Exploration of Lie Symmetry, Bifurcation, Chaos and Exact Solution of the Geophysical KdV Equation

Abstract

This research delves into a deeper investigation of the geophysical KdV equation, which is a crucial mathematical model in the study of nonlinear wave dynamics, particularly in the propagation of oceanic waves. Lie symmetry analysis is used to investigate symmetry reductions, while bifurcation and phase portrait are used to analyze the dynamic behavior. Additionally, chaos theory has been employed to examine the features of the dynamical system. Moreover, by encompassing the recent computational method, namely, the modified Sardar Sub equation (MSSE) approach, we rigorously assess the novel soliton solutions, including dark, bright, singular, combo, periodic, bright-dark, rational forms, and mixed trigonometric. It is observed that while some of the derived solutions align with existing literature, the majority of the solutions obtained in this study differ significantly from previous results, highlighting the novelty of the work. To identify chaotic characteristics, various analytical tools are utilized, such as 3D and 2D phase plots, time series analysis, multistability analysis, Poincaré maps. These findings offer significant insights into the nonlinear behavior and complex dynamics of geophysical wave models, contributing to a deeper understanding of wave propagation and chaotic systems in applied mathematics and physics.