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.
Muhammad Bilal Riaz, Muhammad Iqbal, Muhammad Aziz ur Rehman et al.· Scientific Reports· 0 citations
This article presents a comprehensive analytical and dynamical investigation of the (4+1)-dimensional variable-coefficient generalized Kadomtsev Petviashvili equation (vc-gKP). Exact analytical solutions are constructed using the modified Khater method, constructing a diverse class of localized and propagating wave structures that capture the intrinsic nonlinear characteristics of the governing model. To further examine the system, an extensive dynamical analysis is carried out using phase-plane trajectories, temporal evolution, Poincare maps, bifurcation analysis, and Lyapunov exponents, allowing the identification of distinct dynamical states and their transitions. The obtained results reveal a variety of complex nonlinear behaviors, including periodic oscillations, quasiperiodic motion, chaotic dynamics, and multistable responses associated with both softening and hardening nonlinear effects. To further verify these behaviors, power spectral analysis, recurrence plots, return maps, and fractal dimension calculations are incorporated, providing complementary information regarding the spectral distribution, geometric organization, and complexity of the resulting attractors. The corresponding numerical illustrations clearly demonstrate the evolution of the system from regular to chaotic regimes together with the coexistence of multiple stable attractors under different parameter conditions. Overall, the proposed analytical and computational methodology establishes an effective framework for exploring nonlinear wave phenomena in higher-dimensional evolution equations and contributes to a deeper theoretical understanding of models arising in fluid mechanics, plasma physics, nonlinear optics, and related
branches of mathematical physics.
Muhammad Iqbal, Muhammad Aizaz Ur Rehman, Z. Shah· Punjab University journal of...· 0 citations