Demonstration of Soliton Wave Structures in a Higher-Dimensional Nonlinear Physical Model: Dynamic Insights into Sensitivity, Bifurcation, and Chaos
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.