Bifurcation, chaotic behaviour, multistability and sensitivity analysis: Exact and numerical analysis of nonlinear dispersive wave equation

Abstract

In this research, we examine the equal-width equation, a basic model for one-dimensional wave propagation in nonlinear fluid dynamics. Using the Kudryashov method, we obtain explicit soliton solutions that reflect the equation’s inherent nonlinear nature, modeling different hydrodynamic phenomena like shallow water waves and internal solitons. The solutions are graphically represented using three-dimensional (3D), contour, density, and two-dimensional (2D) plots to gain further insight into wave evolution. To confirm the analytical solutions, we apply the differential transform method (DTM) for numerical simulations, allowing for comparative analysis between theoretical solitons and their discrete approximations. In addition, stability and modulation instability analyses are conducted to determine the robustness of these wave structures under small perturbations, important for understanding turbulence and energy dissipation in fluids. Furthermore, we perform a bifurcation analysis through the building of phase portraits and vector fields, uncovering complex dynamical behaviors like periodic and chaotic motion in nonlinear fluid systems. In order to expand our investigation, we add a periodic perturbation to investigate chaotic wave interactions, represented through phase space trajectories and time series plots. The perturbed system presents a perturbation with elements of intensity δ and frequency ϕ, enabling us to study how small periodic perturbations influence the dynamical behavior and stability of the nonlinear wave solutions. Finally, we investigate multistability and carry out sensitivity analysis, evaluating how initial conditions affect solution trajectories in a fluid system. Our results are helping toward a deeper understanding of nonlinear wave mechanics and their repercussions in fluid physics. This work addresses the lack of a unified framework by combining exact soliton solutions, numerical validation, and nonlinear dynamical analysis for the equal-width equation.