Exploring Nonlinear Dynamics in Intertidal Water Waves: Insights from Fourth-Order Boussinesq Equations

Abstract

The fourth-order nonlinear Boussinesq water wave equation, which describes the propagation of long waves in the intertidal zone, is investigated in this study. The exact wave patterns of the equation were computed using the tanh method. As stability decreased, soliton wave structures were derived using similarity transformations. Numerical simulations supported these findings. The tanh method introduced a Galilean modification, leading to the discovery of several new exact solutions. Subsequently, the fourth-order nonlinear Boussinesq wave equation was transformed into a planar dynamical system using the travelling wave transformation. The quasi-periodic, cyclical, and nonlinear behaviors of the analyzed equation were particularly examined. Numerical simulations revealed that varying the physical parameters impacts the system’s nonlinear behavior. Graphs represent all possible examples of phase portraits in terms of these parameters. Furthermore, the study was proven to be highly beneficial for addressing issues such as shock waves and highly active travelling wave processes. Sensitivity analysis theory and the Lyapunov exponent were employed, offering a wide variety of linear periodic and first-frequency periodic characteristics. Sensitivity analysis and multistability analysis of the Boussinesq water wave equation were thoroughly investigated.

Keywords

• mathematical model
• visualisation
• chaotic behaviors
• sensitivity analysis
• multistability
• dynamical system