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

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 in this subject under the assumption of steady flow. In this work, we provide a numerical approach that allows 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.

Large-Amplitude Steady Solitary Water Waves with Constant Vorticity

“…However, there is as of yet no proof that any of the solutions we construct are truly overturning. The pressure beneath solitary waves with constant vorticity has recently been studied [28], as well as their nonlinear stability [11].…”