Computational and numerical analysis of the soliton solutions to the geophysical KdV equation using two robust analytical methods

Abstract

This paper investigates the soliton dynamics of the geophysical Korteweg–de Vries (GKdV) equation, focussing specifically on different types of soliton solutions that emerge within its framework: trigonometric, hyperbolic and rational solutions. Specifically, the study aims to examine elementary tsunami patterns such as rough waves, singular solitonic waves, periodic waves, sinusoidal waves and kink patterns. The coastal regions have experienced extensive urbanisation and rapid population growth, driven by the advancements of global economy. Consequently, this region is particularly susceptible to severe damages from a range of natural disasters, with tsunamis posing a significant threat. This vulnerability is evident by the occurrence of several devastating tsunami events in the 21st century, which have highlighted the exposure of certain regions to such catastrophic events. In this study, both the first integral method and the sub-ODE method are thoroughly discussed and applied to the GKdV equation. These techniques are employed to derive and analyse exact solutions, providing a deeper understanding of the behaviour and dynamics of the equation in geophysical contexts. The obtained results will enrich the understanding of the dynamics of tsunami models and provide deep insights into the propagation of nonlinear tsunami waves. The Coriolis parameter and the velocity of the travelling wave are considered to have a significant impact on tsunami waves. This study further enhances the understanding of nonlinear wave properties in a geophysical context by integrating phase portrait analysis, waveform characteristics and stability evaluations.