Time-Reversal of Nonlinear Waves - Applicability and Limitations

Time-reversal (TR) refocusing of waves is one of fundamental principles in wave physics. Using the TR approach, "Time-reversal mirrors" can physically create a time-reversed wave that exactly refocus back, in space and time, to its original source regardless of the complexity of the medium as if time were going backwards. Lately, laboratory experiments proved that this approach can be applied not only in acoustics and electromagnetism but also in the field of linear and nonlinear water waves. Studying the range of validity and limitations of the TR approach may determine and quantify its range of applicability in hydrodynamics. In this context, we report a numerical study of hydrodynamic TR using a uni-directional numerical wave tank, implemented by the nonlinear high-order spectral method, known to accurately model the physical processes at play, beyond physical laboratory restrictions. The applicability of the TR approach is assessed over a variety of hydrodynamic localized and pulsating structures' configurations, pointing out the importance of high-order dispersive and particularly nonlinear effects in the refocusing of hydrodynamic stationary envelope solitons and breathers. We expect that the results may motivate similar experiments in other nonlinear dispersive media and encourage several applications with particular emphasis on the field of ocean engineering.Comment: 14 pages, 17 figures ; accepted for publication in Phys. Rev. Fluid

Experimental study of breathers and rogue waves generated by random waves over non-uniform bathymetry

Experimental results describing random, uni-directional, long crested, water waves over non-uniform bathymetry confirm the formation of stable coherent wave packages traveling with almost uniform group velocity. The waves are generated with JONSWAP spectrum for various steepness, height and constant period. A set of statistical procedures were applied to the experimental data, including the space and time variation of kurtosis, skewness, BFI, Fourier and moving Fourier spectra, and probability distribution of wave heights. Stable wave packages formed out of the random field and traveling over shoals, valleys and slopes were compared with exact solutions of the NLS equation resulting in good matches and demonstrating that these packages are very similar to deep water breathers solutions, surviving over the non-uniform bathymetry. We also present events of formation of rogue waves over those regions where the BFI, kurtosis and skewness coefficients have maximal values.Comment: 41 pages, 21 figure

Nonconservative higher-order hydrodynamic modulation instability

The modulation instability (MI) is a universal mechanism that is responsible for the disintegration of weakly nonlinear narrow-banded wave fields and the emergence of localized extreme events in dispersive media. The instability dynamics is naturally triggered, when unstable energy side-bands located around the main energy peak are excited and then follow an exponential growth law. As a consequence of four wave mixing effect, these primary side-bands generate an infinite number of additional side-bands, forming a triangular side-band cascade. After saturation, it is expected that the system experiences a return to initial conditions followed by a spectral recurrence dynamics. Much complex nonlinear wave field motion is expected, when the secondary or successive side-band pair that are created are also located in the finite instability gain range around the main carrier frequency peak. This latter process is referred to as higher-order MI. We report a numerical and experimental study that confirm observation of higher-order MI dynamics in water waves. Furthermore, we show that the presence of weak dissipation may counter-intuitively enhance wave focusing in the second recurrent cycle of wave amplification. The interdisciplinary weakly nonlinear approach in addressing the evolution of unstable nonlinear waves dynamics may find significant resonance in other nonlinear dispersive media in physics, such as optics, solids, superfluids and plasma

Rogue Waves: From Nonlinear Schrödinger Breather Solutions to Sea-Keeping Test

Under suitable assumptions, the nonlinear dynamics of surface gravity waves can be modeled by the one-dimensional nonlinear Schrödinger equation. Besides traveling wave solutions like solitons, this model admits also breather solutions that are now considered as prototypes of rogue waves in ocean. We propose a novel technique to study the interaction between waves and ships/structures during extreme ocean conditions using such breather solutions. In particular, we discuss a state of the art sea-keeping test in a 90-meter long wave tank by creating a Peregrine breather solution hitting a scaled chemical tanker and we discuss its potential devastating effects on the ship

Super rogue waves in simulations based on weakly nonlinear and fully nonlinear hydrodynamic equations

The rogue wave solutions (rational multi-breathers) of the nonlinear Schrodinger equation (NLS) are tested in numerical simulations of weakly nonlinear and fully nonlinear hydrodynamic equations. Only the lowest order solutions from 1 to 5 are considered. A higher accuracy of wave propagation in space is reached using the modified NLS equation (MNLS) also known as the Dysthe equation. This numerical modelling allowed us to directly compare simulations with recent results of laboratory measurements in \cite{Chabchoub2012c}. In order to achieve even higher physical accuracy, we employed fully nonlinear simulations of potential Euler equations. These simulations provided us with basic characteristics of long time evolution of rational solutions of the NLS equation in the case of near breaking conditions. The analytic NLS solutions are found to describe the actual wave dynamics of steep waves reasonably well.Comment: under revision in Physical Review

Superregular breathers in optics and hydrodynamics: Omnipresent modulation instability beyond simple periodicity

Since the 1960s, the Benjamin-Feir (or modulation) instability (MI) has been considered as the self-modulation of the continuous “envelope waves” with respect to small periodic perturbations that precedes the emergence of highly localized wave structures. Nowadays, the universal nature of MI is established through numerous observations in physics. However, even now, 50 years later, more practical but complex forms of this old physical phenomenon at the frontier of nonlinear wave theory have still not been revealed (i.e., when perturbations beyond simple harmonic are involved). Here, we report the evidence of the broadest class of creation and annihilation dynamics of MI, also called superregular breathers. Observations are done in two different branches of wave physics, namely, in optics and hydrodynamics. Based on the common framework of the nonlinear Schrödinger equation, this multidisciplinary approach proves universality and reversibility of nonlinear wave formations from localized perturbations for drastically different spatial and temporal scales

Stabilization of uni-directional water wave trains over an uneven bottom

We study the evolution of nonlinear surface gravity water wave packets developing from modulational instability over an uneven bottom. A nonlinear Schrödinger equation (NLSE) with coefficients varying in space along propagation is used as a reference model. Based on a low-dimensional approximation obtained by considering only three complex harmonic modes, we discuss how to stabilize a one-dimensional pattern in the form of train of large peaks sitting on a background and propagating over a significant distance. Our approach is based on a gradual depth variation, while its conceptual framework is the theory of autoresonance in nonlinear systems and leads to a quasi-frozen state. Three main stages are identified: amplification from small sideband amplitudes, separatrix crossing and adiabatic conversion to orbits oscillating around an elliptic fixed point. Analytical estimates on the three stages are obtained from the low-dimensional approximation and validated by NLSE simulations. Our result will contribute to understand the dynamical stabilization of nonlinear wave packets and the persistence of large undulatory events in hydrodynamics and other nonlinear dispersive media

Persistence of Peregrine Breather in Random Sea States

Rogue waves are widely recognized as great threats to human oceanic activities. The effectiveness of using the Peregrine breather to model rogue waves has been successfully demonstrated in both numerical and physical wave tank. Additionally, its persistence in random seas has been confirmed in some cases, so that it can also be employed to model rogue waves in random seas. However, based on the results obtained by using the fully nonlinear numerical simulations in this paper, it is reported that the persistence of the Peregrine breather will be affected by the nonlinearities of the random seas, i.e., the spectral bandwidth and the background wave steepness. The investigation on the effects of the two parameters indicates that the Peregrine breather cannot persist thus may not be employed to model rogue waves in random seas in some cases. This paper provides knowledge about how to select background wave parameters, in order to model rogue waves by using the Peregrine breather, which can help saving significant time for designing the experiments