Numerical bifurcation and stability for the capillary-gravity Whitham equation

TL;DR

This paper uses advanced numerical methods to explore the bifurcation and stability of periodic traveling waves in the capillary-gravity Whitham equation, enhancing understanding of wave classification and dynamics.

Contribution

It introduces a robust numerical continuation scheme combined with high-accuracy discretization to analyze bifurcation and stability of capillary-gravity waves.

Findings

Classified various capillary-gravity wave solutions.

Identified conditions for orbital stability and instability.

Provided insights into long-term wave dynamics.

Abstract

We adopt a robust numerical continuation scheme to examine the global bifurcation of periodic traveling waves of the capillary-gravity Whitham equation, which combines the dispersion in the linear theory of capillary-gravity waves and a shallow water nonlinearity. We employ a highly accurate numerical method for space discretization and time stepping, to address orbital stability and instability for a rich variety of the solutions. Our findings can help classify capillary-gravity waves and understand their long-term dynamics.