Existence and qualitative theory for stratified solitary water waves

TL;DR

This paper proves the existence of large-amplitude stratified solitary water waves with detailed qualitative properties, including bounds on wave speed, symmetry, and nonexistence of monotone bores, advancing understanding of stratified fluid dynamics.

Contribution

It establishes the existence of a continuous family of stratified solitary waves and characterizes their qualitative features, including bounds, symmetry, and nonexistence of certain wave types.

Findings

Existence of a continuous curve of stratified solitary waves including large amplitudes.

Bounds on wave speed and velocity field, some new even for constant density.

Nonexistence of monotone bores in the stratified water regime.

Abstract

This paper considers two-dimensional gravity solitary waves moving through a body of density stratified water lying below vacuum. The fluid domain is assumed to lie above an impenetrable flat ocean bed, while the interface between the water and vacuum is a free boundary where the pressure is constant. We prove that, for any smooth choice of upstream velocity field and density function, there exists a continuous curve of such solutions that includes large-amplitude surface waves. Furthermore, following this solution curve, one encounters waves that come arbitrarily close to possessing points of horizontal stagnation. We also provide a number of results characterizing the qualitative features of solitary stratified waves. In part, these include bounds on the wave speed from above and below, some of which are new even for constant density flow; an a priori bound on the velocity field and lower bound on the pressure; a proof of the nonexistence of monotone bores in this physical regime; and a theorem ensuring that all supercritical solitary waves of elevation have an axis of even symmetry.