On the existence and qualitative theory of stratified solitary water waves

Abstract

In this note, we announce new results on the existence of two-dimensional solitary waves moving through a body of density stratified water lying beneath air. The fluid domain is assumed to lie above an impenetrable flat ocean bed, while the interface between the air and water is a free boundary where the pressure is constant. We prove that, for any smooth choice of upstream velocity and density distribution, there exists a continuous curve of such solutions that includes large-amplitude waves that come arbitrarily close to having a (horizontal) stagnation point. Additionally, we provide several results characterizing the qualitative features of solitary stratified waves. In part, these include: estimates on the Froude number, velocity, and pressure, some of which are new, even for the constant density case; a proof of the nonexistence of monotone bores in this physical regime; and a theorem ensuring that all supercritical stratified solitary waves of elevation have an axis of even symmetry.