Global bifurcation of waves with multiple critical layers

TL;DR

This paper develops a global bifurcation theory to construct diverse steady periodic water waves with multiple critical layers and interior stagnation points, using an innovative flattening transform.

Contribution

It introduces a novel application of the naive flattening transform for global bifurcation analysis of water waves with complex vorticity structures.

Findings

Construction of families of steady waves with arbitrary critical layers

Demonstration of waves with multiple interior stagnation points

Extension of local bifurcation methods to global analysis

Abstract

Analytic global bifurcation theory is used to construct a large variety of families of steady periodic two-dimensional gravity water waves with real-analytic vorticity distributions, propagating in an incompressible fluid. The waves that are constructed can possess an arbitrary number of interior stagnation points in the fluid, and corresponding critical layers consisting of closed streamlines. This is made possible by the use of the so-called naive flattening transform, which has previously only been used for local bifurcation.