Approximate Lie Symmetries and Conservation Laws of Third-Order Nonlinear Perturbed Korteweg–de Vries Equation

This work analyses the perturbed Korteweg-de Vries (KdV) equation, a third-order nonlinear differential equation that is critical to understanding wave evolution. The emphasis is on discovering and investigating the approximate Lie symmetries and their associated conservation laws with this equation when exposed to different perturbing functions. Using the partial Lagrange approach, the study discovers approximate symmetries and their related conservation laws for the perturbed KdV equation. The goal is to identify particular perturbations that increase the number of approximation symmetries relative to the original KdV equation, exposing previously unknown system properties. The research involves adding various perturbations to the KdV equation, detecting the resulting Lie symmetries, and finding when the perturbed equation exhibits more symmetries than its unperturbed counterpart.