Numerical Bifurcation for the Capillary Whitham Equation

TL;DR

This paper investigates the effects of capillarity on water wave models by analyzing bifurcation patterns of the capillary Whitham equation, introducing a spectral collocation scheme and revealing complex solution structures.

Contribution

It develops a spectral collocation method for the capillary Whitham equation and uncovers intricate bifurcation structures influenced by capillarity effects.

Findings

Identification of rich bifurcation patterns including subharmonic bifurcations

Discovery of solution branches intersecting at secondary bifurcations

Observation of crossing and self-crossing bifurcation curves

Abstract

The so-called Whitham equation arises in the modeling of free surface water waves, and combines a generic nonlinear quadratic term with the exact linear dispersion relation for gravity waves on the free surface of a fluid with finite depth. In this work, the effect of incorporating capillarity into the Whitham equation is in focus. The capillary Whitham equation is a nonlocal equation similar to the usual Whitham equation, but containing an additional term with a coefficient depending on the Bond number T which measures the relative strength of capillary and gravity effects on the wave motion. A spectral collocation scheme for computing approximations to periodic traveling waves for the capillary Whitham equation is put forward. Numerical approximations of periodic traveling waves are computed using a bifurcation approach, and a number of bifurcation curves are found. Our analysis uncovers a rich structure of bifurcation patterns, including subharmonic bifurcations, as well as connecting and crossing branches. Indeed, for some values of the Bond number T, the bifurcation diagram features distinct branches of solutions which intersect at a secondary bifurcation point. The same branches may also cross without connecting, and some bifurcation curves feature self-crossings without self-connections.