Research On Gap Soliton In Saturable Nonlinear Fractional Schr?dinger Equation

The research of soliton is usually inseparable from the support of nonlinear Schr?dinger equation(NLSE).In 1998,Bender and Boettcher proposed parity-time(PT)symmetric potentials.In 2000,Laskin proposed to the fractional Schr?dinger equation(FSE).In 2015,Longhi introduced FSE into the optical system.Through the research and analysis of FSE,its research results are conducive to more diversified research and analysis of beam propagation in FSE.In this article,we derive the normalized NLSE.The derivation of this equation firstly starts from the Maxwell equations,and then goes through a series of calculations and simplifications to further derive the normalized NLSE.After linearizing the NLSE equation,the bandgap structures of optical lattices can be studied by the plane wave expansion method,and then the soliton solution can be obtained by numerical analysis using the Modified SquaredOperator Iteration Method(MOSM).Through the Fourier collocation method and the split-step Fourier method,the stable and unstable regions when propagating in the existence domain can be investigated and further simulation to verify the propagation characteristics of the gap soliton can also be simulated.We report on the existence and the stability of spatial solitons supported by onedimensional(1D)PT-symmetric optical lattices with self-focusing saturable nonlinearity in the fractional Schr?dinger equation.These spatial optical solitons are found to exist in the semiinfinite gap.Here,we mainly study the stability of soliton transmission in the existence domain.Through the Bloch band structure,we can know the range of the semi-infinite gap.Research finds that the spatial gap soliton will alternate between stability and instability when propagating in this existence domain.This paper also discusses the influence of Lévy index(?)and saturation parameter(s)on the existence and stability of spatial optical soliton.With the increase of ?,the stability domain of the soliton in this system gradually widens,and the soliton of unstable transmission changes the transmission path and drifts to the right during the propagation process.The system finds that ? has a threshold value with the increase of ?.Below this threshold,the soliton is unstable,otherwise the soliton can transmit stably.When ?increases,as ? increases from 1 to 2,the stability region is divided into two parts.The soliton oscillation of unstable transmission will appear during the propagation process.As s increases,the system again divides the stable region of soliton into the two parts and the stable and unstable cases appear alternately.The unstable transmission soliton first oscillates in the process of propagation,and then drifts to the right.In view of the above phenomenon,both the Lévy index and saturation parameter can affect the stability of the gap soliton,and can also produce interesting soliton propagation characteristics.