Abstract The study of nonlinear partial differential equations through analytical methods is arguably the most important and fascinating to grasp various complex physical and engineering phenomena. The positive Gardner–Kadomtsev–Petviashvili equation, which combines the dimensions effects of the KP equation with the dispersion properties of the Gardner equation, is investigated in this study. Non-linear wave systems, fluid dynamics, plasma physics, and wave interactions are all explained by this equation. A Hamiltonian formulation is presented, followed by chaos analysis, which visualizes fractal structures and reveals underlying deterministic patterns in seemingly random systems. A new extended algebraic method is introduced to solve the positive–Gardner-KP equation, yielding rational, exponential, and trigonometric solutions. The graphical propagation of soliton solutions is examined, illustrating their behavior under various parametric conditions. The results highlight the efficiency of the proposed method in understanding the dynamics of shallow water waves, particularly in the absence of surface tension and viscosity, with implications for atmospheric and oceanic wave phenomena.
Toseef et al. (Thu,) studied this question.