Periodic gravity-capillary roll wave solutions to the inclined viscous shallow water equations in two dimensions

TL;DR

This paper constructs and analyzes two-dimensional viscous roll wave solutions in shallow water equations with gravity and surface tension, identifying parameter regimes for their existence and employing bifurcation techniques.

Contribution

First mathematical construction and study of properly two-dimensional viscous roll waves in shallow water equations.

Findings

Identified parameter regimes for nontrivial solutions

Established existence of small amplitude wave solutions

Applied bifurcation theory to classify solution curves

Abstract

We study periodic, two-dimensional, gravity-capillary traveling wave solutions to a viscous shallow water system posed on an inclined plane. While thinking of the Reynolds and Bond numbers as fixed and finite, we vary the speed of the traveling frame and the degree of the incline and identify a set of the latter two parameters that classifies from which combinations nontrivial and small amplitude solution curves originate. Our principal technical tools are a combination of the implicit function theorem and a local multiparameter bifurcation theorem. To the best of the author's knowledge, this paper constitutes the first construction and mathematical study of properly two dimensional examples of viscous roll waves.