Application Of Symmetry Perturbed In Perturbative Nonlinear Schr(?)dinger Equation

The ultrashort pulse attracts people's interesting for its broad applications in many files, such as laser processing, medical and bio-optical, opto-electronics, etc. It is well known that nonlinear Schrodinger equation(NLSE) can describe propagation of picosecond optical pulses in single-mode fiber. While picosecond optical pulses, for ultrashort pulses, it is needed to take into account the influence of high-order effects on the light's propagating in the medium. The third-order dispersion, delayed Raman response and self-steepening are three most important high order effects. The modified nonlinear Schrodinger equation(MNLSE)can describe propagation of ultrashort pulses in single-mode fiber by adding high-order effects terms.Many numerical methods and perturbation methods are used to understand the evolution of ultrashort pulses in single-mode fiber. The approximate symmetry perturbation(here-in-after called symmetry perturbation) method is a powerful method for the study of the nonlinear partial differential equations. The symmetry perturbation method, as one of the perturbation methods, has clear thinking and the widely application scope and it is applicable to both integrable and unintegrable systems. For the development of compute, though computing software Maple, we can calculate a mount number of linear differential equations to obtain the symmetries of equations.In this paper we apply the symmetry perturbation method to study the approximate analytical solutions to the NLSE containing the high-order terms. It is obtained that infinity series of similarity reduction equations to the NLSEs containing the third-order dispersion term and delayed Raman response term. Though setting some boundary and initial conditions, the analytical first order perturbation solution is also obtained. the first three order similarity reduction equations to the NLSE with the self-steepening term was obtained. while in our treatment the perturbation solutions, the zero-order solution of perturbation can be anyone of the exact solutions of the unperturbed equation, and different first-order correction can be obtained by solving first-order reduction linear ordinary differential equations under different boundary conditions. Generally, it is a course to determine the integral constants.