Solitonic structures and chaotic behavior in the geophysical Korteweg-de Vries equation: A μ-symmetry and g′-expansion approach

Abstract

This study discusses the mu and Lie symmetries, mu-conservation laws, analytical solutions, chaotic phenomena, and sensitivity analysis of the geophysical Korteweg-de Vries equation (GKdVE). The GKdVE describes the propagation of long waves in geophysical systems like oceans, taking into account the influence of the Coriolis force due to Earth's rotation. We aim to understand the behavior of waves better in geophysical settings and their potential applications across fields like oceanography, meteorology, and climate science. By using the similarity variables, the GKdVE is transformed into a reduced ordinary differential equation (RODE). We employ the (g ')-expansion procedure in one of the RODEs to obtain soliton solutions. Thanks to the (g ')-expansion procedure, we discover six wave solutions. Through the implementation of the variational problem strategy, we derive both the Lagrangian and the mu-conservation law (mu-CL). Additionally, we revisit the planar dynamical system associated with the equation of interest, conducting a sensitive inspection to assess its sensitivity. Moreover, the introduction of a perturbed term reveals chaotic and quasi-periodic behaviors across a range of parameter values. Furthermore, we provide visual demonstrations of these properties through figures depicting the exact solutions.