Revealing hidden nonlinear soliton dynamics, multistable regimes, and chaotic transitions in regularized long-wave equations under complex external forcing.

Abstract

This paper presents a comprehensive investigation of emerging wave structures associated with the regularized long-wave equation formulated in a nonlinear (2+1)-dimensional framework. By employing advanced analytical techniques, namely the modified Khater method and the Sardar subequation technique, a diverse class of exact solutions is constructed. These solutions encompass bright, dark, singular, and periodic soliton profiles, each demonstrating distinct propagation characteristics. The physical nature of these wave forms is illustrated through detailed two-dimensional plots, three-dimensional surfaces, and projected visual representations to enhance interpretability. To further understand the intrinsic dynamics of the model, qualitative analysis of the corresponding unperturbed planar system is conducted through phase-portrait investigation. When an external periodic forcing term is incorporated, the system exhibits complex nonlinear phenomena, including the onset of chaotic motion. This transition is rigorously examined using phase projections, temporal evolution plots, Poincaré sections, and the computation of Lyapunov exponents to confirm the presence of sensitive dependence on initial conditions. Moreover, an extensive multistability analysis is performed by varying initial states, revealing that slight modifications in system parameters can induce significant transitions between stable and unstable dynamical regimes. Numerical simulations implemented via the fourth-order Runge-Kutta algorithm provide strong computational support for the analytical findings. Overall, the integration of symbolic techniques with high-precision numerical simulations establishes a robust framework for exploring intricate behaviors in higher-dimensional nonlinear dynamical systems.