The effective dynamics of interacting waves for coupled Schrödinger-Korteweg-de Vries
equations over a slowly varying random bottom is rigorously studied. One motivation for
studying such a system is better understanding the unidirectional motion of interacting
surface and internal waves for a fluid system that is formed of two immiscible layers. It
was shown recently by Craig-Guyenne-Sulem [1] that
in the regime where the internal wave has a large amplitude and a long wavelength, the
dynamics of the surface of the fluid is described by the Schrödinger equation, while that
of the internal wave is described by the Korteweg-de Vries equation. The purpose of this
letter is to show that in the presence of a slowly varying random bottom, the coupled
waves evolve adiabatically over a long time scale. The analysis covers the cases when the
surface wave is a stable bound state or a long-lived metastable state.