Solitons in the midst of chaos

A system of coupled nonlinear Schrödinger equations describes pulse propagation in weakly birefringent optical fibers. Soliton solutions of this system are found numerically through the shooting method. We employ Poincaré surface of section plots—a standard dynamical systems approach—to analyze the phase space behavior of these solutions and neighboring trajectories. Chaotic behavior around the solitons is apparent and suggests dynamical instability. A Lyapunov stability analysis confirms this result. Thus, solitons exist in the midst of chaos.

Figure 1

The graph of $F\left(q\right)$ from Eq. (12) shown at $k=1∕2$. The dotted horizontal lines correspond to different values of $H$, each of them illustrating a different region of similar solutions to Eq. (9): $a$ and $a^{\prime }$ correspond to $H∊[-k^2∕2,0)$, $b$ and $b^{\prime }$ correspond to $H=0$, and $c$ corresponds to $H>0$.

Figure 2

A phase space diagram of the system in Eq. (11) with trajectories corresponding to the values of the Hamiltonian presented in Fig. 1: $a$ and $a^{\prime }$ correspond to $H∊[-k^2∕2,0)$, $b$ and $b^{\prime }$ correspond to $H=0$, and $c$ corresponds to $H>0$. The actual values are $H=-0.05$, $H=0$, and $H=0.05$. A time domain representation of the solutions corresponding to the trajectories can be seen in Fig. 3.

Figure 3

The time domain representation of several solutions of Eq. (11), corresponding to the phase space trajectories shown in Fig. 2. The same numerical values are used. Note that the soliton solution in (b) corresponds to a homoclinic trajectory in Fig. 2.

Figure 4

Argument for the existence of a symmetric soliton solution to the coupled nonlinear Schrödinger equation in the case $k_1=1$, $k_2=0.7$, and $\beta =2∕3$. (a) Contour plot of the difference between $p_1(t=0,c_1,c_2)$ when integrating from the left and its value when integrating from the right based on initial asymptotic coefficient pairs for the first normal mode; (b) similar contour plot for the second normal mode; (c) zero-level contours for both normal modes, with an intersection at $c_1=2.69$ and $c_2=1.07$; (d) the soliton corresponding to the initial conditions in (c).

Figure 5

(Color online) Numerical solutions with $\beta =2∕3$, $k_1=k_2=1$. The solid and the dashed curves represent $q_1$ and $q_2$, respectively, plotted vs the normalized time $t$.

Figure 6

(Color online) Numerical solutions with $\beta =2∕3$, $k_1=1$, $k_2=0.7$. The solid curve represents $q_1$ and the dashed curve represents $q_2$.

Figure 7

(Color online) Surfaces of section taken at $q_2=0.4$ with $\beta =2∕3$, $k_1=k_2=1$. (a) The entire phase space section is presented; (b) a magnification in the neighborhood of two of the soliton trajectories; (c) and (d) further magnifications with the soliton solution intersects in the middle of the figure. The solitons in (c) and (d) are placed at the intersection of the marker lines, dashed for the soliton in Fig. 5a and solid for the one in Fig. 5b. The diamonds and the crosses in (a) mark the intersections of the solitons in Figs. 5c, 5d, respectively.

Figure 8

Surface of section taken at $q_2=0.4$ with $\beta =2∕3$, $k_1=1$, and $k_2=0.7$. The soliton intersections represented by a circle, rectangle, upper triangles and lower triangles correspond to Figs. 6a, 6b, 6c, and 6d respectively. They are located in the same region of visible chaos.

Figure 9

(Color online) Linear stability analysis showing average normalized deviation vs time. The dotted lines correspond to solitons in the $k_1\ne k_2$ case (Fig. 6) and the dashed lines to solitons in the $k_1=k_2=1$ case, as in Fig. 5; the solid lines represent a soliton solution of the Manakov system [14] (triangles pointing left) and a regular oscillatory solution of the $\beta =2∕3$, $k_1=k_2=1$ system (triangles pointing right).