Capillary Rogue Waves

Abstract

We report the first observation of extreme wave events (rogue waves) in parametrically driven capillary waves. Rogue waves are observed above a certain threshold in forcing. Above this threshold, frequency spectra broaden and develop exponential tails. For the first time we present evidence of strong four-wave coupling in nonlinear waves (high tricoherence), which points to modulation instability as the main mechanism in rogue waves. The generation of rogue waves is identified as the onset of a distinct tail in the probability density function of the wave heights. Their probability is higher than expected from the measured wave background.

Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents

“…In two-dimensional water basin experiments as well as in the related simulations the amplitude and the modular inclination of the RWs were found to be higher [10,13,17,18] than those predicted and observed in one dimension [14,17].…”

“…We hope that this breakthrough in describing large ocean RWs by an analytic dynamical lump soliton with adjustable height, inclination and duration would also be valuable for experimental findings of two-dimensional RWs in other systems, such as capillary fluid waves [13] optical cavity waves [11] and basin water waves [10]. Derivation of our exact lump soliton from the breather solution of the integrable two-dimensional NLS equation presented here in a systematic way as well as to find higher order rational lump solitons would be challenging theoretical problems.…”

“…This elusive freak wave caught the imagination of the broad scientific community quite recently [8][9][10][11][12][13][14][15], triggering off an upsurge in theoretical [7,[16][17][18] and experimental [5,[8][9][10][11][12][13][14][15] studies of this unique phenomenon. For identifying such extreme waves, the suggested signature of these rare events is a deviation of the probability distribution function (PDF) of the wave amplitude from its usual random Gaussian distribution (GD), by having a long-tail, indicating that the appearance of high-intensity pulses more often has much higher probability than that predicted by the GD [19].…”

“…Among various possible factors contributing to the creation of the RW, the modulation instability (MI) supported by the nonlinear effect is believed to play a crucial role, by inducing preliminary amplification of water wave height, which may trigger self-attractive nonlinear interaction, initiating the RW formation [20]. The MI can also cause wave-wave interaction leading to the four-wave mixing at matching frequencies and wave numbers, inducing resonance effect which might also develop into an RW [7,13,21]. Like the four-wave nonlinear interaction, a leading order nonlinear effect in deep-sea waves is found also to be a dominant interaction in the nonlinear Schrödinger (NLS) equation…”

The hierarchy of higher order solutions of the derivative nonlinear Schrödinger equation

“…The concept of rogue wave was first proposed in the studies of deep ocean waves [16,17], and gradually extended to other fields such as optics fibre [18][19][20], Bose-Einstein condensates [21], capillary phenomena [22], and so on. Rogue wave, "which appears from nowhere and disappears without a trace (WANDT)" [23], possesses the following two remarkable characteristics: i) locates in both space and time, ii) exhibits a dominant peak.…”