Analysis of propagation and behavior of waves with Qiao equations: Interactions with singular and peaked soliton types

In this paper, we generate different types of solitary and peaked soliton traveling wave solutions for the Qiao equations, which are closely related to the propagation modeling of shallow water waves. An expansion method has been employed to achieve this. For certain special values of the nonlinearity term in the Qiao equation, solitary waves and peaked solitons are obtained. Subsequently, traveling wave solutions in hyperbolic form, distinct from the existing literature and dependent on the nonlinearity term, are produced for the most general case. Specifically, the responses of parameters representing the degree of nonlinearity and wave velocity for these traveling wave solutions are discussed. The main motivation of this work is to provide a better understanding of the phenomenon of nonlinear wave propagation and in this context to reveal the mathematical richness offered by nonlinear wave equations such as the Qiao equations. The aim is to elucidate the nonlinear wave propagation phenomenon through the generated traveling wave solutions, in accordance with the basic principles of shallow water theory. Additionally, the interactions between solutions are examined separately when the speed of the solitary wave exceeds that of the peaked soliton. It is hoped that this work will shed light on developments in the field of nonlinear dynamics.