Analyzing dispersive optical solitons in nonlinear models using an analytical technique and its applications

The article focuses on exploring three distinct equations: the Jimbo-Miwa equation (JME), the generalized shallow water equation (GSWE), and the Hirota-Satsuma-Ito equation (HSIE). By applying the \(\exp (-\Phi (\eta ))\)-expansion method (EEM), we have successfully obtained novel solutions with trigonometric, elliptic, and hyperbolic properties. The main objective of this study is to identify and explore previously undiscovered soliton solutions within nonlinear wave equations, contributing to a deeper comprehension of wave behaviors and facilitating potential applications across diverse scientific and engineering domains. The Jimbo-Miwa equation is relevant to integrable systems and mathematical physics, potentially finding applications in quantum field theory and condensed matter physics. The generalized shallow water equation extends the classical shallow water equations, enabling better modeling of complex fluid dynamics like ocean currents and tsunamis. The Hirota-Satsuma-Ito equation, likely a soliton-based nonlinear equation, holds importance in nonlinear optics, fluid dynamics, and possibly biological studies, contributing to the comprehension of wave-like behaviors in diverse systems. Soliton and solitary wave structures are extracted as distinct solutions. By selecting appropriate values for arbitrary parameters within the accurate range, we create 3D, 2D, and contour plots to visualize the discovered solutions. Modifying model parameters enables the alteration of the solution dynamics generated by the models. The calculations for this research were exclusively performed using the symbolic software Mathematica. The solutions received encompass a variety of types, such as dark, bright, combo dark-bright, singular, cuspons, peakons, periodic solitary wave solutions, single-soliton solutions, double-soliton solutions, N-soliton solutions, and numerous others. These solutions have real-life applications in areas such as predicting coastal hazards, improving optical communications, studying nonlinear dynamics, enhancing material science, and advancing medical imaging techniques. The complexity and nonlinear nature of the system are underscored by these findings, emphasizing the necessity for additional analysis. Moreover, the obtained results offer valuable insights into understanding and modeling comparable physical systems. This research marks a significant advancement by utilizing the the \(\exp (-\Phi (\eta ))\)-expansion method to reveal solitonic solutions for an unsolved model, thereby expanding the existing literature and introducing a novel mathematical technique to address nonlinear physical models. The proposed method is concise, transparent, and reliable, leading to reduced computations and widespread applicability.