分数衍射系统中部分PT对称孤子的对称破缺

Symmetry Breaking of Partially PT-Symmetric Solitons in Fractional Diffraction Systems

基于分数阶非线性薛定谔方程,采用虚时演化的数值方法研究了部分宇称时间(PT)对称的光孤子及其自发对称破缺现象,通过线性稳定性分析并结合数值模拟研究了分数衍射效应对二维光孤子稳定性和传输的影响。结果表明:分数衍射系统中存在部分PT对称的二维光孤子,分数衍射效应随着莱维指数α的减小而增强,莱维指数α的改变影响光孤子的稳定性。当孤子功率超过特定的临界值Pc时,部分PT对称的光孤子发生自发对称破缺,并转变为复传播常数的不对称态。通过分析莱维指数α与孤子自发对称破缺临界功率Pc之间的关系,发现将莱维指数α从2减小至1时,孤子的自发对称破缺临界功率Pc由1.6降低为0.4。这表明增强分数衍射效应使得部分PT对称的二维光孤子的稳定性变弱,进而在更小的孤子功率情况下发生自发对称破缺。该研究结果为分数衍射的非厄米非线性光学波导中控制孤子的形态和传输提供了理论依据。

Objective 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.Methods 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.Results and Discussions 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 1causes 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.Conclusions 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 PTsymmetric 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.