Traveling gravity water waves with critical layers

TL;DR

This paper proves the existence of small-amplitude steady gravity water waves with critical layers and multiple crests, using bifurcation theory and analyzing the local geometry of the kernel equation.

Contribution

It introduces a new bifurcation approach to construct steady gravity waves with critical layers and arbitrary crest numbers, expanding previous theoretical frameworks.

Findings

Existence of small-amplitude uni- and bimodal steady waves with critical layers.

Detailed analysis of the kernel equation's local geometry.

Asymptotic behavior of bifurcating solutions studied.

Abstract

We establish the existence of small-amplitude uni- and bimodal steady periodic gravity waves with an affine vorticity distribution, using a bifurcation argument that differs slightly from earlier theory. The solutions describe waves with critical layers and an arbitrary number of crests and troughs in each minimal period. An important part of the analysis is a fairly complete description of the local geometry of the so-called kernel equation, and of the small-amplitude solutions. Finally, we investigate the asymptotic behavior of the bifurcating solutions.