Exploring cone-shaped solitons, breather, and lump-forms solutions using the lie symmetry method and unified approach to a coupled breaking soliton model

Abstract In this research article, we investigate the coupled breaking soliton (cBS) model using two distinct analytical methods, namely, the Lie symmetry approach and the Unified method. We start by applying the Lie group technique to the cBS model, allowing us to establish infinitesimals, vector fields, commutative and adjoint tables, and an adjoint transformation matrix. Through the utilization of the adjoint transformation matrix, we identify a one-dimensional optimal system of subalgebras. This essential stage allows the cBS model to be reduced into several collections of ordinary differential equations (ODEs) relating to similarity variables resulting from symmetry reduction. By solving these ODE systems under specific parametric constraints, we successfully obtain exact solutions in terms of closed form. Furthermore, the Unified method is employed to address the governing equation, leading us to deduce polynomial and rational function solutions. The dynamic behaviours and characteristics of these such solutions are comprehensively explored through 3-dimensional (3D) plots and contour plots. The graphics show breather solitons, cone-shaped solitons, lump solitons, and patterns of flower petals, periodic solitons, and solitary waves. Additionally, we have connected our mathematical findings with real-world phenomena, which enrich our research work. Furthermore, breathers and lumps arise in many fields of mathematical physics, including fluid dynamics, plasma physics, ocean engineering, nonlinear optics, and physical sciences, as well as several other areas of nonlinear dynamics.