Influence of the imaginary component of the photonic potential on the properties of solitons in $\mathcal{PT}$-symmetric systems

Existence, stability, and dynamics of $\mathcal{PT}$-symmetric fundamental bright solitons supported by localized super-Gaussian potentials in a focusing Kerr medium are investigated theoretically. We address how the shape and the magnitude of the transverse profile of the loss-gain distribution affect soliton stability. We find the stability region for nonlinear wave packets via a linear stability analysis, interpreting the insurgence of instability as an unbalanced flow of energy on the transverse plane. We confirm our results via numerical simulations, showing that an unstable soliton first undergoes longitudinal oscillations in propagation due to the interference between the soliton and the exponentially growing perturbation modes, eventually forming a highly localized single peak in the gain region.

Figure 1

Ground state $\mu$ vs $V_i$ for the linear potential of type $A$. The top row depicts real part ${\mu }_R$ and the middle row depicts imaginary part ${\mu }_I$, plotted vs $V_i$. Imaginary eigenvalues with equal magnitude but opposite in sign appear at the exceptional point. The stationary profile at $V_i=3$, which falls in a different spectral region for each $l$, is plotted in the last row; the black symmetric solid line is the intensity, the blue dashed line is the real part of $\varphi$, and finally the antisymmetric red solid line is the imaginary part of the field $\varphi$. Here $V_r=1$.

Figure 2

Ground state $\mu$ vs $V_i$ for the linear potential of type $B$. The top row depicts real part ${\mu }_R$ and the bottom row depicts imaginary part ${\mu }_I$, plotted vs $V_i$. As compared to type $A$, the trend of the eigenvalue ${\mu }_I$ vs $V_i$ depends much less on $l$ given that $V_I\left(\xi \right)$ is kept constant. Here $V_r=1$.

Figure 3

Soliton profiles for potentials of type $A$ for $l=1$, 2, and 8 (top to bottom rows) for $V_i$ as marked. Symmetric solid curves represent the normalized real part of the profile for $\mu =1.3$ (solid lines) and $\mu =6$ (dashed lines). Antisymmetric curves represent the imaginary part normalized to the peak of the real part; the higher the magnitude the lower the $\mu$. The behaviors of $V_R$ (blue symmetric lines) and $V_I$ (red antisymmetric lines) vs $\xi$ are shown in the last column for each $l$ for $V_i=3$.

Figure 4

Soliton profiles for potentials of type $B$ for $l=2$ (other values of $l$ provide almost identical results), for $V_i$ as marked. Symmetric blue curves represent the normalized real part of the profile for $\mu =1.8$ (solid lines) and $\mu =6$ (dash-dotted lines). Solid red lines are the imaginary component of the field normalized with respect to the peak of the real part: the higher the magnitude the lower the $\mu$. Symmetric dashed black lines represent the real part of the potential. The antisymmetric dashed green line represents the imaginary part of the potential. Symmetric curves correspond to the left axis and antisymmetric curves correspond to the right axis, respectively.

Figure 5

Features of the fundamental soliton vs $V_i$ for the lowest ${\mu }_R$ allowed, for potentials of both type $A$ (blue solid lines) and type $B$ (red points). The first two rows depict real part ${\mu }_R$ and imaginary part ${\mu }_I$, respectively. The corresponding minimum power $P_{\min }$ and width $w_{\xi }^{\min }$ of the soliton are plotted in the last two rows.

Figure 6

Total power $P$ (top) and the corresponding power carried in the imaginary component $P_I$ (bottom) for potentials of type $A$ (solid lines) and of type $B$ (dashed lines) for $l=2$ (blue), 4 (black), and 8 (red), from top to bottom curves, respectively; $V_i$ values are marked in the figure. For low $V_i$ total power $P$ follows the same trend for both the potentials.

Figure 7

Stability spectrum for $\mu =1.3$ and $l=1$. Solid blue lines correspond to ${\left|p\left(x\right)\right|}^2$ and dashed black lines correspond to ${\left|q\left(x\right)\right|}^2$ for the eigenvalue $\lambda$ with the maximum imaginary part, hereafter named ${\lambda }_{\mathrm{dom}}$. The eigenvalue spectrum is shown in the bottom row. For low $V_i$, the soliton is quasistable due to the smallness of $\text{Im}\left({\lambda }_{\mathrm{dom}}\right)$, whereas for high $V_i$ the linear perturbation modes are asymmetric and the soliton blows up.

Figure 8

Stability eigenspectrum computed from Eq. (8) in the plane $(V_i,\mu )$: $l$ changes as marked in the figure. Top and bottom rows depict results for potentials of type $A$ and $B$, respectively. The region embedded between the two horizontal lines corresponds to the presence of complex eigenvalues in the linear spectrum, white dashed lines and red solid lines representing $V_c$ and $V_t$, respectively. In general, solitons become highly unstable when $V_i>V_c$. Keeping $V_i$ fixed, larger $l$ enhances soliton instability for potentials of type $B$ if compared with type $A$. Vertical blue lines represent the boundary between regions allowing the existence of a solitonic solution (right side) and the forbidden region (left side).

Figure 9

Top row: Intensity evolution of soliton solutions for $\mu =3.4$ and (left panel) $V_i=0.1$ showing the quasistable soliton and (right panel) $V_i=0.5$ showing the unstable soliton. Bottom row: Transverse profiles of the solution for $V_i=1$ and $\mu =3.4$ for potential $A$ are shown for $\zeta =0$ (blue solid line), $\zeta =100$ (black dotted line), and $\zeta =110$ (red dashed line) for $l=1$; $\zeta =\left[0140150\right]$ for $l=2$; $\zeta =\left[0150200\right]$ for $l=8$. Transverse profiles of the solution for $V_i=1$ and $\mu =3.4$ for potential $B$ (last two columns) are shown for $\zeta =\left[0130140\right]$ for $l=2$ and $\zeta =\left[0129135\right]$ for $l=8$ showing narrowing and peak oscillation upon propagation. The profiles are normalized with power for better visibility.

Figure 10

Evolution of soliton trajectory $\langle \xi \rangle$ for $V_i=0.5$ and 1 for $l=1$ and $\mu =3.4$ showing the oscillations in the nonlinear wave trajectory.

Figure 11

Evolution of trajectory $\langle \xi \rangle$ and width $w_{\xi }$ vs $\zeta$ for $V_i=0.5$ (top two rows) and $V_i=1$ (bottom two rows) for potentials of type $A$ (second and third column) and potentials of type $B$ (last two columns) for different $l$ as marked; for $l=1$ both the potentials are equivalent. Dotted lines correspond to the propagation of stationary nonlinear modes, whereas solid lines correspond to wave evolution initially perturbed by $1\%$ random noise in amplitude for $\mu =0.7$ $(\square )$, $\mu =3.4$ $(\diamond )$, and $\mu =6$ $\left(▵\right)$.

Figure 12

Evolution of wave radius vs $\zeta$ for $l=1$ (first row) and potentials of type $A$ (second row, first two columns) and potentials of type $B$ (second row, last two columns) for different $l$ and $V_i=1$; for $l=1$ both the potentials are equivalent. Solid lines correspond to the propagation of unperturbed solitons, whereas dashed lines correspond to evolution with $1.01$ of the soliton power for $\mu =0.7$ $(\square )$, $\mu =3.4$ $(\diamond )$, and $\mu =6$ $\left(▵\right)$.