A numerical investigation of the Whitham Equation for solitary waves propagating on conducting flows

We introduce the Whitham equation in the context of electrohydrodynamic (EHD) flows, which incorporates the nonlinearity of the Korteweg-de Vries (KdV) and the full linear dispersion relation associated with electrohydrodynamic effects, extending the classical Whitham approach to electrical regimes. {\color{black} This electrohydrodynamic extension will be referred to as the e-Whitham equation.} To assess its performance, we conduct numerical simulations comparing the e-Whitham equation to the Korteweg-de Vries-Benjamin-Ono (KdV-BO) across various electric field strengths. We investigate travelling wave profiles, solitary wave collisions, and trapped wave phenomena. The numerical experiments demonstrate strong agreement with asymptotic predictions. {\color{black} The model reduces to the KdV-BO equation in the weakly dispersive regime, confirming its consistency with known asymptotics and ensuring accuracy where asymptotic models are valid. Its main novelty lies in extending the Whitham framework to electrohydrodynamic flows, making it suitable for exploring parameter regimes beyond the reach of KdV-BO.