Dynamics of Nonlinear Kink Waves and Multi-Kink Wave for (2+1)-Dimensional Extensions of Boussinesq Equation

Abstract

The (2+1)-dimensional extended Boussinesq equation considering higher-order dispersion and nonlinear coupling terms is a paradigm for complex wave representation that lower-order equations in shallow water, nonlinear optics, and dispersive media are not sufficiently captured. Although this class of models, the joint effects of higher-order terms on multi-wave diversity taken as separate and interacting structures remain insufficiently explored. This study addresses these gaps by structuring the complexity of information related to the exact wave with a rich hierarchical model of exact wave solutions, and methodically considering how higher-order dispersion and nonlinearity control wave structure, stability, and propagation behavior. Different family of solutions showing different and yet qualitatively the same dynamical features localized, periodic, resonant, and fusion–fission wave structures is constructed and depicted using exhaustive two- and three-dimensional graphical descriptions using different parameter regimes. The results show how we can actually modulate wave amplitude, wave width, wave interaction topology, and energy across solution types. The novelty in this work is in the unified analytical treatment of multi-wave interactions including soliton resonance, lump dynamics, breather propagation, and Y-shape/X-shape fusion (fission) phenomena and all based on one higher-dimensional framework, providing novel physical perspective which is directly applicable to wave engineering, plasma physics, and nonlinear optics.