Dark soliton solutions of cubic-quartic non-linear Schrödinger equation via Sumudu HPM

Abstract

This research provides study about the cubic-quartic nonlinear Schrödinger equation (CQNLSE), a basic mathematical model with applications in nonlinear optics, plasma physics, and Bose-Einstein condensates. Accurate solutions to such equations are important to understand the behavior of nonlinear waves, including formation of dark solitons. Traditional numerical methods contain discretization and linearization errors, which limits their accuracy and efficiency. To overcome such challenges, a novel hybrid semi-analytical method is proposed in this study, named as Sumudu-HPM, which combines the Sumudu transform with the homotopy perturbation method (HPM). The Sumudu transform simplifies tackling of initial conditions and converts governing equation into an algebraic form, while HPM generates a recursive approximation to the solution. This hybrid approach preserves the nonlinearity of the system without discretization or linearization. The fetched semi-analytical dark soliton solutions claim excellent agreement with exact solutions across a wide range of time levels, demonstrating accuracy, efficiency, and robustness of proposed method.

The key novelty of this work is in the application of Sumudu-HPM approach to solve cubic-quartic nonlinear Schrödinger equation, which bridges the gap between transform-based techniques and perturbation methods for complex nonlinear wave models. The proposed method provides a computationally efficient, stable, and accurate prototype to tackle highly nonlinear partial differential equations, which makes it a valuable tool for researchers across nonlinear science, engineering, and applied mathematics. The main novelty of this work is to develop and implement an efficient semi-analytical technique which is free from discretization error and is computationally efficient.