The study on peaked waves and nonlinear Jacobi elliptic function based solitons formulation with mixed dispersion to asymptotic scaling system

Abstract In this study, we explore the Dullin–Gottwald–Holm (DGH) equation, an integrable nonlinear model that captures the intricate interplay between nonlinearity, dispersion, nonlocality, and memory effects, having key features governing wave phenomena in fluid dynamics, nonlinear optics, and plasma physics. A comprehensive dynamical analysis of the DGH equation is performed using a hybrid analytical framework that combines the Kumar–Malik approach with the generalized Arnous method. By employing a suitable traveling wave transformation, the governing partial differential equation is reduced to an ordinary differential equation (ODE), which is analytically solved for the first time through the synergistic use of these methods. The derived solutions encompass a rich spectrum of nonlinear wave structures, including Jacobi elliptic, trigonometric, hyperbolic, exponential, and periodic soliton forms, each exhibiting distinct solitonic characteristics. To further elucidate the physical behavior and intrinsic dynamics of the system, the obtained analytical solutions are graphically illustrated using Maple software, incorporating 3D surface plots, 2D time-evolution profiles, contour maps, and density visualizations. The findings not only reaffirm the versatility of the DGH equation in describing diverse nonlinear wave patterns but also demonstrate the efficacy and generality of the combined Kumar–Malik and generalized Arnous techniques in solving complex nonlinear evolution equations.