Nonconservative higher-order hydrodynamic modulation instability

Abstract

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 sideband 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.One possible explanation for the formation of extreme wave events for instance in the ocean and nonlinear optical media is the modulation instability (MI) [1][2][3]. Understanding the wave dynamics of modulationally unstable waves is of major significance for the sake of accurate modeling and prediction of localized structures as well as of rogue waves in particular [4,5]. The MI describes the disintegration of uni-directional and narrow-banded wave fields. Physically, the instability is driven, when sidebands that are located around the main carrier energy peak in a specific instability range are excited. The progressive focusing of the wave field is translated in spectral domain with an advancing formation of an infinite number of side-bands in form of a triangular cascade [6,7]. One deterministic way to study the MI is by use of the nonlinear Schrödinger equation (NLSE) [8,9]. The latter weakly nonlinear evolution equation is indeed very useful in the study of the problem, in view of its integrability [10]. In fact, the NLSE admits a family of exact solutions that model stationary, pulsating and modulationally unstable wave fields [11]. The standard model that describes the MI process are the family of Akhmediev breathers (ABs) [12]. Indeed, for each unstable modulation frequency, or unstable side-band, one can assign an exact analytical AB expression to study the spatiotemporal evolution of the wave field. From an experimental perspective these universal type of solutions are very valuable, since the complex nonlinear physical processes can be controlled in time and space a...

Nonlinear spectral analysis of Peregrine solitons observed in optics and in hydrodynamic experiments

“…[80] and [83]. Such observations are also known to be related to the phenomenon of higher modulation instability recently observed in optics and in hydrodynamics [82,[84][85][86], wherein non-ideal excitation of first-order breathers (ABs, PS...) or propagation losses were identified as perturbations that introduce deviations from the expected nonlinear dynamics on longer propagation distances (simply due the unstable nature of NLS breathers). In particular, it was shown that higher order modulation instability arises from perturbations on breathers close to the PS limit, thus resulting in a nonlinear superposition (i.e., complex arrangement) of several elementary breathers.…”

A comparison of frequency downshift models of wave trains on deep water

“…Carter & Govan [4] showed that the vDysthe equation accurately models data from experiments in which frequency downshift was observed and experiments in which frequency downshift was not observed. Kimmoun et al [14] showed that the vDysthe equation accurately models data from experiments conducted in a much larger experimental tank. When δ = ǫ = 0, the viscous Dysthe equation reduces to the NLS equation,…”

“…However, Segur et al [22] noted that in their experiments, the FD in the spectral peak sense was always accompanied by a decrease in P. Thus, our conclusions for FD in the spectral peak sense are based on the evolution of P. These conclusions are consistent with our numerical results that are discussed in Section 5. We also note that in experiments, there is damping, which causes a nearly exponential decay of both M and P. Thus, we seek a model that predicts both this decay as well as the FD exhibited by an evolving P. The vDysthe equation, (8), the IS equation, (14), and the dGT equation, (15), have all of these ingredients. The vDysthe equation does not preserve M, P, or ω m in χ (dimensionless distance down the tank).…”

Experiments on higher-order and degenerate Akhmediev breather-type rogue water waves

“…Furthermore, these results can be considered to be analogous to the observations of MI and the Fermi-Pasta-Ulam recurrence, starting from three wave systems, as reported in Tulin and Waseda (1999). More details can be found in Erkintalo et al (2011, Chabchoub and Grimshaw (2016), Kimmoun et al (2016Kimmoun et al ( , 2017.…”