Lie Symmetries, Soliton Dynamics, Conservation Laws and Stability Analysis of Bogoyavlensky–Konoplechenko System

Nonlinear waves are pivotal in analyzing the propagation of electromagnetic waves and the dynamics of oceanic systems. These waves are indispensable for modeling the long-term impacts of wave energy in coastal regions, addressing critical issues such as climate change adaptation, erosion and coastal flooding. The Bogoyavlensky–Konoplechenko system describes the interaction of the Riemann wave and the long wave propagating in two dimensions. The present work aims to elaborate symmetry reductions and derive invariant solutions of the proposed system. Meanwhile, the infinitesimal generators under one-parameter transformation are constructed, which render the system invariant. Therefore, a repeated process of reductions results in an equivalent system of ordinary differential equations and hence, leads to exact solutions. The solutions have rich physical significance and are efficient in defining several phenomena due to existing arbitrary functions and constants. To examine the physical nature of these solutions, numerical simulation is performed and thus, wave structures like bright and dark lumps, multisoliton, line multisoliton, periodic and annihilation profiles are analyzed. The bifurcation theory has been used to investigate the stability of dynamical system and examine corresponding phase portraits. Furthermore, the conserved vectors with underlying symmetries are constructed using Noether’s theorem.