<![CDATA[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. 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. Kerr nonlinearity supports sharper, narrower solitons, whereas quadratic-cubic nonlinearity yields broader profiles with enhanced stability. This study obtains soliton solutions of the NLSE with Kerr and QC nonlinearities using the RBSODE and BSODE methods, analyzes the qualitative differences in the obtained profiles, and examines the conservation laws characterizing the dynamics. The RBSODE and BSODE methods are chosen for their algebraic flexibility and their ability to handle the nonlinearODEs derived from the traveling-wave reduction of the NLSE. Furthermore, we use the multiplier method to derive 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 systems. Our research contributes to the understanding of soliton behavior in nonlinear media, with potential applications in optical signal transmission and ultrafast laser propagation.]]>
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