Three-Dimensional Structures of the Spatiotemporal Nonlinear Schrödinger Equation with Power-Law Nonlinearity in PT-Symmetric Potentials

The spatiotemporal nonlinear Schrödinger equation with power-law nonlinearity in -symmetric potentials is investigated, and two families of analytical three-dimensional spatiotemporal structure solutions are obtained. The stability of these solutions is tested by the linear stability analysis and the direct numerical simulation. Results indicate that solutions are stable below some thresholds for the imaginary part of -symmetric potentials in the self-focusing medium, while they are always unstable for all parameters in the self-defocusing medium. Moreover, some dynamical properties of these solutions are discussed, such as the phase switch, power and transverse power-flow density. The span of phase switch gradually enlarges with the decrease of the competing parameter k in -symmetric potentials. The power and power-flow density are all positive, which implies that the power flow and exchange from the gain toward the loss domains in the cell.


Introduction
In the last few decades, there has been a surge of interest in obtaining exact analytical solutions of nonlinear partial differential equations (NPDEs) to describe the natural physical phenomena in numerous branches from mathematical physics, engineering sciences, chemistry to biology [1][2][3]. Exact solutions often facilitate the testing of numerical solvers as well as aiding in the stability analysis.
The nonlinear Schrödinger equation (NLSE), as one of important nonlinear models, has now become an intensely studied subjects due to its potential applications in physics, biology and other fields. Abundant mathematical solutions and physical localized structures for various NLSEs have been reported. For example, bright and dark solitons and similaritons [4][5][6], rogue waves [7], nonautonomous solitons [8] and light bullets [9] etc. have been predicted theoretically and observed experimentally in different domains.
Recently, two-dimensional accessible solitons [10] and nonautonomous solitons [11] for NLSE in parity-time (PT )-symmetric potentials have been reported. The PT -symmetry originates from quantum mechanics [12], and was introduced into optical field since the important development on the application of PT symmetry in optics was initiated by the key contributions of Christodoulides and co-workers [13]. Quite recently, various nonlinear localized structures in PT -symmetric potentials have been extensively studied. Nonlinear localized modes in PTsymmetric optical media with competing gain and loss were studied [14]. The dynamical behaviors of (1+1)-dimensional solitons in PT -symmetric potential with competing nonlinearity were investigated [15]. Bright spatial solitons in Kerr media with PT -symmetric potentials have also been reported [16]. Dark solitons and vortices in PT -symmetric nonlinear media were discussed, too [17]. Moreover, Ruter et al. [18] and Guo et al. [19] studied the experimental realizations of such PT systems. However, three-dimensional (3D) spatiotemporal structures in PT -symmetric potentials are less studied. Especially, 3D spatiotemporal structures in PT -symmetric potentials with power-law nonlinearities are hardly reported.
The aim of this paper is to present 3D spatiotemporal structures of 3DNLSE with power-law nonlinearity in PT -symmetric potentials. Two issues are firstly investigated in this present paper: i) analytical spatiotemporal structure solutions are firstly reported in PT -symmetric power-law nonlinear media, and ii) linear stability analysis for exact solutions and direct simulation are firstly carried out in PT -symmetric power-law nonlinear media. Our results will rich the localized structures of NLSE in the field of mathematical physics, and might also provide useful information for potential applications of synthetic PT -symmetric systems in nonlinear optics and condensed matter physics.

