Stable parity-time-symmetric nonlinear modes and excitations in a derivative nonlinear Schrödinger equation

The effect of derivative nonlinearity and parity-time-symmetric ($\mathcal{PT}$-symmetric) potentials on the wave propagation dynamics is explored in the derivative nonlinear Schrödinger equation, where the physically interesting Scarf-II and harmonic-Hermite-Gaussian potentials are chosen. We study numerically the regions of unbroken and broken linear $\mathcal{PT}$-symmetric phases and find some stable bright solitons of this model in a wide range of potential parameters even though the corresponding linear $\mathcal{PT}$-symmetric phases are broken. The semielastic interactions between particular bright solitons and exotic incident waves are illustrated such that we find that particular nonlinear modes almost keep their shapes after interactions even if the exotic incident waves have evidently been changed. Moreover, we exert the adiabatic switching on $\mathcal{PT}$-symmetric potential parameters such that a stable nonlinear mode with the unbroken linear $\mathcal{PT}$-symmetric phase can be excited to another stable nonlinear mode belonging to the broken linear $\mathcal{PT}$-symmetric phase.

Figure 1

(a) The unbroken (broken) $\mathcal{PT}$-symmetric phase is in the domain inside (outside) two phase breaking lines (the theoretically dash-dotted green and the numerically dotted red coincide by and large) for the linear operator $L$ in Eq. (13) with $\mathcal{PT}$-symmetric Scarf-II potentials (12), where the dashed blue parabola is $W_0^2+4V_0+3=0$ (similarly hereinafter), whose tangent points with the two phase-breaking lines above are $(V_0,W_0)=(-1.75,\pm 2)$, and the solid parabola is $4W_0^2+12V_0+9=0$ (similarly hereinafter), which is tangent with the dashed parabola at $(V_0,W_0)=(-0.75,0)$. (b) Real and (c) imaginary parts of the first two eigenvalues $\lambda$ of the linear problem (13) with $\mathcal{PT}$-symmetric potential (12) as a function of $V_0$ at $W_0=2$. The phase transition threshold is approximately $V_0=-1.75$, which well coincides with the theoretic result $|W_0|=0.25-V_0$.

Figure 2

(a) Stable (darker areas with smaller color values) and unstable (brighter areas with larger color values) regions of nonlinear modes (14) [determined by the maximal absolute value of imaginary parts of the linearized eigenvalue $\delta$ in Eq. (11) in the $(V_0,W_0)$ space (common logarithmic scale), similarly hereinafter]. (b) The existence region (the shaded red and green areas) for Scarf-II-Case-1. Profile and evolution of nonlinear modes for (c), (d) $V_0=-1,W_0=-1.1$ (unbroken linear $\mathcal{PT}$ symmetry), (e), (f) $V_0=-1,W_0=-1.4$ (broken linear $\mathcal{PT}$ symmetry), (g), (h) $V_0=-1,W_0=-1.5$ (broken linear $\mathcal{PT}$ symmetry).

Figure 3

(a) Stable and unstable regions of nonlinear modes (14). (b) The existence region (the shaded red area) for Scarf-II-Case-2. Profile and evolution of nonlinear modes with unbroken linear $\mathcal{PT}$ symmetry for (c), (d) $V_0=-0.9,W_0=0.74$; (e), (f) $V_0=-0.9,W_0=0.78$; (g), (h) $V_0=-0.91,W_0=0.78$.

Figure 4

Interactions of two solitary waves in Eq. (3) with the Scarf-II potential (12). (a) The solution (14) for Scarf-II-Case-1 with the wave $\sqrt{\frac{2}{3}{\varphi }_0\mathrm{sech}(x+20)}{e}^{4ix}$ and $V_0=-1,W_0=-1.1$. For $V_0=-1$, the solution (14) for Scarf-II-Case-1 with the wave $\sqrt{\frac{2}{3}{\varphi }_0\mathrm{sech}(x+40)}{e}^{10ix}$ and (b) $W_0=-1.2$, (c) $W_0=-1.3$, (d) $W_0=-1.4$. (e) The solution (14) for Scarf-II-Case-2 with the wave $\sqrt{\frac{3}{2}{\varphi }_0\mathrm{sech}(x+40)}{e}^{4ix}$ and $V_0=-0.9,W_0=0.74$. (f) The solution (14) for Scarf-II-Case-2 with the wave $\sqrt{\frac{3}{2}{\varphi }_0\mathrm{sech}(x+40)}{e}^{4ix}$ and $V_0=-0.9,W_0=0.78$.

Figure 5

Signs and directions of the transverse power flow $S\left(x\right)$ with regard to nonlinear modes (14). The dashed parabola $W_0^2+4V_0+3=0$, solid parabola $4W_0^2+12V_0+9=0$, horizontal line $W_0=0$ and vertical line $V_0=-3/4$ divide the existent region of particular soliton solutions (14) into six small domains (I, II, III, IV, ${\mathrm{V}}_{1,2}$ for Scarf-II-Case-1, and only ${\mathrm{V}}_{1,2}$ for Scarf-II-Case-2, where $+$ ($-$) denotes the positive (negative) sign of $S\left(x\right)$, and the rightward (leftward) arrow denotes the direction of power flow is from loss (gain) to gain (loss).

Figure 6

Excitation of stable nonlinear localized states [cf. Eq. (18)]. (a) $V_0=-1,W_{01}=-1.1,W_{02}=-1.4$, from stable nonlinear state belonging to unbroken linear $\mathcal{PT}$ symmetry to stable nonlinear state belonging to broken linear $\mathcal{PT}$ symmetry for Scarf-II-Case-1; (b) $V_0=-0.9,W_{01}=0.78,W_{02}=0.74$, (c) $V_{01}=-0.9,V_{02}=-0.91,W_0=0.78$, (d) $V_{01}=-0.9,V_{02}=-0.91,W_{01}=0.78,W_{02}=0.74$, from stable nonlinear state to stable nonlinear state with unbroken linear $\mathcal{PT}$ symmetry for Scarf-II-Case-2.

