Periodic traveling interfacial hydroelastic waves with or without mass II: Multiple bifurcations and ripples

TL;DR

This paper extends the analysis of hydroelastic interfacial traveling waves to cases with two-dimensional kernels, demonstrating existence and computing solutions in resonant and non-resonant scenarios.

Contribution

It introduces a method to handle two-dimensional kernels in bifurcation analysis, expanding the understanding of wave solutions beyond the one-dimensional kernel case.

Findings

Existence of traveling waves with two-dimensional kernels proven.

Numerical computation of waves in resonant and non-resonant cases.

Identification of bifurcation structures in complex parameter regimes.

Abstract

In a prior work, the authors proved a global bifurcation theorem for spatially periodic interfacial hydroelastic traveling waves on infinite depth, and computed such traveling waves. The formulation of the traveling wave problem used both analytically and numerically allows for waves with multi-valued height. The global bifurcation theorem required a one-dimensional kernel in the linearization of the relevant mapping, but for some parameter values, the kernel is instead two-dimensional. In the present work, we study these cases with two-dimensional kernels, which occur in resonant and non-resonant variants. We apply an implicit function theorem argument to prove existence of traveling waves in both of these situations. We compute the waves numerically as well, in both the resonant and non-resonant cases.