Optical Solitons with Parabolic and Weakly Nonlocal Law of Self-Phase Modulation by Laplace–Adomian Decomposition Method

Computational modeling plays a vital role in advancing our understanding and application of soliton theory. It allows researchers to both simulate and analyze complex soliton phenomena and discover new types of soliton solutions. In the present study, we computationally derive the bright and dark optical solitons for a Schrödinger equation that contains a specific type of nonlinearity. This nonlinearity in the model is the result of the combination of the parabolic law and the non-local law of self-phase modulation structures. The numerical simulation is accomplished through the application of an algorithm that integrates the classical Adomian method with the Laplace transform. The results obtained have not been previously reported for this type of nonlinearity. Additionally, for the purpose of comparison, the numerical examination has taken into account some scenarios with fixed parameter values. Notably, the numerical derivation of solitons without the assistance of an exact solution is an exceptional take-home lesson from this study. Furthermore, the proposed approach is demonstrated to possess optimal computational accuracy in the results presentation, which includes error tables and graphs. It is important to mention that the methodology employed in this study does not involve any form of linearization, discretization, or perturbation. Consequently, the physical nature of the problem to be solved remains unaltered, which is one of the main advantages.