Breather-to-soliton transitions, nonlinear wave interactions, and modulational instability in a higher-order generalized nonlinear Schrödinger equation

We study the nonlinear waves on constant backgrounds of the higher-order generalized nonlinear Schrödinger (HGNLS) equation describing the propagation of ultrashort optical pulse in optical fibers. We derive the breather, rogue wave, and semirational solutions of the HGNLS equation. Our results show that these three types of solutions can be converted into the nonpulsating soliton solutions. In particular, we present the explicit conditions for the transitions between breathers and solitons with different structures. Further, we investigate the characteristics of the collisions between the soliton and breathers. Especially, based on the semirational solutions of the HGNLS equation, we display the novel interactions between the rogue waves and other nonlinear waves. In addition, we reveal the explicit relation between the transition and the distribution characteristics of the modulation instability growth rate.

Figure 1

First-order breather with $a=0.5,c=0.6,{\gamma }_1=0.1$, and  ${\lambda }_1={\lambda }_2^{*}=0.1+0.65i.$

Figure 2

Second-order breather with  $a=0.4,c=0.4,{\gamma }_1=0.5,{\lambda }_1={\lambda }_2^{*}=0.1+0.5i$, and  ${\lambda }_3={\lambda }_4^{*}=0.5i.$

Figure 3

First-order rogue wave with  $a=0.1,c=0.8,{\gamma }_1=0.1$, and ${\lambda }_0=-0.05+0.8i.$

Figure 4

Second-order semirational rogue wave with  $a=0.4,c=0.6,{\gamma }_1=0.1,{\lambda }_0=-0.2+0.6i$, and  ${\lambda }_1={\lambda }_2^{*}=0.1+0.6i.$

Figure 5

Solutions of Eq. (5) on a complex plane of $\lambda =\alpha +\beta i.$ (a): $a=0.5,c=1$, and${\gamma }_1=-\frac{6}{11}.$ (b): $a=0.5,c=1$, and${\gamma }_1=-\frac{17}{11}.$ (c): $a=0.5,c=1$, and${\gamma }_1=-\frac{19}{55}.$

Figure 6

A breather transformed into an oscillation W-shaped soliton with  $a=0.5,c=1,{\gamma }_1=-0.6$, and ${\lambda }_1={\lambda }_2^{*}=0.504+0.7i.$ (b) is the contour plot of (a). (c) is the cross-sectional view of (a) at $t=0$.

Figure 7

A breather transformed into an oscillation M-shaped soliton with  $a=0.5,c=1,{\gamma }_1=-0.6,$ and ${\lambda }_1={\lambda }_2^{*}=0.504-0.7i.$ (b) is the contour plot of (a). (c) is the cross-sectional view of (a) at $t=0$.

Figure 8

A breather transformed into a W-shaped soliton with  $a=0.5,c=0.9,{\gamma }_1=-0.243$, and ${\lambda }_1={\lambda }_2^{*}=-0.25-i.$ (b) is the contour plot of (a). (c) is the cross-sectional view of (a) at $t=0$.

Figure 9

A breather transformed into a M-shaped soliton with  $a=0.5,c=0.6,{\gamma }_1=-0.6$, and ${\lambda }_1={\lambda }_2^{*}=0.349-0.7i.$ (b) is the contour plot of (a). (c) is the cross-sectional view of (a) at $t=0$.

Figure 10

A breather transformed into an oscillation W-shaped soliton with  $a=0.5,c=1,{\gamma }_1=-4.382$, and ${\lambda }_1={\lambda }_2^{*}=0.504+0.36i.$ (b) is the contour plot of (a). (c) is the cross-sectional view of (a) at $t=0$.

Figure 11

A breather transformed into an antidark soliton with  $a=0.5,c=0.9,{\gamma }_1=-0.076$, and ${\lambda }_1={\lambda }_2^{*}=-0.25+1.8i.$ (b) is the contour plot of (a). (c) is the cross-sectional view of (a) at $t=0$.

Figure 12

A breather transformed into a periodic wave with  $a=0.5,c=0.9,{\gamma }_1=-0.373$, and ${\lambda }_1={\lambda }_2^{*}=-0.25+0.8i.$ (b) is the contour plot of (a). (c) is the cross-sectional view of (a) at $t=0$.

