Truncated-bloch-wave Soliton In The Nonlinear Fractional Schr(?)dinger Equation With An Optical Lattice

Spatial soliton is a physical phenomenon that the shape and energy of the beam are kept constant during the transmission.It could be used in many technology fileds like the optical communication.The soliton is different with another because of the different system which could create a situation for stable solitons.There is a special soliton named Truncated-Bloch-wave soliton that is support by the nonlinear periodic system.Truncated-Bloch-wave soliton bridges nonlinear-bloch-waves and gap solitions.The fractional Schr?dinger equation(FSE)generalized by N.Laskin describes the physics when the Brownian trajectories in Feynman path integrals are replaced by Lévy flights.One-dimensional Lévy crystal in a condensed-matter environment was proposed as a probable candidate for the experimental realization of space-fractional quantum mechanics.In 2015,the first optical realization of FSE was proposed by S.Longhi,who put forward an effective optical scheme to simulate the fractional quantum harmonic oscillator.The fractional quantum mechanics system highlights the academic world especially the nonlinear optics of physics.There are so many researches show that the fractional effect can effects the instability of the spatial soliton.So it is necessary to study the fractional effect under the nonlinear optics.The article is based on the nonlinear fractional Schr?dinger equation.We discover the stability solutions by Square operator iteration and analysis the linear instability of the solutions by Split-Step Fourier Methods.The unstable growth rate isalso reached.The linear stability is proved by the simulated communication method.Before we start the core work we need do something that is important and necessary which is to study how comes the truncated-bloch-wave solitons and analysis how is the research work.The foundation part is to get the domain of propagation constant by equating the spectrum of the nonlinear fractional Shr?dinger equation.When the foundation work is well done we begin the core work.This article includes the following core content:We investigate the properties of truncated-Bloch-wave solitons in nonlinear fractional Schr?dinger equation with a periodic potential.Nonlinear modes with different number of peaks are found in the finite gaps of the corresponding linear system.At the same propagation constant,the truncated-Bloch-wave soliton coincides exactly with the corresponding nonlinear Bloch wave.Out-of-phase truncated-Bloch-wave solitons with different number peaks are entirely unstable and in-phase solitons can propagate stably in almost their whole existence domains.Particularly,the instability of truncated-Bloch-wave solitons can be effectively suppressed by the decrease of Lévy index.