Figure 7

(a) The unbroken (broken) $\mathcal{PT}$-symmetric phases are in the domain below (above) the phase-breaking curves [$n=0$ (solid), $n=1$ (dashed), $n=2$ (dotted)] for the linear operator $L$ in Eq. (13) with $\mathcal{PT}$-symmetric harmonic potentials (20) and gain-and-loss distributions (21). (b), (d), (f) Real and (c), (e), (g) imaginary parts of the eigenvalues $\lambda$ of the linear problem (13) with $\mathcal{PT}$-symmetric potential (20) and (21) as a function of $\omega$, at ($\sigma =4$,  $n=0$), ($\sigma =2.59$,  $n=1$), and ($\sigma =0.81$,  $n=2$), respectively. The three phase transition thresholds are all approximately $\omega =1$, in accord with the phase-breaking curves in (a).

Figure 8

Linear stability of nonlinear modes (23) for (a1) $n=0$, (b1) $n=1$, and (c1) $n=2$. Evolutions of nonlinear modes (23) for one-hump ($n=0$) with unbroken linear $\mathcal{PT}$ symmetry [(a2) $\omega =1,\sigma =0.1$ (stable), (a3) $\omega =1,\sigma =1.1$ (stable), (a4) $\omega =1,\sigma =1.2$ (unstable), (a5) $\omega =2,\sigma =0.1$ (periodically varying)], two-hump ($n=1$) with unbroken linear $\mathcal{PT}$ symmetry [(b2) $\omega =2,\sigma =0.1$ (stable), (b3) $\omega =2,\sigma =0.8$ (stable), (b4) $\omega =2,\sigma =1$ (unstable), (b5) $\omega =3,\sigma =0.1$ (stable)], three-hump ($n=2$) with unbroken linear $\mathcal{PT}$ symmetry [(c2) $\omega =2,\sigma =0.1$ (stable), (c3) $\omega =2,\sigma =0.2$ (stable), (c4) $\omega =2,\sigma =0.3$ (unstable), (c5) $\omega =3,\sigma =0.1$ (stable)].

Figure 9

Interactions of two solitary waves in Eq. (3) with the potential (20) and (21). (a) The solution (23) for $n=0$ with the wave $0.8\sqrt{\sigma }{e}^{-\omega {(x+10)}^2/2}{e}^{i\phi \left(x\right)}$ and $\omega =1,\sigma =0.1$. (b) The solution (23) for $n=0$ with the wave $0.5\sqrt{\sigma }{e}^{-\omega {(x+10)}^2/2}{e}^{i\phi \left(x\right)}$ and $\omega =1,\sigma =1.1$. (c) The solution (23) for $n=1$ with the wave $\sqrt{\sigma \omega }(x+5)e^{-\omega {(x+5)}^2/2}{e}^{i\phi \left(x\right)}$ and $\omega =2,\sigma =0.1$. (d) The solution (23) for $n=1$ with the wave $\sqrt{\sigma \omega }(x+5)e^{-\omega {(x+5)}^2/2}{e}^{i\phi \left(x\right)}$ and $\omega =2,\sigma =0.8$. (e) The solution (23) for $n=2$ with the wave $0.5\sqrt{\sigma }{e}^{-\omega {(x+5)}^2/2}{e}^{i\phi \left(x\right)}$ and $\omega =2,\sigma =0.1$. (f) The solution (23) for $n=2$ with the wave $\sqrt{\sigma \omega }(x+5)e^{-\omega {(x+5)}^2/2}{e}^{i\phi \left(x\right)}$ and $\omega =2,\sigma =0.2$.

Figure 10

(a) The direction of the transverse power flow for $n=0$ is from the gain to loss. (b) The direction of the transverse power flow for $n=1$ is first from the gain to loss, then from the loss to gain, finally from the gain to loss again (from left to right). Here, $W_0\left(x\right)$ only has one root $x=0$ and $W_1\left(x\right)$ has three roots $x=0,\pm 1$. The left arrows and its lengths denote respectively negative directions and strengths of the transverse power flow $S_{0,1}\left(x\right)$ with regard to nonlinear modes (23). G (L) denotes the gain (loss) distribution of $W_{0,1}\left(x\right)$. Other parameters are $\omega =\sigma =1$.

Figure 11

Excitation of stable nonlinear localized states [cf. Eq. (18)] for $n=0$ [(a1) $\omega =1,{\sigma }_1=0.1,{\sigma }_2=1.1$, (a2) ${\omega }_1=1,{\omega }_2=2,\sigma =0.1$, (a3) ${\omega }_1=1,{\omega }_2=2,{\sigma }_1=0.1,{\sigma }_2=1.1$]; $n=1$ [(b1) $\omega =2,{\sigma }_1=0.1,{\sigma }_2=0.8$, (b2) ${\omega }_1=2,{\omega }_2=3,\sigma =0.1$, (b3) ${\omega }_1=2,{\omega }_2=3,{\sigma }_1=0.1,{\sigma }_2=0.8$]; $n=2$ [(c1) $\omega =2,{\sigma }_1=0.1,{\sigma }_2=0.2$, (c2) ${\omega }_1=2,{\omega }_2=3,\sigma =0.1$, (c3) ${\omega }_1=2,{\omega }_2=3,{\sigma }_1=0.1,{\sigma }_2=0.2$].