Research On Multipole Solitons And Defect Solitons In Fractional Schr?dinger Equation

The spatial optical soliton is a very representative physical phenomenon.Because the diffraction effect is balanced by the nonlinear effect,the shape and energy of the beam remain unchanged during the transmission.This phenomenon has been applied in many fields in reality,such as optical routing,etc.Specific systems based on the traditional nonlinear Schr?dinger equation(NLSE)or the nonlinear fractional Schr?dinger equation(NLFSE)can support a variety of solitons,but the propagation characteristics of solitons supported by the NLFSE are obviously different from that supported by the traditional NLSE when the instability is suppressed.This paper studies the spatial solitons in optical lattices with parity-time(PT)symmetry based on the NLFSE.Firstly,the linear part of the equation is analyzed by using the plane wave expansion method,and the band structure of the PT-symmetric optical lattices is obtained.Secondly,the soliton solutions of the equation are obtained by the modified squared-operator iteration method.Thirdly,by perturbing the stationary solitons,the Fourier collocation method is used to analyze the linear stability of the solitons and determine their stable range.Finally,the split-step Fourier method is used to verify the transmission characteristics of solitons.The primary contents of this paper are as follows:1.This thesis investigates the existence and stability of in-phase three-pole and four-pole gap solitons in the NLFSE supported by one-dimensional PT-symmetric optical lattices with defocusing Kerr nonlinearity.These solitons in the first finite gap are stable in the medium power region.As the Lévy index is reduced,the stable regions of these in-phase multipole gap solitons shrink.Below a Lévy index threshold,the effective width of the multipole solitons reduces when the Lévy index raise.Above the threshold,the degree of localization of these solitons decreases with the increase of the Lévy index.The Lévy index cannot change the phase transition point of the PT-symmetric optical lattices.In addition,the transverse power flow of these multipole gap solitons is also studied in the thesis.2.When both linearity and nonlinearity exist in PT-symmetric optical lattices,the propagation properties of defect solitons are also studied in the NLFSE.For the positive defect,the solitons only exist in the semi-infinite gap,and can propagate stably in a wide area.For the negative defect,the solitons can not only exist in the semi-infinite gap,but also in the first finite gap where the defect solitons are stable.In the semi-infinite gap,the negative defect solitons are stable in a wide area,but cannot transmit stably near the Bloch band.When fixing propagation constant,for the negative defect,solitons can become stable by increasing Lévy index.Moreover,for larger propagation constant,the region of the Lévy index for stable solitons grows in the semi-infinite gap,while it shrinks in the first finite gap.