Investigating Rogue Wave Dynamics and Interaction Structures in the KPBBM Model Within the Oceanic Atmosphere

This study discusses the (2+1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona– Mahony equation, which emerges in weakly nonlinear dispersive plasma waves and shallow water dynamics in ocean engineering. Logarithmic dependent-variable transformations are applied to reconstruct a one-exponential tau function as a common one-soliton profile and derive its dispersion relation. This is a standard transformed solution listed as three normalized logarithmic maps, but not a new family of solutions. Only a one-exponential soliton is claimed, no two-soliton family and no arbitrary-N soliton family. The explicit rational rogue-wave families of the first, second, and third orders are derived using a modified version of a well-known center-shifted polynomial tau-function method that is applied to the KPBBM bilinear form, with both center parameters β and γ independent. The novelty is thus limited to the specific model and is not based on a new KPBBM equation or a fundamentally novel symbolic algorithm. The rogue-wave center translates in the longitudinal and transverse directions through β and γ, respectively, for a fixed order N and fixed model parameters. They leave the pattern, localization width, background, and the arrangement of inner patterns unchanged. Lump solutions and lump–soliton interaction structures are also obtained and investigated. The auxiliary Hirota bilinear constraint and its reduced bilinear representation are explicitly given. The higher-degree equations found in the directional logarithmic maps are not new multilinear equations, but rather the denominator-cleared differential polynomial residuals. The validity of each solution family retained is guaranteed by means of analytical substitution or vanishing of symbolically identical-to-zero residual in the original KP–BBM equation. The two- and three-dimensional plots are used only to demonstrate the amplitude profile, localization, and propagation of the solutions, as verified by the analysis. In the weakly nonlinear, long-wave and weakly transverse regime where the KPBBM reduction is valid, these solutions give idealized mathematical representations of localization and interaction mechanisms. They are not predictive of coastal instability or offshore hydrodynamic loading, for which dimensional calibration and experimental/field validation would be necessary.