Bifurcation, Quasi-Periodic, Chaotic Pattern, and Soliton Solutions to Dual-Mode Gardner Equation

This study aims to investigate various dynamical aspects of the dual-mode Gardner equation derived from an ideal fluid model. By applying a specific wave transformation, the model is reduced to a planar dynamical system, which corresponds to a conservative Hamiltonian system with one degree of freedom. Using Hamiltonian concepts, phase portraits are introduced and briefly discussed. Additionally, the conditions for the existence of periodic, super-periodic, and solitary solutions are summarized in tabular form. These solutions are explicitly constructed, with some graphically represented through their $2D$ and $3D$ profiles. Furthermore, the influence of specific physical parameters on these solutions is analyzed, highlighting their effects on amplitude and width. By introducing a more general periodic external influence into the model, quasi-periodic and chaotic behavior are explored. This is achieved through the presentation of $2D$ and $3D$ phase portraits, along with time-series analyses. To further examine chaotic patterns, the Poincaré surface of section and sensitivity analysis are employed. Numerical simulations reveal that variations in frequency and amplitude significantly alter the dynamical characteristics of the system.