Nonlinear dynamics of the Dullin-Gottwald-Holm equation: solitons, kinks, and freak waves

The aim of this research work is to explore soliton solutions to nonlinear space-time conformable Dullin-Gottwald-Holm equation arising in water waves by the extended direct algebraic method. The proposed methodology solves the nonlinear fractional partial differential equations by transforming them into nonlinear ordinary differential equations through the use of a finite series formulation of solutions by means of Riccati ordinary differential equation. The newly discovered soliton solutions have diverse mathematical forms including, rational, exponential, trigonometric, and hyperbolic functions and allow an in-depth study of the underlying wave behaviours. A number of 3D, contour and 2D graphical representations are provided which clearly show the existence of shock, double shock, kink, chirped freak, bell-shaped kink and other soliton structures in the intended model. Our results indicate greater level of variability compared to the past solutions, and this provides essential information on processes that exist in the model under study. Furthermore, due to the high count of soliton solutions generated by our method, this method proves to be valuable as it provides significant information about the dynamics of the model under consideration, as well as provides potential opportunity to use it in the management of other nonlinear models.