Modulations of viscous fluid conduit periodic waves

TL;DR

This paper investigates the modulation of periodic interfacial waves in viscous fluid conduits using Whitham and NLS theories, revealing stability properties, soliton persistence, and modulational instability in nonlocal, non-integrable systems.

Contribution

It develops a modulation theory for viscous conduit waves that does not require integrability, including large amplitude regimes and stability analysis, expanding understanding of dispersive wave phenomena.

Findings

Dark and bright envelope solitons persist in the conduit equation.

Modulational instability occurs above a critical wavenumber due to non-convex dispersion.

Structural properties of Whitham equations are characterized, including hyperbolicity and nonlinearity.

Abstract

In this work, modulation of periodic interfacial waves on a conduit of viscous liquid is explored utilizing Whitham theory and Nonlinear Schr\"odinger (NLS) theory. Large amplitude periodic wave modulation theory does not require integrability of the underlying model equation, yet in practice, either integrable equations are studied or the full extent of Whitham (wave-averaging) theory is not developed. The governing conduit equation is nonlocal with nonlinear dispersion and is not integrable. Via a scaling symmetry, periodic waves can be characterized by their wavenumber and amplitude. In the weakly nonlinear regime, both the defocusing and focusing variants of the NLS equation are derived, depending on the wavenumber. Dark and bright envelope solitons are found to persist in the conduit equation. Due to non-convex dispersion, modulational instability for periodic waves above a critical wavenumber is predicted. In the large amplitude regime, structural properties of the Whitham modulation equations are computed, including strict hyperbolicity, genuine nonlinearity, and linear degeneracy. Bifurcating from the NLS critical wavenumber at zero amplitude is an amplitude-dependent elliptic region for the Whitham equations within which a maximally unstable periodic wave is identified. These results have implications for dispersive shock waves, recently observed experimentally.