Models for Internal Waves in Deep Water

We study properties of solitary wave solutions of three evolution equations arising in the modeling of internal waves Our experiments indicate that broad classes of initial data resolve into solitary waves but also suggest that solitary waves do not interact exactly thus suggesting two of these equa tions are not integrable In the course of our numerical simulations interesting meta stable quasi periodic structures have also come to light Introduction In this paper consideration is given to long crested unidirec tional waves at the interface of a two layer system of incompressible inviscid uids The top layer is assumed to be in nitely deep while the heavier bottom layer has a nite depth h Attention is restricted to waves whose wavelength is large com pared to the depth h of the lower layer and whose amplitude a is small compared to h Moreover the two small quantities h and a h are supposed to be of the same order Let x y z connote a standard Cartesian coordinate system with z the vertical direction and z located at the interface between the two uids in their rest position In this situation the Benjamin Ono equation ut ux uux Huxx was rst proposed by Benjamin as an approximate model equation for waves on the interface whose primary direction of propagation is that of increasing values of x which do not vary signi cantly in the y direction and for which the e ects of surface tension viscosity and molecular di usion may be safely ignored As mentioned x is proportional to distance in the direction of propagation t is proportional to elapsed time and u x t is proportional to the vertical deviation of the interface from its rest position at the point x at time t The operator H is the Hilbert transform applied Mathematics Subject Classi cation S Q K K G M B B