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

TL;DR

This paper investigates the recurrence behavior of modulationally unstable deep-water waves using fully nonlinear simulations, revealing that complete recurrence is prevented by interactions with waves of different lengths, leading to a quasi-breather phenomenon.

Contribution

It demonstrates through numerical simulations that rogue waves in deep water do not fully recur due to wave interactions, introducing the concept of a quasi-breather in Euler equations.

Findings

Complete recurrence of rogue waves is prevented by wave interactions.

Wave envelope exhibits quasi-periodic breathing rather than full recurrence.

The breathing period can be thousands of dominant wave periods.

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.