Special solutions to a compact equation for deep-water gravity waves

Fedele, Francesco; Dutykh, Denys

doi:10.1017/jfm.2012.447

Cited by 24 publications

(26 citation statements)

References 43 publications

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“…It might be elastic or nonelastic. In the papers Fedele and Dutykh, 2012) one collision of two breathers was considered. -First breather has the following parameters: Ω 1 = 4.01, V 1 = 1/16.…”

Section: Discussionmentioning

confidence: 99%

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Collisions of two breathers at the surface of deep water

Dyachenko

Kachulin

Захаров

2013

Nat. Hazards Earth Syst. Sci.

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Abstract. We present results of numerical experiments on long-term evolution and collisions of breathers (which correspond to envelope solitons in the NLSE approximation) at the surface of deep ideal fluid. The collisions happen to be nonelastic. In the numerical experiment it can be observed only after many acts of interactions. This supports the hypothesis of "deep water nonintegrability". The experiments were performed in the framework of the new and refined version of the Zakharov equation free of nonessential terms in the quartic Hamiltonian. Simplification is possible due to exact cancellation of nonelastic four-wave interaction.

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Section: Discussionmentioning

confidence: 99%

“…It might be elastic or nonelastic. In the papers Fedele and Dutykh, 2012) there were considered one collision of two breathers. This collision was seamed to be elastic.…”

Section: Compact Equationmentioning

confidence: 99%

Collisions of two breathers at the surface of deep water

Dyachenko

Kachulin

Захаров

2013

Nat. Hazards Earth Syst. Sci.

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“…The compact form follows from a canonical transformation of the Z equation and eliminates trivial resonant quartet interactions (Dyachenko & Zakharov (2011)). As a result, the Z model reduces to a generalized derivative NLS type equation (Fedele & Dutykh (2012)).…”

Section: Introductionmentioning

confidence: 99%

On certain properties of the compact Zakharov equation

Fedele

2014

J. Fluid Mech.

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Long-time evolution of a weakly perturbed wavetrain near the modulational instability threshold is examined within the framework of the compact Zakharov equation for unidirectional deep-water waves (Dyachenko & Zakharov (2011)). Multiple-scale solutions reveal that a perturbation to a slightly unstable uniform wavetrain of steepness µ slowly evolves according to a nonlinear Schrodinger equation. In particular, for small carrier wave steepness µ < µ 1 ≈ 0.27 the perturbation dynamics is of focusing type and the long-time behavior is characterized by the Fermi-Pasta-Ulam recurrence, the signature of breather interactions. However, the amplitude of breathers and their likelihood of occurrence tend to diminish as µ increases while the Benjamin-Feir index decreases and becomes nil at µ 1 . This indicates that homoclinic orbits persist only for small values of wave steepness µ ≪ µ 1 , in agreement with recent experimental and numerical observations of breathers.When the compact Zakharov equation is beyond its nominal range of validity, i.e. for µ > µ 1 , predictions seem to foreshadow a dynamical trend to wave breaking. In particular, the perturbation dynamics becomes of defocusing type, and nonlinearities tend to stabilize a linearly unstable wavetrain as the Fermi-Pasta-Ulam recurrence is suppressed. At µ = µ c ≈ 0.577, subharmonic perturbations restabilize and superharmonic instability appears, possibly indicating that wave dynamical behavior changes at large steepness, in qualitative agreement with the numerical simulations of Longuet-Higgins & Cokelet (1978) for steep waves. Indeed, for µ > µ c a multiple-scale perturbation analysis reveals that a weak narrowband perturbation to a uniform wavetrain evolves in accord with a modified Korteweg-de Vries/Camassa-Holm type equation, again implying a possible mechanism conducive to wave breaking.

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“…The main aim of this work is to verify in the fully nonlinear model the effects of breather interactions predicted recently by Kachulin and Gelash [12] by using a modification of the Zakharov model-the so called super compact Dyachenko-Zakharov equation (see [13][14][15] and also [16][17][18]). The work [12] focuses on how the breather interaction dynamics depends on their relative phase.…”

Section: Introductionmentioning

confidence: 95%

Interactions of Coherent Structures on the Surface of Deep Water

Kachulin

Dyachenko

Gelash

2019

Fluids

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We numerically investigate pairwise collisions of solitary wave structures on the surface of deep water-breathers. These breathers are spatially localised coherent groups of surface gravity waves which propagate so that their envelopes are stable and demonstrate weak oscillations. We perform numerical simulations of breather mutual collisions by using fully nonlinear equations for the potential flow of ideal incompressible fluid with a free surface written in conformal variables. The breather collisions are inelastic. However, the breathers can still propagate as stable localised wave groups after the interaction. To generate initial conditions in the form of separate breathers we use the reduced model-the Zakharov equation. We present an explicit expression for the four-wave interaction coefficient and third order accuracy formulas to recover physical variables in the Zakharov model. The suggested procedure allows the generation of breathers of controlled phase which propagate stably in the fully nonlinear model, demonstrating only minor radiation of incoherent waves. We perform a detailed study of breather collision dynamics depending on their relative phase. In 2018 Kachulin and Gelash predicted new effects of breather interactions using the Dyachenko-Zakharov equation. Here we show that all these effects can be observed in the fully nonlinear model. Namely, we report that the relative phase controls the process of energy exchange between breathers, level of energy loses, and space positions of breathers after the collision.

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Special solutions to a compact equation for deep-water gravity waves

Cited by 24 publications

References 43 publications

Collisions of two breathers at the surface of deep water

Collisions of two breathers at the surface of deep water

On certain properties of the compact Zakharov equation

Interactions of Coherent Structures on the Surface of Deep Water

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