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The Boussinesq equation has drawn significant interest in models for coastline and oceanic engineering, as it can simulate various phenomena such as shallow water waves and harbors, tsunami transmission, and near-shore wave mechanisms. This study examines different approaches for solving the (3+1)-dimensional integrable Boussinesq equation. For this purpose, the Bäcklund transformation is derived by utilizing the Hirota bilinear representation. The understanding of the equation is improved by this transformation, which yields solutions for exponential functions. Furthermore, the model's bilinear form is used to construct its two-, three-, and multi-wave solutions. The features and behavior of the wave solutions to the equation are clarified by this investigation. Additionally, the concerned equation is transformed into an ordinary differential equation by means of a traveling wave transformation, and the results consisting of solutions for rational and polynomial functions are extracted by means of the unified technique. The graphical representations are an essential visual assistance for comprehending the intricate dynamics and behaviors displayed by the governing equation's solutions.
Raza et al. (Thu,) studied this question.