Unraveling the interplay of gravity and surface tension in driving waves on water's surface

The aim of this paper is to incorporate modern computational and numerical methods to give analytical and approximate soliton solutions for the generalized fractional Hietarinta–type (GFH) model in (2+1) dimensions. Solitary wave solutions for the suggested model, which mathematically shows how gravity and surface tension make waves move again on the water's surface, are identified and illustrated. The phase velocity of waves with different wavelengths is described by the difference in frequency in fluid dynamics. A new modern numerical technique is used to show how accurate the computational solution is, along with analytical solutions. Hamiltonian system features are also utilized to regulate the solutions’ stability. All findings are double-checked by reinserting them into Mathematica 13.1’s core model.