Symmetry breaking of PT-symmetric solitons in self-defocusing saturable nonlinear Schrödinger equation

. The symmetry breaking phenomenon of the parity-time (PT) symmetric solitons in self-defocusing saturable nonlinear Schrödinger equation is studied. As the soliton power increases, branches of asymmetric solitons are separated from antisymmetric solitons, and they coexist with both symmetric and antisymmetric solitons. The anti-symmetric solitons require different power thresholds when they are under different saturable nonlinear strength. The stronger the saturable nonlinearity is, the larger the power threshold is. The saturable nonlinear strength has obvious modulation effect on the symmetry breaking of antisymmetric solitons and the bifurcation of the power curve. However, when the modulation strength of PT- symmetric potential increases, the effect of this modulation effect weakens. The antisymmetric solitons are only stable in the low power region, and the stability of symmetric and asymmetric solitons is less affected by the soliton power. The increase of the saturable nonlinear strength leads to the increase of the critical power of the symmetry breaking. When a beam propagates in a PT-symmetric optical waveguide, the symmetry breaking of antisymmetric solitons can be controlled by changing the saturable nonlinear strength.

general dissipative system. It has the property of a conservative system before the occurrence of the symmetry breaking, and shows the property of a dissipative system after the occurrence of the symmetry breaking. Therefore, the PT-symmetric system establishes the relationship between the conservative system and the dissipative system. The necessary condition for PT-symmetric potential x is the normalized horizontal coordinate, " * "denotes complex conjugate.
Thus, the imaginary part of () Uxrequires odd function and the real part of () Uxrequires even function.
In nonlinear optical waveguides, the transmission of the paraxial ray can be described by the nonlinear Schrödinger equation (NLSE) [23,24]. The refractive index distribution of the optical waveguide is described by the real potential function, while the loss and gain in the optical waveguide are represented by the imaginary potential function [16,17,22]. Recently, the asymmetric solitons in the PT-symmetric potential have attracted more and more attention. Because the symmetry of solitons is broken, their contours become asymmetric. Perturbation analysis shows that in the PT-symmetric potential, such symmetry breaking needs to satisfy an infinite number of nontrivial conditions simultaneously, so it is impossible to occur in the general PT-symmetric potential [25]. But in a special type of PT-symmetric potential, there are stable PT-symmetry breaking solitons [13]. It is proved that the symmetry breaking of solitons exist in a special type of PT-symmetric potential.
Recently, the symmetry breaking of the soliton in the PT-symmetric potential were intensely investigated. The symmetry breaking of solitons was found in PT-symmetric optical waveguides with focused saturable nonlinearity [26]. And the existence and stability of solitons in PT-symmetric optical lattices in fractional NLSE models were studied [27]. Subsequently, the fork bifurcation of the symmetry breaking for fractional optical solitons were reported [28], and the symmetry breaking behavior of solitons were found in partial PT-symmetric potential [29]. The soliton solutions of the (1+1) dimensional nonhomogeneous cubic-quintic-septimal NLSE with PT-symmetric potential were discussed [30]. Then the symmetric-broken and symmetric-unbroken soliton solutions of the semi-discrete nonlocal NLSE were also discussed [31]. In addition, the phenomena of dark solitons and vortices from spontaneous symmetry breaking to nonlinear PT phase transition were found in PT-symmetric nonlinear system [32]. Although there have been many studies on the NLSEs under the PT-symmetric potential and, the solitons of the NLSE supported by the PT-symmetric potential and the saturable defocusing nonlinearity is less studied.
In previous studies, the saturable nonlinear strength and the shape and parameters of the potential function have certain effects on the symmetry breaking of the soliton under the PTsymmetric potential. But the modulating relationship between them has never been discussed. In this paper, the symmetry breaking of solitons is studied based on the self-defocusing saturable NLSE, and the effects of symmetry breaking and bifurcation of solitons are analyzed in detail. Furthermore, the relationship among the modulating strength of the PT-symmetric potential, the modulation strength of PT-symmetric potential, the saturable nonlinear strength and the power threshold leading to the symmetry breaking is explored. It is found that when the modulation strength of PTsymmetric potential function reaches a certain threshold, the power threshold leading to the symmetry breaking will no longer change with the add of saturable nonlinear strength. Beyond the modulation threshold of potential function, saturable nonlinearity can no longer modulate symmetry breaking. This provides a new way to control the transformation of antisymmetric solitons into asymmetric solitons in the PT-symmetric system.

Model
In the approximately paraxial conditions, the propagation of optical wave in a saturable nonlinear waveguide is represented by the following normalized NLSE [29] 22 represents the distribution of the light field and ( ) is the normalized PT-symmetric potential, =+1  represents the focusing nonlinearity and =1  − represents selfdefocus nonlinearity, the parameter S defines the nonlinear saturable. When the self-defocusing nonlinear parameter =0  , Eq. (1) gives a description of the PT-symmetric system. In this paper, we consider the self-defocusing saturable nonlinearity with =1  − . In order to explore the symmetry breaking phenomenon of solitons in Eq.(1), the form of the steady-state solution is expressed as ( In our research, we select the following PT-symmetric potential function [29] where () ux is a real even symmetric function. We choose the form of () ux as 00 0 00 where 0  describes the modulation strength of the PT-symmetric potential, 0 x represents the distance between the two peaks of the potential function, and 0  represents the width of a single peak of the potential function. Thus, the real and imaginary parts of PT-symmetric potential are respectively 2 2 00 0 00

