Physical vs mathematical origin of the extended KdV and mKdV equations

The higher-order Korteweg-de Vries (KdV) and modified KdV (mKdV) equations are derived from a physical model describing a three-component plasma composed of cold fluid ions and two species of Boltzmann electrons at different temperatures. While the higher-order KdV equation is well established, the corresponding mKdV equation is typically derived using the system's integrability properties. In this work, we present the extended mKdV equation, derived directly from the physical system, offering a fundamentally different form from its integrable counterpart. We explore the connections between the two equations via Miura transformations and analyze their solutions within the framework of asymptotic integrability.