Stability of solitary waves on deep water with constant vorticity

Waves on deep water with constant vorticity propagating in the direction of the shear are known to be weakly dispersive in the long wave limit. Weakly-nonlinear evolution of such waves can be described by the Benjamin-Ono equation, which is integrable and has stable soliton solutions. In the present study we investigate behaviour of finite-amplitude counterparts of Benjamin-Ono solitons by modelling their dynamics within exact equations of motion (Euler equations). Due to the solitons having a near-Lorentzian shape with slowly decaying tails, we need to approach them by examining periodic waves, whose crests, indeed, become more and more localised as the period increases. We perform a parameter space study and analyse how stability of very long waves depends on their amplitude and period. We show that large-amplitude solitary waves are unstable.
This research was supported by RFBR (grant No. 16-05-00839) and by the President of Russian Federation (grant No. MK-2041.2017.5). Numerical experiments were supported by RSF grant No. 14-17-00667, data processing was supported by RSF grant No. 15-17-20009.