Traveling water waves -- the ebb and flow of two centuries

TL;DR

This survey reviews the recent mathematical advances in the theory of steady water waves, covering variational methods, wave types, and complex phenomena like overhanging waves and stratification.

Contribution

It provides a comprehensive overview of current research topics in steady water wave theory, highlighting recent analytical and numerical developments.

Findings

Advances in variational characterizations of water waves

Existence and nonexistence results for solitary waves

Numerical studies of overhanging and stratified waves

Abstract

This survey covers the mathematical theory of steady water waves with an emphasis on topics that are at the forefront of current research. These areas include: variational characterizations of traveling water waves; analytical and numerical studies of periodic waves with critical layers that may overhang; existence, nonexistence, and qualitative theory of solitary waves and fronts; traveling waves with localized vorticity or density stratification; and waves in three dimensions.