Controlling of blow-up responses by nonlinear $\mathcal{PT}$-symmetric coupling

We investigate the dynamics of a coupled waveguide system with competing linear and nonlinear loss-gain profiles which can facilitate power saturation. We show the usefulness of the model in achieving unidirectional beam propagation. In this regard, the considered type of coupled waveguide system has two drawbacks: (i) difficulty in achieving perfect isolation of light in a waveguide and (ii) existence of blow-up-type behavior for certain input power situations. We here show a nonlinear $\mathcal{PT}$-symmetric coupling that helps to overcome these two drawbacks. Such a nonlinear coupling has close connection with the phenomenon of stimulated Raman scattering. In particular, we have elucidated the role of this nonlinear coupling using an integrable $\mathcal{PT}$-symmetric situation. In particular, using the integrals of motion, we have reduced this coupled waveguide problem to the problem of dynamics of a particle in a potential. With the latter picture, we have clearly illustrated the role of the considered nonlinear coupling. The above $\mathcal{PT}$-symmetric case corresponds to a limiting form of a general equation describing the phenomenon of stimulated Raman scattering. We also point out the ability to transport light unidirectionally even in this general case.

Figure 1

Limit cycle behaviors of ${\varphi }_1$ and ${\varphi }_2$: Figures are plotted for $\gamma =1.0,\alpha =0.5$, and $\chi =0.0$. Panels (a), (b), and (c) are plotted for $k=0.0,0.4,$ and 0.5, respectively. Here the dynamics of the system is represented in terms of the variables $x_j$ and $y_j$, where ${\varphi }_j=x_j+i{y}_j$, and ${\varphi }_1$ is represented by the dashed red line and ${\varphi }_2$ is represented by the continuous black line. The inset in panel (b) shows the variation in $|{\varphi }_1{|}^2$ and $|{\varphi }_2^2|$ with respect to $z$.

Figure 2

Dynamics in the unbroken $\mathcal{PT}$ region: The figures are plotted for $\gamma =1.0,\alpha =0.5,\chi =0.0,$ and $k=0.51$. Panel (a) is plotted for the initial condition ${\varphi }_1=1.0$ and ${\varphi }_2=0.0$. From the figure, we find that the oscillations become quasiperiodic (where ${\varphi }_j=x_j+i{y}_j$) and their corresponding power oscillations are given in panel (b). ${\varphi }_1$ is represented by the dashed red line and ${\varphi }_2$ is represented by the continuous black line in panels (a) and (b). Panel (b) shows the equal distribution of power among the two waveguides so that the symmetry is now unbroken. By considering different initial conditions ${\varphi }_1=\sqrt{0.1},1.0,$ and $\sqrt{2.0}$ with ${\varphi }_2=0.0$, we have plotted variation of $|{\varphi }_1{|}^2$ in panel (c) and the existence of nonisolated power oscillations are shown in the figure.

Figure 3

Values of power in the first and second waveguides for different values of $\chi$ plotted for $k=0.5,\gamma =1.0,$ and $\alpha =0.5$. From the figure, we can observe the isolation of light in the second waveguide with increase of $\chi$.

Figure 4

Dynamics in the broken $\mathcal{PT}$-symmetric region for $\chi =0.0$: potential curve of $\tilde{V}\left(p\right)$ for $\gamma =1.0,\alpha =0.5,k=0.4$, and $\chi =0.0$, obtained for the initial condition ${\varphi }_1=1.0+0.0i$ and ${\varphi }_2=0.0$. The empty circles along the $\tilde{V}\left(p\right)$ curve represent the dynamics of $p$ with respect to $z$.

Figure 5

Blow-up responses in the broken $\mathcal{PT}$-symmetric region: Panel (a) is plotted for the same values of parameters as that of Fig. 4 but for initial conditions ${\varphi }_1=\sqrt{2.5}$ and ${\varphi }_2=0.0$ and it also shows the existence of a blow-up solution for certain input power configurations. Panel (b) shows the curves of $\tilde{V}\left(p\right)$ for different values of $\chi$, namely, 0.0,0.3, and 0.4, for the same initial conditions and parametric values as considered in panel (a).

Figure 6

Dynamics in the unbroken $\mathcal{PT}$-symmetric region for $\chi =0.0$: The figure shows the curves of $\tilde{V}\left(p\right)$ for $\gamma =1.0,\alpha =0.5,k=0.8,\chi =0.0,{\varphi }_1\left(0\right)=1.0+0.0i,$ and ${\varphi }_2\left(0\right)=0.0$. The dynamics in this region is the same as the dynamics observed in the unbroken $\mathcal{PT}$ region of the coupled waveguide problem.

Figure 7

Effect of nonlinear $\mathcal{PT}$-symmetric coupling in the unbroken $\mathcal{PT}$ region: The figure shows the curves of $\tilde{V}\left(p\right)$ for $\gamma =1.0,\alpha =0.5,k=0.8,\chi =0.0,0.6$, and 0.8 where ${\varphi }_1\left(0\right)=\sqrt{3.0}+0.0i$ and ${\varphi }_2\left(0\right)=0.0$.

Figure 8

The ability of nonlinear coupling in localizing light in the case ${\omega }_1\ne {\omega }_2$. The figure is plotted for two values of $\chi$, the black curves correspond to the case $\chi =0.0,$ and the blue (gray) curves correspond to the case $\chi =0.3$ with ${\omega }_1=1.0$ and ${\omega }_2=0.8$. Note that in Eq. (1) ${\chi }_1=\frac{{\omega }_1}{{\omega }_2}\chi$ and ${\chi }_2=\chi$. In the figure, the curve corresponding to $|{\varphi }_1{|}^2$ is represented by dotted lines and the curve corresponding to $|{\varphi }_2{|}^2$ is represented by continuous lines. The figure is plotted for $\gamma =1.0,\alpha =0.5,$ and $k=0.5$.

Figure 9

Role of nonlinear coupling in controlling blow-up responses. Panel (a) is plotted for $\chi =0.0$ and panel (b) is for $\chi =0.3$ and in both the figures we considered $\gamma =1.0,\alpha =0.5,{\omega }_1=1.0,{\omega }_2=0.8,$ and $k=0.5$ (the same parametric values as chosen in Fig. 8). We here varied the initial conditions and pictured out the input power configurations that lead to blow-up. To draw panels (a) and (b), we considered ${\varphi }_1\left(0\right)=u\left(0\right)+0i$ and ${\varphi }_2\left(0\right)=v\left(0\right)+0i$.