Stable long-distance propagation and on-off switching of colliding soliton sequences with dissipative interaction

We study propagation and on-off switching of two colliding soliton sequences in the presence of second-order dispersion, Kerr nonlinearity, linear loss, cubic gain, and quintic loss. Employing a Lotka-Volterra (LV) model for dynamics of soliton amplitudes along with simulations with two perturbed coupled nonlinear Schrödinger (NLS) equations, we show that stable long-distance propagation can be achieved for a wide range of the gain-loss coefficients, including values that are outside of the perturbative regime. Furthermore, we demonstrate robust on-off and off-on switching of one of the sequences by an abrupt change in the ratio of cubic gain and quintic loss coefficients, and extend the results to pulse sequences with periodically alternating phases. Our study significantly strengthens the recently found relation between collision dynamics of sequences of NLS solitons and population dynamics in LV models, and indicates that the relation might be further extended to solitary waves of the cubic-quintic Ginzburg-Landau equation.

Figure 1

(a) $z$ dependence of ${\eta }_1$ for various ${\epsilon }_5$ values and $T=20$, $\kappa =1.5$, ${\eta }_1\left(0\right)=1.25$, and ${\eta }_2\left(0\right)=0.7$. The solid black, dotted blue, and dashed red lines correspond to ${\eta }_1\left(z\right)$ values obtained by numerical solution of the coupled-NLS model (2) with ${\epsilon }_5=0.01$, ${\epsilon }_5=0.06$, and ${\epsilon }_5=0.1$, respectively. The dashed-dotted green, short dashed purple, and short dashed-dotted orange lines represent ${\eta }_1\left(z\right)$ values predicted by the LV model (4) with ${\epsilon }_5=0.01$, ${\epsilon }_5=0.06$, and ${\epsilon }_5=0.1$. (b) The $z$ dependence of ${\eta }_1$ and ${\eta }_2$ for ${\epsilon }_5=0.5$ and the same values of $T$, $\kappa$, ${\eta }_1\left(0\right)$, and ${\eta }_2\left(0\right)$ as in (a). The solid red and dashed blue lines represent ${\eta }_1\left(z\right)$ and ${\eta }_2\left(z\right)$ as obtained by numerical solution of Eq. (2), while the dashed and short dashed black lines correspond to ${\eta }_1\left(z\right)$ and ${\eta }_2\left(z\right)$ values as obtained by the LV model (4). The inset shows the final pulse patterns $|{\psi }_1(t,z_f)|$ (solid red) and $|{\psi }_2(t,z_f)|$ (dashed blue) obtained in the coupled-NLS simulation.

Figure 2

$z$ dependence of soliton amplitudes in on-off switching (a), and in off-on switching (b) with $z_s=175$ and parameter values as described in the text. The red circles and blue squares represent ${\eta }_1\left(z\right)$ and ${\eta }_2\left(z\right)$ values obtained with the coupled-NLS model (2), while the solid and dashed black lines correspond to ${\eta }_1\left(z\right)$ and ${\eta }_2\left(z\right)$ values obtained with the LV model (4).

Figure 3

$z$ dependence of pulse amplitudes for soliton sequences with alternating initial phases and parameter values $T=20$, $\kappa =1.5$, ${\epsilon }_5=0.01$, ${\eta }_1\left(0\right)=1.05$, and ${\eta }_2\left(0\right)=0.9$. (a) ${\eta }_j$ vs $z$ for ${\alpha }_{jk}\left(0\right)=0$ and ${\alpha }_{jk+1}\left(0\right)=\pi$. (b) ${\eta }_j$ vs $z$ for ${\alpha }_{jk}\left(0\right)=0$ and ${\alpha }_{jk+1}\left(0\right)=\pi /2$. The solid red and dashed blue lines represent ${\eta }_1\left(z\right)$ and ${\eta }_2\left(z\right)$ values obtained with the coupled-NLS model (2), while the solid and dashed black lines correspond to ${\eta }_1\left(z\right)$ and ${\eta }_2\left(z\right)$ values obtained with the LV model (4).