically predicted) pulses are given in Figs. 7 and 8.

Figure 16

Three patterns of an order-3 b-positon $|q_{\mathrm{b}-\mathrm{positon}}^{\left[3\right]}{|}^2$ with $c=\frac{1}{2},{\lambda }_1=\frac{2}{5}i,a=s_0=0$. (a) The fundamental pattern with $s_1=s_2=0$ (density plot), (b) the fundamental pattern with $s_1=s_2=0$ (three dimensional profile), (c) the triangular pattern with $s_1=50,s_2=0$ (density plot), (d) the triangular pattern with $s_1=50,s_2=0$ (three dimensional profile), (e) the circular pattern with $s_1=0,s_2=500$ (density plot), (f) the circular pattern with $s_1=0,s_2=500$ (three dimensional profile). The ideal initial input and output(theoretically predicted) pulses are given in Figs. 9–11.

Figure 17

Numerical simulation of the fundamental pattern in central region of an order-2 b-positon $|q_{\mathrm{b}-\mathrm{positon}}^{\left[2\right]}{|}^2$ with $s_0=s_1=0,a=0,c=\frac{1}{2},{\lambda }_1=\frac{2}{5}i$. Red line denotes the theoretical (and exact) result, and square point denotes the result of numerical simulation. The significant discrepancies occur at the two ends of period due to the reflection of simulation. (a) The initial signal of fundamental pattern without noise at $z=-6.80$, (b) The observe signal of fundamental pattern without noise at $z=0$, (c) The initial signal of fundamental pattern with noise at $z=-6.80$ (SNR=100), (d) The observe signal of fundamental pattern with noise at $z=0$ (SNR = 100).

Figure 18

Numerical simulation of three periods from periodic extension of Fig. 17. In order to get recognizable curve, theoretical results are not added. (a) The initial signal of fundamental pattern without noise at $z=-6.80$, (b) The observe signal of fundamental pattern without noise at $z=0$, (c) The initial signal of fundamental pattern with noise at $z=-6.80$ (SNR = 100), (d) The observe signal of fundamental pattern with noise at $z=0$ (SNR = 100).

Figure 19

Numerical simulation of the triangular pattern in central region of an order-2 b-positon $|q_{\mathrm{b}-\mathrm{positon}}^{\left[2\right]}{|}^2$ with $s_0=0,s_1=50,a=0,c=\frac{1}{2},{\lambda }_1=\frac{2}{5}i$. Red line denotes the theoretical (and exact) result, and square point denotes the result of numerical simulation. The significant discrepancies occur at the two ends of period due to the reflection of simulation. (a) The initial signal of triangular pattern without noise at $z=-7.43$, (b) The observe signal of triangular pattern without noise at $z=-4.81$, (c) The observe signal of triangular pattern without noise at $z=4.42$, (d) The initial signal of triangular pattern with noise at $z=-7.43$ (SNR = 100), (e) The observe signal of triangular pattern with noise at $z=-4.81$ (SNR = 100), (f) The observe signal of triangular pattern with noise at $z=4.42$ (SNR = 100).

Figure 20

Numerical simulation of three periods from periodic extension of Fig. 19. In order to get recognizable curve, theoretical results are not added. (a) The initial signal of triangular pattern without noise at $z=-7.43$, (b) The observe signal of triangular pattern without noise at $z=-4.81$, (c) The observe signal of triangular pattern without noise at $z=4.42$, (d) The initial signal of triangular pattern with noise at $z=-7.43$ (SNR = 100), (e) The observe signal of triangular pattern with noise at $z=-4.81$ (SNR = 100), (f) The observe signal of triangular pattern with noise at $z=4.42$ (SNR = 100).

Figure 21

The increasing trend of instability for an order-2 b-positon $|q_{\mathrm{b}-\mathrm{positon}}^{\left[2\right]}{|}^2$ with SNR = 100 and parameters ${\lambda }_0=-\frac{a}{2}+ic,a=0,c=\frac{1}{2},s_0=0,s_1=0$. From top to bottom, this b-positon is approaching an order-2 rogue wave as ${\lambda }_1$ tends to ${\lambda }_0$, which corresponds to the left column of Fig. 2. The right column is the corresponding density plot of the left. The parameter ${\lambda }_1$ of b-positons is $\frac{2}{5}i$ in (a), (b), $\frac{6}{13}i$ in (c), (d), $\frac{12}{25}i$ in (e), (f), and $\frac{1}{2}i$ in (g), (h).