In this paper, we examined the (2+1)-dimensional generalized breaking soliton system (GBSS), an adaptable framework that accurately describes the three-dimensional, wave-dominated interactions occurring in many non-linear media, i.e., fluids, plasmas, and optical fibers. We used an improved F-expansion technique to generate new families of exact solitonic and periodic wave solutions, significantly enlarging the well-studied solution space and providing insight into the complicated interaction between multi-pulse solitons. Validation of these results and an assessment of their stability were carried out by developing a numerical scheme based on finite difference and undertaking a detailed error and stability analysis, demonstrating unconditional stability across a range of parameter values. The results provide new insights into the interplay of dispersion, non-linearity, and cross-wave coupling in governing soliton formation and energy transport in multidimensional systems. In addition to its theoretical importance, this work can provide valuable practical information on engineering applications such as soliton-based communications and wave control applications in fluid systems. This study offers a new methodology to investigate more complex non-linear wave phenomena by integrating the power of symbolic computation with that of robust numerical verification, opening new opportunities for further developments in soliton-driven technologies.

This paper is concerned with the analytical and numerical study of the improved Boussinesq (IB) equation, a nonlinear dispersive model for applications in fluid dynamics, elasticity, geophysics, and nonlinear optics. Two systematic symbolic algorithms, i.e., the generalized tanh method and the $(1/{\mathrm{\Theta }}^{\prime })$-expansion method, are used for the recovery of analytical traveling-wave solutions of the IB equation. These solutions reveal a vast taxonomy of nonlinear waveforms corresponding to solitary, rational, and periodic profiles, governed by parameter combinations that regulate dispersion, wave amplitude, and phase. As a complement to the analytical study, we use an $r$-adaptive numerical method built from the Parabolic Monge-Ampère (PMA) moving mesh method and discretized by central differences in space and BDF2 in time. An adaptive algorithm automatically relocates the mesh nodes toward locations where sharp gradients are present, thereby ensuring accuracy and efficiency and preventing unnecessary computational cost. Numerical experiments evidence second-order convergence and stability and demonstrate the ability of the method to resolve sharp wave interaction without spurious oscillations. In total, the combination of exact benchmarks and adaptive simulation provides a practical framework for simulating nonlinear dispersive waves with impact in applications such as tsunami simulation, earthquake wave propagation, and optical signal pulse transmission.