Research On Gap Solitons And Gray Solitons In Fractional Schr?dinger Equation

Nonlinear optics mainly research the interaction between strongly coherent light and nonlinear medium,which has certain application value in the fields of all optoelectronic devices,optoelectronic switches,and all optical communication.Many researches on spatial optical solitons in the past have been carried out,and many unique and novel properties have been found.In this thesis,starting from the nonlinear Schr?dinger equation,we first use a series of numerical calculation methods(calculating the bandgap structure by the plane wave expansion method,solving the soliton solution by the modified squared-operator iteration method,solving the linear-stability spectra by the Fourier collocation method,and simulating the soliton propagation numerically by the symmetric split-step Fourier method),then the existence and stability range of solitons are determined.The primary study contents of the paper include:1.The existence and stability of in-phase dipole gap solitons in two-dimensional(2D)parity-time(PT)-symmetric optical lattices with defocusing saturable nonlinearity and fourth-order diffraction are investigated.The critical threshold(phase transition point)of the2 D PT symmetric optical lattice keeps unchanged when the value of the coupling constant of the fourth order diffraction changes.The effects of different coupling constants of the fourthorder diffraction and saturation parameters on the existence and stability of dipole solitons are discussed.These in-phase dipole solitons exist in the first finite gap and are stable in a moderate power region.In addition,we also research the transverse power-flow vector of these gap solitons.2.Moreover,we also study the existence and stability of gray solitons in the nonlinear fractional Schr?dinger equation with a PT symmetric localized potential.In this system,we find that gray solitons can exist stably.The influences of the Lévy index and the real and imaginary parts of the PT symmetric localized potential on the grayness,existence,and stability of the gray solitons are also discussed.In particular,when the signs of both the real and imaginary parts of the PT-symmetric localized potential are changed,the gray solitons will become anti-dark solitons.These anti-dark solitons can also be stable.Finally,we also learn the transverse power-flow density in the gray and anti-dark solitons.