Discovering diverse soliton solutions in the modified Schrödinger’s equation via innovative approaches

Within the context of this article, we undertake a comprehensive exploration of the paraxial wave dynamical model, a modified version of Schrödinger’s equation. This equation is a well-known nonlinear partial differential equation that has garnered significant attention in modeling of several essential phenomena specifically, wave dynamics in the optical fibers. In a unified framework, our innovative method harmoniously integrates simplicity and strength, enabling us to obtain a diverse set of exact solutions. Through the implementation of this approach, our research unveils fresh perspectives on the characteristics of the model, broadening the scope of knowledge beyond previous works. Among the examined frameworks explored in this paper for predetermined structures, the utilization of the Jacobi elliptic functions is prominent. These functions assume a crucial role in solving nonlinear wave equations and analyzing soliton solutions, particularly within the realm of optics, where they find various applications. To enhance the understanding of the obtained results, we present graphical representations that illustrate the dynamic characteristics of the solutions. These visualizations demonstrate the effectiveness of our proposed method. Furthermore, the practical implications of our research extend beyond its theoretical contributions. Our findings have applications in various fields such as fluid mechanics, nonlinear optics, and plasma physics. Additionally, our approach facilitates the identification of soliton solutions for various other partial differential equations. The utilization of Mathematica software has been employed for computational purposes as well as for generating graphical representations.