On the incomplete recurrence of modulationally unstable deep-water surface gravity waves

The issue of a recurrence of modulationally unstable water wave trains within the framework of the fully nonlinear potential Euler equations is addressed. It is examined, in particular, if a modulation, which appears from nowhere (i.e., is infinitesimal initially) and generates a rogue wave, can disappear with no trace. If so, this wave solution would be a breather solution of the primitive hydrodynamic equations. It is shown with the help of the fully nonlinear numerical simulation that when a rogue wave emerges from a uniform Stokes wave train, it excites other waves which have different lengths. This process prevents the complete recurrence and, eventually, results in a quasi-periodic breathing of the wave envelope. Meanwhile the discovered effects are rather small in magnitude, and the period of the modulation breathing may be thousands of the dominant wave periods. Thus, the obtained solution may be called a quasi-breather of the Euler equations.