Lax pairs and Bäcklund transformations for a new (3+1)-dimensional integrable equation utilizing symbolic computation

This article discusses a new integrable wave equation in (3+1)-dimensions that describes shallow water waves. The auto–Bäcklund and Cole-Hopf transformations are constructed by means of the homogeneous balance approach, which results in numerous new solitary wave-type solutions. Based on Hirota's method, Bäcklund transformation is derived, and the respective lax pairs are also calculated. The exponential, trigonometric, and hyperbolic wave functions generate numerous kinds of soliton-type solutions. To further capitalize on the potential and physical behavior of the equation, the findings are also presented through phase portraits in the form of 3D, density, and 2D plots. This research may advance knowledge of the characteristics of nonlinear waves that form in seas and oceans.