Numerical stability of solitary waves in flows with constant vorticity for the Euler equations

TL;DR

This paper develops a numerical method to compute and analyze the stability of solitary waves in flows with constant vorticity using the Euler equations, revealing instability at high vorticity levels.

Contribution

It introduces a conformal mapping-based numerical approach to compute steady solitary waves and assesses their stability in time-dependent Euler flows with constant vorticity.

Findings

Solitary waves are unstable for vorticities with absolute value much greater than one.

Large waves are unstable even at small vorticity levels.

The numerical method effectively computes solitary waves in flows with constant vorticity.

Abstract

The study of the Euler equations in flows with constant vorticity has piqued the curiosity of a considerable number of researchers over the years. Much research has been conducted on this subject under the assumption of steady flow. In this work, we provide a numerical approach that allows us to compute solitary waves in flows with constant vorticity and analyse their stability. Through a conformal mapping technique, we compute solutions of the steady Euler equations, then feed them as initial data for the time-dependent Euler equations. We focus on analysing to what extent the steady solitary waves are stable within the time-dependent framework. Our numerical simulations indicate that although it is possible to compute solitary waves for the steady Euler equations in flows with large values of vorticity, such waves are not numerically stable for vorticities with absolute value much greater than one. Besides, we notice that large waves are unstable even for small values of vorticity.