Integrable model of the interaction of counter-propagating weakly nonlinear waves on the fluid boundary in a horizontal electric field

We consider the nonlinear dynamics of the free surface of a high-permittivity dielectric fluid in a strong horizontal electric field. In the framework of the weakly nonlinear approximation where we assume that the inclination angles of the boundary are small and take only the terms quadratically nonlinear in a small parameter into account in the equations of motion, we obtain a compact model equation that describes nonlinear wave processes in the system. We use this equation to investigate the interaction of counterpropagating solitary surface waves analytically and numerically. In particular, we demonstrate that the counter-propagating waves restore their shape after the interaction and thus acquire a certain phase shift. We also show that these properties of the model originate from its integrability.