Analysis of peakon-like soliton solutions: (3+1)-dimensional Fractional Klein-Gordon equation

In this study, we investigate the fundamental properties of ( $3+1$)- $D$ Fractional Klein-Gordon equation using the sophisticated techniques of Riccatti-Bornoulli sub-ODE approach with Backlund transformation. Using a more stringent criterion, our study reveals new soliton solutions that have peakon-like properties and unique cusp features. This research provides significant understanding of the dynamic behaviours and odd events related to these solutions. This work is important because it helps to elucidate the complex dynamics that exist within physical systems, which will benefit many different scientific fields. Our method is used to examine the existence and stability of compactons and kinks in the context of actual physical systems. Under a double-well on-site potential, these structures are made up of a network of connected nonlinear pendulums. Both $2D$ and contour plots produced by parameter changes provide as clear examples of the efficiency, simplicity, and conciseness of the computational method used. The results highlight how flexible this approach is, and demonstrate how symbolic calculations broaden its application to more complex events. This work offers a useful framework and studying intricate physical systems, as well as a flexible computational tool that may be used in a variety of scientific fields.