Fractional diffraction effects and various novel phenomena produced by parity-time (PT) symmetric optics systems have become research hotspots in the field of optics. A large amount of theoretical research has proven the existence of the optical soliton in the fractional nonlinear Schr?dinger equation containing PT-symmetric potentials. However, the existence, stability, and dynamics of partially PT-symmetric solitons in non-Hermitian nonlinear optical waveguides with fractional diffraction effect have not been explored yet. The phenomenon and mechanism of spontaneous symmetry breaking of the partially PT-symmetric solitons are still unclear. Meanwhile, the obtained research results provide new insights into the propagation and controlling of the partially PT-symmetric solitons in the non-Hermitian nonlinear optical waveguides with fractional diffraction.

We numerically solve partially PT-symmetric soliton solutions and asymmetric solutions. Specifically, the accelerated imaginary time evolution method is used to solve the stationary fractional nonlinear Schr?dinger equation. Two types of solutions are obtained. The first type is the partially PT-symmetric solitons with real propagation constants, and the second is the asymmetric solutions with complex propagation constants. Then, the solutions of the perturbation are linearized through linear stability analysis, and the eigenvalue problem of the perturbation modes is transformed into the spectral space by using the Fourier collocation method. The spectrum of the eigenvalue problem of the perturbation modes is numerically solved. The propagations of the partially PT-symmetric solitons and the asymmetric solutions are numerically simulated using the split-step Fourier method. Finally, the obtained results are compared with the results of linear stability analysis.

First, two types of solutions are confirmed to exist in the fractional nonlinear Schr?dinger equation with the partially PT-symmetric potential. The first type of solution is the partially PT-symmetric solitons with real propagation constants, and the second type of solution is the asymmetric solutions with complex propagation constants. The results are shown in Fig. 2 and Fig. 3, respectively. Then, the critical power of the symmetry breaking bifurcation point of the partially PT-symmetric solitons is numerically determined and verified with the linear stability analysis, and the results are shown in Fig. 4(c) and Fig. 5(b), respectively. The reduction of the Lévy index from 2 to 1 causes the critical power of the spontaneous symmetry breaking for the partially PT-symmetric solitons to decrease from 1.6 to 0.4. The numerical simulations of the transmissions of the partially PT-symmetric solitons and the asymmetric solutions are shown in Fig. 6, Fig. 7, and Fig. 8, respectively. It is found that the stable partially PT-symmetric solitons obtained by linear stability analysis are robust, as shown in Fig. 6. The amplitude oscillates periodically during the propagations for the unstable partially PT-symmetric solitons in Fig. 7. In Fig. 8, the amplitude and light field distribution of the asymmetric solution change significantly.

In summary, the partially PT-symmetric optical solitons and spontaneous symmetry breaking phenomenon in the fractional nonlinear Schr?dinger equation are numerically studied. The research results show that there exist partially PT-symmetric solitons. The soliton power exceeds the critical value, and the partially PT-symmetric solitons turn into the asymmetric state. The enhanced fractional diffraction effect weakens the stability of the partially PT-symmetric solitons, and then spontaneous symmetry breaking occurs under the smaller soliton power. The critical power of the partially PT-symmetric soliton decreases to 0.409, when the Lévy index decreases to 1. The stable partially PT-symmetric solitons are robust and can be transmitted stably up to 1000 times the diffraction length, even in the presence of the perturbation. The research results of this work may be used to control optical solitons in the non-Hermitian nonlinear optical waveguides with fractional diffraction.