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This study introduces an analysis of bifurcations, chaotic dynamics, and sensitivity using the Galilean transformation applied to the perturbed non-linear Schrödinger equation (NLSE) with Kerr-law nonlinearity. Additionally, we employ the efficient Sardar sub-equation (Sse) method to derive various optical soliton solutions. Initially, we outline the general framework of the Sse method proposed. Next, we utilize a traveling wave transform on the NLSE, transforming it into a system of nonlinear ordinary differential equations (NLODEs) which are further separated into their real and imaginary components. Furthermore, we apply this method to derive new optical solitons for the NLSE equation with Kerr law. We perform exact analytical simulations of these optical soliton solutions and investigate the effects of different parameters on the soliton waves.

The goal of this paper is to obtain the new explicit solutions of the Atangana conformable (AC) fractional Biswas–Milovic (BM) equation with Parabolic law nonlinearity (PLN) which is a class of Schrödinger equation (SE) using a new methodology. This methodology, which is an extension of the direct algebraic method, provides some new solutions that are the advantage of this proposed approach. To exhibit the dynamical performance of this equation, some solutions are provided and drawn.. Usage of the new extended direct algebraic method (NEDAM), it is obtained new optical solitons for the proposed model. The proposed method has a better efficiency in comparison to the existing methods and is straightforward for nonlinear evolution equations (NLEEs). Moreover, the introduced method can be programmed using Maple software to obtain new obvious solutions for Atangana conformable nonlinear evolution equations (ACNLEEs).