The exact solutions of the shynaray-iia equation along with analysis of bifurcation and chaotic behaviors

Abstract

In this paper, the Shynaray-IIA equation, which characterizes phenomena like tidal waves and tsunamis, is considered. In the first stage, the relevant equation is transformed into a planer dynamic system by using the Galilean transformation. Thanks to the well-known bifurcation theory, phase portraits for nonlinear traveling wave solutions have been investigated. As far as is known, before this study, there is no research in which this review is conducted. In addition, an external force is added to the resulting dynamic system and its effect is examined. The model's dynamic behaviour is investigated through bifurcation, periodic quasi-periodic and chaotic behaviour, and sensitivity. These include methods like phase portrait rendering, time series scrutiny, Lyapunov exponents calculation, and the assessment of multi-stability. To examine the solitonic wave solutions to the underlying equation, the improved tan (phi/2)-expansion method has been utilized. The solutions retrieved by this method are expressed in the form of hyperbolic, trigonometric, rational and exponential functions. In addition, some of the various kinds of solitons such as dark and bright wave obtained have been visualized with 3D graphics for different values of the parameters in order to better understand their dynamic behavior. Furthermore, the modulation instability of the governing equation is investigated through the application of linear stability analysis.