Exploring Nonlinear Dynamics and Solitary Waves in the Fractional Klein–Gordon Model

Various nonlinear evolution equations reveal the inner characteristics of numerous real-life complex phenomena. Using the extended fractional Riccati expansion method, we investigate optical soliton solutions of the fractional Klein–Gordon equation within this modified framework. This model in physics claims the existence of energy particles and defines the relativistic wave. The proposed procedures provide insight into wave spread and optical solitons, which enterprises in current broadcast communications can utilize to empower fast and long-distance information transmission with minimal signal degradation. They are important for their reliability and optical communication networking. This method can analytically formulate optical soliton solutions using rational, hyperbolic, and trigonometric functions. The interaction between the breather and the king wave, as well as the bright and dark bell-shaped singular soliton waves, are the numerical forms of the obtained solutions, examined using three- and two-dimensional diagrams. These solutions are obtained using the proposed method. For [α, β = 0.1, 0.5, 0.9], we illustrate the impact of conformable and beta fractional parameters in two-dimensional graphs. Understanding and clarifying the physical characteristics of waves may be made easier by the collected results. Nonlinear optics, optical communications, and engineering all rely on unique and precise soliton solutions, and the aforementioned applied techniques may, therefore, serve as a valuable tool for this purpose.