The Exact Solutions To The X⁽²⁾ And X⁽³⁾ Competing Nonlinearity Equation And Their Stability

Optical solitons have been under intensive studies due to their novel physics go with their potential applications,such as all-optic driving,optical-switching, optical communication,data store,capture,control and steer particles and atom cooling.The study of the dynamics of optical solitons is of great importance in both theory and practice.This thesis mainly focus on analytically finding the solutions of the equations describing material with quadratic and cubic nonlinearity,as well as some simulation of optical soliton propagation and propagation stability.First we introduce the history of the soliton being discovered and studied.We also provide some of the classifications of the optical solitons,and emphasize discussing the study of the spatial optical solitons.We derive three typical equations that describe the super short pulse propagation in the media,which include the nonlinear Schr(o|¨)dinger equation,cascading equation and competing nonlinearity equation. So far as I know,many numerical calculation methods have been used to study the competing nonlinearity system,but only one exact analytical solution of this system has been obtained by S.Trillo etc.,in 1995 and shown in Ref.[1]in special case and some of the constraints were also needed to be fulfilled.Lie Group theoretical method is used to study the equations which share some of the nice properties of soliton equations.We obtain large families of analytical solutions,in special cases,we have the periodic,kink and solitary solutions of the equations.Furthermore,we investigate the stability of these solutions under the perturbation of amplitude noises by numerical simulation. Finally,we employ the same methods to solve the 3W system which includes two fundamental waves and one second harmonic wave in competing nonlinearity equations,and obtain large families of analytical solutions.