Coupling between internal and surface waves

In a fluid system in which two immiscible layers are separated by a sharp free interface, there can be strong coupling between large amplitude nonlinear waves on the interface and waves in the overlying free surface. We study the regime where long waves propagate in the interfacial mode, which are coupled to a modulational regime for the free-surface mode. This is a system of Boussinesq equations for the internal mode, coupled to the linear Schrödinger equations for wave propagation on the free surface, and respectively a version of the Korteweg-de Vries equation for the internal mode in case of unidirectional motions. The perturbation methods are based on the Hamiltonian formulation for the original system of irrotational Euler’s equations, as described in (Benjamin and Bridges, J Fluid Mech 333:301–325, 1997, Craig et al., Comm Pure Appl Math 58:1587–1641, 2005a, Zakharov, J Appl Mech Tech Phys 9:190–194, 1968), using the perturbation theory for the modulational regime that is given in (Craig et al. to appear). We focus in particular on the situation in which the internal wave gives rise to localized bound states for the Schrödinger equation, which are interpreted as surface wave patterns that give a characteristic signature of the presence of an internal wave soliton. We also comment on the discrepancies between the free interface-free surface cases and the approximation of the upper boundary condition by a rigid lid.