Exploring Nonlinear Wave Phenomena in (3+1)-Dimensional Jimbo–Miwa Equation: A Comprehensive Study of Singular, Kink-Type, and Periodic Traveling Dynamics

The objective of this manuscript is to examine new soliton solutions for the (3+1)-dimensional extended Jimbo-Miwa model with nonlinear properties. The model represents a wide range of scientific processes in domains such as mathematical biology, nonlinear optics, and quantum field theory, which include complex wave interactions. To begin, we employ the homogeneous balance method to create the original Auto-Bäcklund and Cole-Hopf transformations for the given model, leading to the derivation of various soliton-like solutions that exhibit hyperbolic, trigonometric, and exponential wave functions. After that, a Bäcklund transformation is generated, which has an equal number of arbitrary parameters and bilinear equations. Then, this formation is used to generate two categories of exponential and rational traveling wave solutions with arbitrary wave numbers, resulting in numerous soliton-like solutions. This study also constructs new complexiton solutions using the extended transform rational function method combined with the Hirota bilinear form. To provide further insight into the physical aspects of these solutions, we depict them through a range of visual methods, including 3-D, 2-D, and density plots. The results of this research are innovative and contribute significantly to the ongoing investigation of the equation, providing useful guidance for researchers in future studies. Also, the obtained exact solutions and their physical interpretations are expected to attract considerable interest in the study of nonlinear evolution equations, integrable models, and soliton theory within theoretical physics.