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

Abstract

The issue of a recurrence of the 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 which then disappears 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 occurs from a uniform Stokes wave train, it excites other waves which have different lengths, what 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.

Drifting breathers and Fermi–Pasta–Ulam paradox for water waves

“…It might be an interesting task to simplify the model of the D-cascade using a single drifting ABs rather than a set of fundamental ABs. Future work will be also devoted to investigate the corresponding drift and FPU mechanisms in the fully nonlinear water wave context [43,44].…”

Numerical Simulations of Modulated Waves in a Higher-Order Dysthe Equation

“…The strongly nonlinear stage of the modulational instability is the most interesting as then the wave dynamics may deviate essentially from the weakly nonlinear frameworks. The dynamics of unstable modulations and also breather solutions of the NLS equation have been tested against fully nonlinear simulations and laboratory measurements in plenty of publications [Henderson et al, 1999;Chabchoub et al, 2011Chabchoub et al, , 2012Shemer & Alperovich, 2013;Slunyaev & Shrira, 2013;Slunyaev & Dosaev, 2019).…”