Energy Analysis, Soliton Dynamics, Chaos, and Sensitivity Analysis for a Forced Damped Gardner Model

In this study, the complete discrimination system for the polynomial method (CDSPM) is employed to analyze the integrable Gardner Equation (IGE). Through a traveling wave transformation, the model is reduced to a nonlinear ordinary differential equation, enabling the derivation of a wide class of exact solutions, including trigonometric, hyperbolic, rational, and Jacobi elliptic functions. For example, a bright soliton solution is obtained for parameters A=1.3, β=−0.1, and γ=0.8. Qualitative analysis reveals diverse phase portraits, indicating the presence of saddle points, centers, and cuspidal points depending on parameter values. Chaos and quasi-periodic dynamics are investigated via Poincaré maps and time-series analysis, where chaotic patterns emerge for values like ν1=1.45, ν2=2.18, Ξ0=4, and λ=2π. Sensitivity analysis confirms the model’s sensitivity to initial conditions χ=2.2,2.4,2.6, reflecting real-world unpredictability. Additionally, the energy balance method (EBM) is applied to approximate periodic solutions by conserving kinetic and potential energies. These results highlight the IGE’s ability to capture complex nonlinear behaviors relevant to fluid dynamics, plasma waves, and nonlinear optics.