The numerical results
We aim to find the symmetric, asymmetric and antisymmetric soliton solutions, and explore the corresponding symmetry breaking phenomenon. First, we study the self-defocus strongly saturable nonlinearity 1 S = . The function () ux , and two parts of the PT-symmetric potential function, () VR x and () VI x , are shown in Fig. 1(a).
These steady-state solutions can be obtained by using the Power-conserving square operator method [24], in which the soliton power is defined as The numerical results show that there are two kinds of soliton solutions, namely PT-symmetric soliton solutions and non-PT-symmetric soliton solutions in Eq. (2). The bifurcation diagram of the power curve in Fig. 1(b) shows that symmetric soliton solutions, antisymmetric soliton solutions and asymmetric soliton solutions all exist in the case of self-defocusing saturable nonlinearity. From . Beyond the critical point, the symmetry of the antisymmetric solitons is broken, but the symmetry of the symmetric solitons maintains. In a conservative system, if the slopes of antisymmetric and asymmetric solitons have same signs at bifurcation points, then antisymmetric and asymmetric solitons will have opposite stability or instability [33], which is similar to the results from the linear stability analysis and dynamic evolution in Figs. 5 and 6. We can see the real and imaginary parts of the antisymmetric and asymmetric solitons in Fig. 1(c) and Fig. 1(d) (corresponding to points c and d in Fig. 1(b)), respectively. The real and imaginary parts of antisymmetric solitons are odd symmetric and even symmetric respectively, while the real and imaginary parts of asymmetrical solitons are both asymmetric.
The profiles of symmetric, asymmetric, and antisymmetric solitons with varying power are shown in Figure 2. In Fig.2(b), asymmetric soliton exists between 0.8 P = and 3 P = . As the power increases, the energy is gradually concentrated on one of the peaks.
Next, the influence of saturable nonlinear parameters on the soliton bifurcation is discussed for different saturable nonlinear parameters. The soliton bifurcations for strongly weakly, moderately and weakly saturable nonlinearities are given in Fig.1(b)、Figs.3(b) 、Figs.3 (c), respectively. As shown in Fig. 1(b), under the condition of strongly saturable nonlinearity ( 1 S = ), when the soliton power exceeds 0.8, asymmetric soliton is separated from antisymmetric solitons, the symmetry breaking phenomenon occurs at the same time, and the corresponding bifurcation of powerpropagation constant is generated. Figs. 3(a) and 3(b) show that the power required to generate the symmetry breaking is 0.7 P = and 0.6 P = respectively, when the saturable nonlinear strength is

Linear stability and dynamics
The linear stability of solitons will be analyzed by adding perturbations to the model as: with the unstable growth rate  -(complex values). The linear stability is represented by unstable growth rate  . The real and imaginary parts of  represent the unstable growth rate of the soliton solution, and the shape oscillation in the soliton transmission process, respectively. When the absolute value of the real part of  is zero, the soliton solution is linearly stable. The dependence between solitons power and maximum instability growth rates  We also find that with the increase of the saturable nonlinear strength, the stable region of the symmetric soliton increases, but that of the asymmetric soliton does not change significantly and the asymmetric soliton remains weakly stable. When the power In order to prove the above results, the dynamical evolution behaviors of symmetric and asymmetric solitons will be studied with the 3% random noise added in the initial soliton solution in Figs. 6(b) and 6(d). The numerical simulation results show that the symmetric soliton with the weak stable state can propagate stably, but the asymmetric soliton will oscillate and break up after propagating a long distance because of its weak instability. The linear-stability spectrum for antisymmetric soliton in Fig. 6(e) indicates that the antisymmetric soliton is in an unstable state at power 1.0 P = . The corresponding evolution diagram is shown in Fig. 6(f). The antisymmetric soliton can propagate stably at the initial stage, and its amplitude oscillation is enhanced due to its instability, and finally the waveform is completely destroyed. can be seen that the symmetric soliton at 2.0 P = in Fig. 7(a) is more stable than the asymmetric soliton in the same case, and both the symmetric and the asymmetric solitons show the weak stability.
In order to verify its stability, the initial soliton solution is interfered by 3% random noise, and the corresponding evolution diagrams are shown in Figs.7 (b) and7 (d). In Fig. 7(b), the symmetric soliton oscillates and ruptures after the stable propagation over a long distance. In Fig. 7(d), the wave of the asymmetric soliton is broken after the stable transmission over a short distance. In Fig.   7(e), the linear-stability spectrum of antisymmetric soliton shows that the antisymmetric soliton is in an unstable state when the power 2.0 P = . From the evolution diagram in Fig. 7(f), antisymmetric soliton can only propagate a relatively short distance, and then oscillate and finally lead to waveform rupture.

Conclusion
In conclusion, the NLSE with the self-defocus saturable nonlinearity is studied systematically.
We prove that the model supports symmetric, asymmetric and antisymmetric solitons by numerical methods. With the increase of soliton power, asymmetric soliton is separated from antisymmetric soliton and coexists with symmetric and antisymmetric solitons.
The power curves of solitons show that the threshold of power required for the symmetry breaking bifurcations generated by different saturable nonlinear strengths is different, and this threshold will not change with the increase of the modulation strength of the potential function. This shows that although the modulation strength 0  of the potential function can adjust the power threshold leading to the symmetry breaking bifurcations generated by different saturable nonlinear strengths, it has a certain limit in its adjustment capacity, and beyond this limit, the adjustment capacity will be lost. Soliton stability is studied from the linear stability analysis and verified by the direct numerical simulation. Symmetric and asymmetric solitons can propagate stably with the