From Symmetries to Solitons: Unveiling the Dynamics of the KdV-CDG Model through Painlevê Analysis and Modulation Instability

Abstract

The main aim of this paper is to perform an extensive investigation of the Korteweg-de Vries-Caudrey-Dodd-Gibbon dynamical model, which is of great importance in modeling different phenomena of ocean dynamics. We start by examining the integrable version of the Korteweg-de Vries-Caudrey-Dodd-Gibbon model and carry out a thorough symmetry analysis to investigate its underlying mathematical structure. The Painlevê analysis is performed to investigate the integrability characteristics of the model to ensure its compatibility with analytical solution techniques. In addition, we obtain exact soliton solutions based on the modified Sardar subequation method, illustrating its efficiency in yielding explicit wave solutions. The solutions are next visualized based on multiple graphical plots such as 3D surface plots with projections, polar plots, and 2D line plots, offering deeper insight into the wave behavior of the model. Finally, the modulation instability analysis is performed to examine the stability of wave propagation for small perturbations. A comparative study with existing work is provided, with emphasis on the originality and relevance of our results. The findings presented in this work are additions to what is already known about the model and its usage in nonlinear wave theory, making it even more relevant in the research of oceanic and fluid dynamics.

Keywords: Korteweg-de Vries-Caudrey-Dodd-Gibbon (KdV-CDG), Optimal System, Integrablility, Soliton, Lie Symmetries