Nonlinear stage of Benjamin-Feir instability in forced/damped deep-water waves

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 approxi

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“…We follow the threewave truncation proposed in Ref. [18], that was proven effective also in higher-order generalizations of the NLSE [19,33].…”

Section: Nonlinear Regimementioning

confidence: 99%

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

et al. 2020

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“…The three-wave truncation allows the system to be described more thoroughly as it enables to study its dynamics explicitly. Below we briefly recall the relevant notation and conclusions from an earlier work [ 17 ].…”

Section: Theorymentioning

confidence: 99%

“…2 gives a system of three ordinary differential equations for (Eq. (5) of [ 17 ]). Writing allows to construct a closed system of equations, consisting of an evolution equation in for each of the following quantities (Eq.…”

Section: Theorymentioning

confidence: 99%

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Separatrix crossing and symmetry breaking in NLSE-like systems due to forcing and damping

et al. 2020

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We theoretically and experimentally examine the effect of forcing and damping on systems that can be described by the nonlinear Schrödinger equation (NLSE), by making use of the phase-space predictions of the three-wave truncation. In the latter, the spectrum is truncated to only the fundamental frequency and the upper and lower sidebands. Our experiments are performed on deep water waves, which are better described by the higher-order NLSE, the Dysthe equation. We therefore extend our analysis to this system. However, our conclusions are general for NLSE systems. By means of experimentally obtained phase-space trajectories, we demonstrate that forcing and damping cause a separatrix crossing during the evolution. When the system is damped, it is pulled outside the separatrix, which in the real space corresponds to a phase-shift of the envelope and therefore doubles the period of the Fermi–Pasta–Ulam–Tsingou recurrence cycle. When the system is forced by the wind, it is pulled inside the separatrix, lifting the phase-shift. Furthermore, we observe a growth and decay cycle for modulated plane waves that are conventionally considered stable. Finally, we give a theoretical demonstration that forcing the NLSE system can induce symmetry breaking during the evolution.

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“…This behavior does not qualitatively change if we include Dysthe nonlinear terms in Eq. (17), as in, e.g., [4]: the downshift is associated to a transient upshift at each recurrence cycle [40].…”

Section: A Space-like Propagation Equationmentioning

confidence: 99%

Viscous damping of gravity-capillary waves: Dispersion relations and nonlinear corrections

et al. 2018

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We discuss the impact of viscosity on nonlinear propagation of surface waves at the interface of air and a fluid of large depth. After a survey of the available approximations of the dispersion relation, we propose to modify the hydrodynamic boundary conditions to model both short and long waves. From them we derive a nonlinear Schrödinger equation where both linear and nonlinear parts are modified by dissipation and show that the former plays the main role in both gravity and capillarygravity waves while, in most situations, the latter represents only small corrections. This provides a justification of the conventional approaches to damped propagation found in the literature.

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Nonlinear stage of Benjamin-Feir instability in forced/damped deep-water waves

Cited by 21 publications

References 43 publications

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

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

Separatrix crossing and symmetry breaking in NLSE-like systems due to forcing and damping

Viscous damping of gravity-capillary waves: Dispersion relations and nonlinear corrections

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