An Efficient Analytical Approach for Soliton Solutions to a Variant of the Sasa–Satsuma Equation

The different forms of the Schrödinger equation have been successfully applied in various physical disciplines, including nonlinear optics, plasma physics, and fluid dynamics. In this paper, we consider a generalized form of the Schrödinger equation, as a variant of the Sasa–Satsuma equation, using an efficient analytical approach. To the best of our knowledge, the retrieved solutions have not been previously reported in the existing literature. To solve this model, we employ a robust mathematical approach, called the modified generalized exponential rational function method (GERFM), to derive a wide range of analytical solutions for this model. To enhance understanding of the results, the article provides a variety of supplementary visual aids, including 3D and 2D graphics, and explores the equation’s parameter sensitivity through illustrative figures, allowing readers to better grasp the nuances of the findings. The study also includes a chaotic analysis of the model’s solutions, providing further insight into the behavior of the system. The utilized approach and its resulting outcomes in this study lay a foundation for the development and evaluation of similar models in various fields, enabling future researchers to build upon and expand this work. The main novelty of this work lies in introducing a novel symbolic structure within the framework of the GERFM, enabling new solutions not previously reported in the literature.