Interaction between long internal waves and free surface waves in deep water

TL;DR

This paper models the resonant interaction between long internal waves and free surface waves in deep water, using coupled high-order Benjamin-Ono and Schrödinger equations derived from Hamiltonian perturbation methods.

Contribution

It introduces a novel coupled BO-Schrödinger system to describe internal and surface wave interactions in stratified deep water.

Findings

Derived a coupled high-order Benjamin-Ono and Schrödinger system.

Established local well-posedness for the coupled system.

Applied Hamiltonian perturbation methods to fluid wave interactions.

Abstract

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 Schr\"{o}dinger 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, 1997] and Craig-Guyenne-Kalisch [Comm. Pure Appl. Math. 58, 2005]. We also establish a local wellposedness result for a reduced BO-Schr\"{o}dinger system using an approach developed by Linares-Ponce-Pilod [J. Diff. Eqs. 250, 2011].