Exploring solitary wave structures and bifurcation dynamics in the (2+1)-dimensional generalized Hietarinta equation

This study investigates the $(2+1)$-dimensional generalized Hietarinta equation, which models the propagation of waves on water surfaces in the presence of gravity and surface tension. Solitary wave solutions are obtained using the $exp(-w\left(x\right))$ method and the $F$-expansion method, and are expressed in terms of hyperbolic, trigonometric, exponential and rational functions. Two- and three-dimensional plots illustrate various wave structures, such as dark, kinked, and singular kinked waves, highlighting their dynamic behaviors under different parameter settings. Hamiltonian functions and bifurcation theory are employed to analyze phase portraits and nonlinear wave dynamics, including chaotic behavior. Numerical simulations has been conducted using Mathematica and Maple confirm the theoretical findings. Additionally, the results have been compared with other existing results in the literature to show their uniqueness. The proposed techniques are effective, computationally efficient and reliable. In this context, considering previous studies, the findings of this research contribute to the existing literature.