Investigating Soliton-Wave Dynamics Using the Focusing Nonlinear Schr¨odinger Equation

This research undertakes a comprehensive investigation of the optical soliton solutions of the Focusing Non-linear Schr¨odinger Equation (NLSE), a fundamental model describing the propagation of optical solitons in nonlinear media. To achieve this, we employ two versatile and efficient methods: the Ricatti-Bernoulli Sub Ordinary Differential Equation (RBSODE) method and the Bernoulli Sub Ordinary Differential Equation (BSODE) method. These methods enable us to derive a wide range of optical soliton solutions. We examine two distinct nonlinearities: the Kerr law nonlinearity and the quadratic-cubic nonlinearity. These nonlinearities are crucial in determining the behavior of optical solitons in various nonlinear optical media. Our analysis reveals that the derived soliton solutions exhibit distinct characteristics, such as amplitude, width, and velocity, which are influenced by the type of nonlinearity and the parameters of the NLSE. Furthermore, we utilize the multiplier approach to obtain the conservation laws of the NLSE. These conservation laws provide valuable insights into the underlying dynamics of the optical solitons and have significant implications for the design and optimization of nonlinear optical system.