Analyzing wave structure and bifurcation in geophysical Boussinesq-type equations

This article investigates the traveling wave solution for a geophysical Boussinesq-type equation that models equatorial tsunami waves. The discussed structure exhibits explicit traveling wave solutions characterized by speeds surpassing the linear propagation speed and small amplitude wave near-field variables. A combination of traveling wave transformation, tanh method, extended tanh method, and a modified form of extended tanh method are implemented, leading to some new traveling wave solutions for the referred nonlinear model. Through the appropriate selection of parameters, the research employs two-dimensional, three-dimensional, and contour plots to showcase the characteristics of specific solutions. The presented visual representation serves as an efficient means to understand the nature of these solutions. This research further extends its investigation by transforming the considered equation into a planar dynamical structure. Through this transformation, all potential phase portraits of the dynamical system are thoroughly examined, utilizing the theory of bifurcation. In addition, this work investigates the modulation of instability in the governing equation using the linear stability analysis function. Importantly, all the newly derived solutions conform to the main equation when substituted into it. The obtained results demonstrate the effectiveness, conciseness, and efficiency of the applied techniques. These strategies have the potential to be useful in scrutinizing more complex models that appear in modern science and engineering.