The two dimensional shallow water waves and its stability

Abstract This study around shallow water wave dynamics, deriving a KP equation for weakly nonlinear flows and evaluating the transverse stability of its soliton solutions. Using multi-scale asymptotics, the work rigorously derives a KP equation from the incompressible, inviscid, weakly nonlinear shallow-water model. Perturbation and eigenfunction analyses yield dispersion relations and stability criteria for solitary waves under transverse disturbances. The findings confirm lowest-order soliton stability and show consistency with related plasma and shallow-water studies, demonstrating the universality of the theoretical framework. The study advances understanding of nonlinear shallow-water dynamics and supplies theoretical support for wave prediction and engineering applications.