Comparative analysis of lump, breather, and interaction solutions using a bidirectional data mapping approach.

This study analyzes the [Formula: see text]-dimensional Boussinesq equation, a fundamental model in coastal and ocean engineering for describing the propagation of long waves in shallow water. Understanding the nonlinear wave structures of this equation is essential for predicting energy localization, wave stability, and extreme events such as rogue waves. To this end, the Hirota bilinear method is employed to derive explicit [Formula: see text]-soliton solutions, explicitly classifying them into bright and dark types according to parameter criteria. Breather solutions in different planes are constructed using the complex conjugate approach, while the long-wave limit method is applied to obtain first- and second-order lump waves, representing rationally localized structures. Furthermore, four hybrid solutions combining solitons, lumps, and breathers are developed, and their interaction dynamics (e.g. soliton-soliton and soliton-lump collisions) are systematically analyzed. The interactions are shown to be elastic, and all structures retain their identities after collision. A novel contribution of this work is the use of a bidirectional scatter plot technique to compare the behaviors of these solutions across parameter ranges, providing a unified framework for identifying conditions under which different solutions exhibit similar dynamics. The results demonstrate several practical insights: for example, lump solutions preserve their localization over time, modeling stable energy concentrations, while soliton-breather interactions capture oscillatory instabilities relevant for predicting extreme wave events. These contributions extend beyond previous studies by offering both a systematic taxonomy of nonlinear wave structures and a diagnostic tool for engineers to evaluate wave interactions under varying oceanic conditions.