Interaction Between Long Internal Waves and Free Surface Waves in Deep Water

We consider a density-stratified fluid composed of two immiscible layers separated by a sharp interface. We study the regime of long internal waves interacting with modulated surface wave packets and describe their resonant interaction by a system of equations where the internal wave solves a high-order Benjamin–Ono (BO) equation coupled to a linear Schrödinger equation for the envelope of the free surface. The perturbation methods are based on the Hamiltonian formulation for the original system of irrotational Euler’s equations as described in Benjamin–Bridges (J Fluid Mech 333:301–325, 1997) and Craig et al. (Pure Appl Math 58:1587–1641, 2005). We also establish a local well-posedness result for a reduced BO–Schrödinger system using an approach developed by Linares et al. (J Diff Eqs 250:450–475, 2011).