Analytical spatiotemporal structure solutions
The propagation of spatiotemporal structures in a PTsymmetric nonlinear medium of non-Kerr index is governed by the following 3DNLSE iu z zb 1 D \ uzb 2 u tt zc m DuD 2m uz½V (r)ziW (r)u~0, ð1Þ where D \~( L 2 x ,L 2 y ),r:(r 1 ,r 2 ,r 3 ):(x,y,t), the complex envelope of the electrical field u(z,r) is normalized as (k 0 w 0 ) {1 (n 2 =n 0 ) {1=2 with linear index n 0 and Kerr index n 2 , longitudinal z, transverse x,y coordinates and comoving time t are respectively scaled to the diffraction length L D :k 0 w 2 0 , the input width unit w 0 with the wavenumber k 0 :2pn 0 =l at the input wavelength l and ffiffiffiffiffiffi ffi L D p . Parameters b 1 and b 2 are respectively the coefficients of the diffraction and dispersion, and c m for m~1,:::,n stand for the nonlinearities of orders up to 2n+1. For m = 1 one has the simple Kerr nonlinearity, for m = 2 the quintic, for m = 3 the septic, and so on. Functions V (r):k 2 0 w 2 0 dn R (r) and W (r):k 2 0 w 2 0 dn I (r), with the perturbation of index by a complex profile n~n 0 ½1zn R (r)zin I (r), are the real and imaginary components of the complex PT -symmetric potential, and correspond to the index guiding and the gain or loss distribution of the optical potential respectively. V and W satisfy V (r)~V ({r) and W (r)~{W ({r).
Case 1 First type of extended PT -symmetric potential. Considering the PT -symmetric potential with real parameters V 1j ,V 2j ,V 3 and W j , and the competing parameter k, the localization condition WR0 as rR6' yields solution of Eqs. (3) and (4) sech(x)sech(y)sech(t), where j j~s inh(r j ) and 2 F 1 (a,b,c,o) is the Hypergeometric function [20]. The parameters in the potential (4) and (5) satisfy with three arbitrary constants V 3 , W 1 and W 3 . Parameter m has a serious impact on the nature of the gain and loss profile W(r). The value of k as zero or nonzero leads to W(r) as asymptotically non-vanishing or localized (asymptotically vanishing), respectively. For instance, if k = 0, it is the first type of extended Rosen-Morse potential, and if k = 1, it is the first type of extended hyperbolic Scarf potential.
From (6), V 3 c m v0, thus solution (6) exists in self-focusing (SF) media with positive nonlinearity (c m w0) if V 3 v0, as well as in self- Specially, if the value of k is chosen as 0-3, h(r) has the different forms shown in Table 1.
Case 2 Second type of extended PT -symmetric potential. In the following PT -symmetric potential with real parameters V 1j , V 2j , V 3 and W j , and the competing parameter k, the localization condition WR0 as rR6' leads to solution of Eqs. (3) and (4) in the form ,m~(2b 1 zb 2 )=m 2 and three arbitrary constants V 3 ,W 1 , and W 3 . Moreover, j j~s inh(r j ) and 2 F 1 (a,b,c,o) is the Hypergeometric function.
Specially, when the value of k is chosen as 0-3, the expressions of h(r) are shown in Table 2.

Properties of spatiotemporal structure solutions
The even and odd functions for the real part V and imaginary part W of the PT -symmetric potential (7) are shown in Fig. 1 in regard to x,y and t for different k. Figs. 1(c) and (d) show V for different k at z~30, y~0, t~10 when m = 2 and 1, respectively. Fig. 1(e) shows W for different k at z~30, y~0, t~10 when m = 2. From the yellow dash lines in Figs. 1(c)-(e), V is localized when m = 1 or 2, while W is asymptotically non-vanishing in the 2D extended Rosen-Morse potential. It possesses unbroken PTsymmetry [18]. From red crosses, blue lines and black circles in Figs. 1(c)-(e), the peaks and widths of V and W gradually decrease when k increases. Compared red crosses, blue lines and black circles in Fig. 1(c) with those in Fig. 1(d), the amplitudes of V attenuate when m adds.
In the PT -symmetric potentials above, we can find the phenomena of phase switch of solutions (6) and (8). Fig. 2(a) exhibits switches of phase in (8) for different PT -symmetric potentials (7) with m = 2. When k decreases, phases switch from smaller to bigger values along x, and the spans of switch gradually enlarge. However, in the Rosen-Morse potential with k = 0, no phase switch appears. In the PT -symmetric potential (5) with m = 1,2, the phase switch also exists. We omit the related plots.
The power P can be expressed as P~Ð z?
{? Du(z,r)D 2 dr~8({V 3 =c m ) 1=m for solution (6). For solution (8) (6) and (8) have the formS S1 , which are related to the competing parameter k and parameter m. Obviously, due to V 3 c m v0, S is everywhere positive, which indicates that the power flow and exchange for solutions (6) and (8) in the PT cells are always from the gain toward the loss domains (one direction). An example to this case is shown in Fig. 2(b) for k = 3. The similar results also exist when other k and m are chosen.