Figure 13

A rogue wave transformed into a W-shaped soliton with  $a=0.5,c=0.9,{\gamma }_1=-0.298$, and ${\lambda }_0=-0.25+0.9i.$ (b) is the contour plot of (a). (c) is the cross-sectional view of (a) at $t=0$.

Figure 14

Collision between a W-shaped soliton and a breather with  $a=0.3,c=0.7,{\gamma }_1=-0.257,{\lambda }_1={\lambda }_2^{*}=-0.12+0.5i$, and ${\lambda }_3={\lambda }_4^{*}=0.2+0.9i.$ (b) is the contour plot of (a).

Figure 15

Collision between two W-shaped solitons with  $a=0,c=0.5,{\gamma }_1=0.355,{\lambda }_1={\lambda }_2^{*}=-0.6+0.5i$, and ${\lambda }_3={\lambda }_4^{*}=0.6+0.5i.$ (b) is the contour plot of (a). (c) is the cross-sectional view of (a).

Figure 16

Elastic collision between a W-shaped soliton and an oscillation soliton with  $a=0.7,c=0.5,{\gamma }_1=0.613,{\lambda }_1={\lambda }_2^{*}=-0.37+0.51i$, and ${\lambda }_3={\lambda }_4^{*}=0.6-0.5i.$ (b) is the contour plot of (a). (c) is the cross-sectional view of (a).

Figure 17

Collision between a periodic wave and M-shaped solitons with  $a=0.7,c=0.5,{\gamma }_1=0.539,{\lambda }_1={\lambda }_2^{*}=-0.35+0.382i$, and ${\lambda }_3={\lambda }_4^{*}=0.6+0.44i.$ (b) is the contour plot of (a).

Figure 18

Collision between a periodic wave and oscillation solitons with  $a=0.4,c=0.7,{\gamma }_1=-0.555,{\lambda }_1={\lambda }_2^{*}=-0.2+0.667i$, and ${\lambda }_3={\lambda }_4^{*}=0.18-0.52i.$ (b) is the contour plot of (a).

Figure 19

Collision between a breather and a W-shaped soliton with  $S_0=0,a=0.4,c=0.6,{\gamma }_1=-0.833,{\lambda }_0=-0.2+0.6i$, and ${\lambda }_1={\lambda }_2^{*}=0.1+0.6i.$

Figure 20

Inelastic collision between a breather and a W-shaped soliton with  $S_0=1.5,a=0.4,c=0.6,{\gamma }_1=-0.833,{\lambda }_0=-0.2+0.6i$, and ${\lambda }_1={\lambda }_2^{*}=0.1+0.6i.$

Figure 21

Collision between a W-shaped soliton and a rogue wave with  $a=0.5,c=0.6,{\gamma }_1=-0.188,{\lambda }_0=-0.25+0.6i$, and ${\lambda }_1={\lambda }_2^{*}=-1.1i.$

Figure 22

Collision between an oscillation soliton and a rogue wave with  $a=0.3,c=0.9,{\gamma }_1=-0.737,{\lambda }_0=-0.15+0.9i$, and ${\lambda }_1={\lambda }_2^{*}=0.42-0.6i.$

Figure 23

Collision between an antidark soliton and a rogue wave with  $a=0.5,c=0.6,{\gamma }_1=-0.188,{\lambda }_0=-0.25+0.6i$, and ${\lambda }_1={\lambda }_2^{*}=1.1i.$

Figure 24

The effect of ${\gamma }_1$ on the maximum growth rate of the instability with $c=0.9$ and $a=0.5$.

Figure 25

Characteristics of MI growth rate $\mathrm{\Omega }$ on $(a,\mathrm{\Lambda })$ plane with c=1. (a) ${\gamma }_1=0$; (b) ${\gamma }_1=0.01$. Here the dashed red lines represent the stability region in the perturbation frequency region $-2c<\mathrm{\Lambda }<2c$, which is given as  $a=a_s=\pm \sqrt{\frac{1}{6{\gamma }_1}+c^2}$.