Optical solitons of the perturbed nonlinear Schrödinger equation using Lie symmetry method

This paper presents an analytical treatment of the perturbed nonlinear Schrödinger equation (P-NLSE) using the Lie symmetry method. The NLSE is a fundamental equation in nonlinear optics and describes the propagation of optical pulses in a nonlinear medium. In the presence of perturbations, the solution to the NLSE becomes more complicated and requires more sophisticated mathematical techniques to analyze. The Lie symmetry method is a powerful tool for finding exact solutions to nonlinear differential equations and has been applied successfully to a wide range of problems in physics and engineering. In this paper, we use the Lie symmetry method to obtain exact analytical solutions for the P-NLSE and study their properties. Our results show that the perturbations can have a significant effect on the behavior of the optical pulses, leading to phenomena such as soliton fission and dispersive shock waves. The Lie symmetry method is used to derive three generators for the Lie algebra, and two reductions are examined through combinations of vector generators. The Improved generalized Riccati equation and Nucci reduction method are then employed to obtain exact solutions, including bright soliton, multiple singular soliton, and exponential solutions.