Linear stability analysis of analytical solutions
We study the linear stability of solutions (2) with (6) where s is an eigenvalue, R and I are eigenfunctions with Hermitian operators L +~{ b 1 (L 2 x zL 2 y ){b 2 L 2 t {g + c 1 u n (r) 2 {(V ziW )zm with g + = 3 and g 2 = 1 for m = 1 and The eigenvalues s of solutions (6) and (8) in the SF and DF media under the 2D extended Rosen-Morse potential have many imaginary parts, and thus solutions (6) and (8) are always unstable in these nonlinear media. Fig. 3 shows some examples of the eigenvalue s in the SF and DF media. From Figs. 3(a) and 3(b), the eigenvalues s for both SF and SD nonlinearities have many imaginary parts, and thus solutions (6) and (8)    leads to the linear instability of solutions (6) and (8)    Furthermore, when k = 2,3 in the 2D extended PT -symmetric potentials (5) and (7), solutions (6) and (8) with m = 1 and m = 2 are stable below some thresholds for W 1 and W 3 in the SF medium because the eigenvalues s of solutions (6) and (8) (6) and (8) with m = 1,2 are always unstable because there also exist some imaginary parts of the eigenvalues s for all parameters.
When k is bigger, we have the similar results. Solutions (6) and (8) with m = 1 and m = 2 are stable below some thresholds for W 1 and W 3 in the SF medium, while they are always unstable for all parameters in the SD medium. Here we omit these discussions.

Numerical rerun of analytical solutions
Based on the linear stability analysis, we know the stable domains of analytical solutions under different 2D extended PTsymmetric potentials. In the following, we further test the stability of these solutions by the direct numerical simulation. Here we use a split-step Fourier pulse technique. In real application, the analytical cases are not exactly satisfied, thus we consider the stability of solutions with respect to finite perturbations. The perturbations of 5% white noise are added to initial fields coming from solutions (6) and (8) (6) and (8) are both unstable in the 2D extended hyperbolic Scarf potential, which is shown in Figs. 6(c), (e) and (g). They can not maintain their original shapes, change from distortion to collapse, and ultimately decay into noise. Figure 7 displays other examples of stable analytical solutions, and it is the numerical reruns corresponding to Figs. 5(a),(b),(d),(f) in the 2D extended PT -symmetric potential. In the SF medium, we can obtain stable spatiotemporal structures. From Figs. 5(a),(b),(c),(e), the influence of initial 5% white noise is suppressed, and these spatiotemporal structures (6) and (8) stably propagate over tens of diffraction/dispersion lengths and only some small oscillations appear when k is chosen 2 or 3 in the 2D extended PT -symmetric potential. However, in the DF medium, spatiotemporal structures are unstable and broken down propagating after tens of diffraction/dispersion lengths, and at last turn into noise. Compared Fig. 6(d) with Fig. 7 (c) or Fig. 6 (f) with Fig. 7 (e) respectively, spatiotemporal structures are more stable in the 2D extended PT -symmetric potential with k = 3 than those with k = 1.

Conclusions
We conclude the main points offered in this paper: N Analytical spatiotemporal structure solutions are firstly reported in PT -symmetric power-law nonlinear media.
N We obtain two families of analytical three-dimensional spatiotemporal structure solutions of a spatiotemporal NLSE with power-law nonlinearity in PT -symmetric potentials. Some dynamical characteristics of these solutions are discussed, such as the phase switch, power and power-flow density. The spans of phase switch gradually enlarge with the decrease of the competing parameter k in PT -symmetric potentials. The power and power-flow density are all positive, which implies that the power flow and exchange from the gain toward the loss domain in the PT cell.
N Linear stability analysis for exact solutions and direct simulation are firstly carried out in PT -symmetric power-law nonlinear media.  N Our results will rich the localized structures of NLSE in the field of mathematical physics, and might also provide useful information for potential applications of synthetic PTsymmetric systems in nonlinear optics and condensed matter physics.