Spatial solitons and stability in the one-dimensional and the two-dimensional generalized nonlinear Schrödinger equation with fourth-order diffraction and parity-time-symmetric potentials

We investigate the existence and stability of solitons in parity-time ($\mathcal{PT}$)-symmetric optical media characterized by a generic complex hyperbolic refractive index distribution and fourth-order diffraction (FOD). For the linear case, we demonstrate numerically that the FOD parameter can alter the $\mathcal{PT}$-breaking points. For nonlinear cases, the exact analytical expressions of the localized modes are obtained both in one- and two-dimensional nonlinear Schrödinger equations with self-focusing and self-defocusing Kerr nonlinearity. The effect of FOD on the stability structure of these localized modes is discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. Examples of stable and unstable solutions are given. The transverse power flow density associated with these localized modes is also discussed. It is found that the relative strength of the FOD coefficient can utterly change the direction of the power flow, which may be used to control the energy exchange among gain or loss regions.

Figure 1

(a) The unbroken or broken $\mathcal{PT}$-symmetric phase produced by linear eigenvalue problem (4) with $\mathcal{PT}$-symmetric potential (3). (b) Imaginary part of the eingavalue $\lambda$ as a function of the FOD coefficient $\beta$ at $V_0=1$. The parameter $W_0=2$.

Figure 2

The perturbation growth rate of solitons versus $\beta$ with different values of $W_0$: (a) $W_0=0.01$, (b) $W_0=0.1$, (c) $W_0=1$ in the focusing nonlinearity $g=1$. In all cases we have considered $V_0=-0.5$.

Figure 3

The perturbation growth rate of solitons versus $\beta$ with different values of $W_0$: (a) $W_0=0.01$, (b) $W_0=0.1$, (c) $W_0=1$ in the defocusing nonlinearity $g=-1$. In all cases we have considered $V_0=-0.5$.

Figure 4

(a1, a2) Real and imaginary parts of $\mathcal{PT}$-symmetric potentials given in Eq. (3). (b1, b2) Numerically computed linear stability spectra in the case of self-focusing nonlinearity $g=1$. (c1) Stable and (c2) unstable propagations of nonlinear modes described in Eq. (2). (a1, b1, c1) $V_0=-0.5$, $W_0=0.1$, $\beta =0.04$. (a2, b2, c2) $V_0=-0.5$, $W_0=0.1$, $\beta =-0.04$.

Figure 5

(a1, a2) Real and imaginary parts of $\mathcal{PT}$-symmetric potentials given in Eq. (3). (b1, b2) Numerically computed linear stability spectra in the case of self-defocusing nonlinearity $g=-1$. (c1) Stable and (c2) unstable propagations of nonlinear modes described in Eq. (2). (a1, b1, c1) $V_0=-0.5$, $W_0=0.1$, $\beta =0.4$. (a2, b2, c2) $V_0=-0.5$, $W_0=0.1$, $\beta =0.8$.

Figure 6

Transverse power flow vector indicating the power flow (a) from gain towards loss regions when $\beta =0.1$; (b) from loss towards gain regions when $\beta =0.25$. The other parameters are $V_0=-4$, $W_0=1$, $g=1$.

Figure 7

(a1, a2) Plots of the linear stability eigenvalue spectra of the 2D solitons in the case of self-focusing nonlinearity $g=1$. (b1) and (b2) plots of the corresponding stable and unstable intensity evolution. The parameters are (a1, b1) $V_0=-2$, $W_0=0.1$, $\beta =0.15$; (a2, b2) $V_0=-1$, $W_0=1$, $\beta =0.03$.

Figure 8

(a1, a2) Plots of the linear stability eigenvalue spectra of the 2D solitons in the case of self-defocusing nonlinearity $g=-1$. (b1, b2) Plots of the corresponding stable and unstable intensity evolution. The parameters are (a1, b1) $V_0=0.1$, $W_0=0.5$, $\beta =0.1$; (a2, b2) $V_0=5$, $W_0=0.1$, $\beta =-0.14$.