Communications in Mathematical Sciences

Volume 13 (2015)

Number 4

Special Issue in Honor of George Papanicolaou’s 70th Birthday

Guest Editors: Liliana Borcea, Jean-Pierre Fouque, Shi Jin, Lenya Ryzhik, and Jack Xin

Internal waves coupled to surface gravity waves in three dimensions

Pages: 893 – 910



Walter Craig (The Fields Institute, Toronto, Ontario, Canada; and Department of Mathematics, McMaster University, Hamilton, Ontario, Canada)

Philippe Guyenne (Department of Mathematical Sciences, University of Delaware, Newark, Del., U.S.A.)

Catherine Sulem (Department of Mathematics, University of Toronto, Ontario, Canada)


We consider the nonlinear interaction of internal waves and surface waves in a threedimensional fluid composed of two distinct layers. Using Hamiltonian perturbation theory, we show that long internal waves are modeled by the KPII equation and generate a resonant interaction with modulated surface waves at resonant wavenumbers. The surface wave envelope is described by a linear Schrödinger equation in two space dimensions with a potential given by the internal wave. We review the two-dimensional case where an analysis of the model equations, in analogy with radiative absorption in the semi-classical limit, provides an explanation of characteristic features observed on the sea surface due to the presence of an internal wave. In the three-dimensional case, for an internal wave in the form of an oblique line soliton, it is possible to relax the resonance condition to one admitting families of carrier frequencies. We also discuss open problems related to more general KP internal waves.


Hamiltonian systems, internal waves, surface water waves, three-dimensional flows

2010 Mathematics Subject Classification

37K05, 76B07, 76B15, 76B55

Published 12 March 2015