Novel Nonlinear Dynamical Solutions to the (2 + 1)-Dimensional Variable Coefficients Equation Arise in Oceanography

This article examines the (2 + 1)-dimensional variable coefficient KdV-type equation that arises in oceanography. Oceanography is the science of studying the oceans. It is an earth science that addresses many different subjects, such as the dynamics of ecosystems, ocean currents, waves, and geophysical fluid dynamics. Fluid dynamics is a nonlinear science that studies the flow of liquids and gases in physics, physical chemistry, and engineering. The assistance of the Hirota bilinear method secures different forms of multiple solitons and M-lump waves, such as one-, two-, and three-soliton, by using different functions of variable coefficients. Moreover, the long-wave technique is under consideration when analyzing hybrid solutions, namely mixed soliton-lump, two-soliton-lump, and soliton-two-lump solutions. The results of this exploration exhibit the physical characteristics of solutions and collision-related aspects within the variety of nonlinear systems. The physical significance of the extracted solutions is meticulously analyzed by presenting a variety of profiles that display the behavior of the solutions for specific parameter values. Our results indicate their potential use in future endeavors to discover unique and assorted solutions for nonlinear evolution equations encountered in engineering and mathematical physics. The reported results in this study are the first of their kind to be reported in the literature. These results may be helpful in explaining the physical behavior of various nonlinear physical models.