We investigate the one-dimensional nonlinear SchrSdinger equation with a perturbation of polynomial type. The approximate symmetries and approximate symmetry reduction equations are obtained with the approximate symmetry perturbation theory.
So far, Lou＇s direct perturbation method has been applied successfully to solve the nonlinear SchrSdinger equation（NLSE） hierarchy, such as the NLSE, the coupled NLSE, the critical NLSE, and the derivative NLSE. But to our knowledge, this method for other types of perturbed nonlinear evolution equations has still been lacking. In this paper, Lou＇s direct perturbation method is applied to the study of perturbed complex Burgers equation. By this method, we calculate not only the zero-order adiabatic solution, but also the first order modification.
The Lie group theoretical method is used to study the equations describing materials with competing quadratic and cubic nonlinearities. The equations shave some of the nice properties of soliton equations. From the elliptic functions expansion method, 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.
The generalized nonlinear SchrSdinger equation （NLSE）, which governs the dynamics of dispersion-managed （DM） solitons, is considered. A novel transformation is constructed such that the DM fibre system equation with optical loss （gain） is transformed to the standard NLSE under a restricted condition. Abundant new soliton and periodic wave solutions are obtained by using the transformation and the solutions of standard NLSE. Further, we discuss their main properties and the interaction scenario between two neighbouring solitons by using direct computer simulation.
