Zamir Hussain

In this article, the modified $\alpha$ equation is solved using the direct algebraic approach. As a result, numerous new and more generalized exact solutions for such equations have been found, taking into account the wide range of travelling structures. The rational, trigonometric, hyperbolic, and exponential functions with a couple of licentious parameters are thus included in these exact answers. Analytical solutions feature a variety of physical structures, which are visually studied to demonstrate their dynamic behavior in 2D and 3D. Considering the parameters, all feasible phase portraits are shown. Furthermore, we used numerical approaches to determine the nonlinear periodic structures of the mentioned model, and the data are graphically displayed. Additionally, we employed numerical approaches to determine the nonlinear conditions that contribute to the presented model, and the data are graphically displayed. After evaluating the influence of frequency following the application of an external periodic factor, sensitivity exploration is used to study quasi-periodic and chaotic behavior for several starting value problems. Furthermore, the function of physical characteristics is investigated using an external periodic force. Quasi-periodic and quasi-periodic-chaotic patterns are described with the inclusion of a perturbation term. The direct algebraic methodology would be used to derive the soliton solution of modified $\alpha$ equation, from which the Galilean transformation derives traveling wave solutions of the considered and a bifurcation behavior is reported. Analytical and numerical methods have been used to have the condition of the travelling wave phase transformation. The well-judged values of parameters are enhanced well with a graphically formal analysis of such specific solutions to illustrate their propagation. Then a planer dynamical system is introduced, and a bifurcation analysis is utilized to identify the bifurcation structures of the dynamical model’s nonlinear wave propagation solutions. Additionally, the periodic and quasi-periodic behavior of the discussed equation is analyzed using sensitivity analysis for a range of beginning values. To further comprehend the dynamical behaviors of the resultant solutions, a graphic analysis is conducted.

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.