Collisions of two breathers at the surface of deep water

Dyachenko, Alexander; Kachulin, Dmitry; Захаров, В. Е.

doi:10.5194/nhess-13-3205-2013

Cited by 12 publications

(3 citation statements)

References 20 publications

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“…Further, the interaction of two traveling waves with a ground state appears also elastic (see Figures 12 -13). This suggests the integrability of the cDZ equation (2.3) in agreement with the recent results of [10]. However, the associated Hamiltonian version of the Dysthe equation (2.7) does not support elastic collisions as shown in Figure 14.…”

Section: Numerical Resultssupporting

confidence: 82%

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

Fedele

Dutykh

2012

J. Fluid Mech.

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Abstract. Recently, Dyachenko & Zakharov (2011) [9] have derived a compact form of the well known Zakharov integro-differential equation for the third order Hamiltonian dynamics of a potential flow of an incompressible, infinitely deep fluid with a free surface. Special traveling wave solutions of this compact equation are numerically constructed using the Petviashvili method. Their stability properties are also investigated. In particular, unstable traveling waves with wedge-type singularities, viz. peakons, are numerically discovered. To gain insights into the properties of these singular solutions, we also consider the academic case of a perturbed version of the compact equation, for which analytical peakons with exponential shape are derived. Finally, by means of an accurate Fouriertype spectral scheme it is found that smooth solitary waves appear to collide elastically, suggesting the integrability of the Zakharov equation.

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Section: Numerical Resultssupporting

confidence: 82%

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

Fedele

Dutykh

2012

J. Fluid Mech.

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“…It can be found numerically only by the Petviashvili method (see details in [13]). It was found in [21] that the SCDZE is not integrable so that the collision of a pair of breathers is not purely elastic and is accompanied by an exchange of energy between breathers and the radiation of incoherent waves (see earlier papers [21,29], and also [20,24]).…”

Section: Bound Structures In the Super Compact Dyachenko-zakharov Equmentioning

confidence: 99%

Bound Coherent Structures Propagating on the Free Surface of Deep Water

Kachulin

Dremov

Dyachenko

2021

Fluids

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This article presents a study of bound periodically oscillating coherent structures arising on the free surface of deep water. Such structures resemble the well known bi-soliton solution of the nonlinear Schrödinger equation. The research was carried out in the super-compact Dyachenko-Zakharov equation model for unidirectional deep water waves and the full system of nonlinear equations for potential flows of an ideal incompressible fluid written in conformal variables. The special numerical algorithm that includes a damping procedure of radiation and velocity adjusting was used for obtaining such bound structures. The results showed that in both nonlinear models for deep water waves after the damping is turned off, a periodically oscillating bound structure remains on the fluid surface and propagates stably over hundreds of thousands of characteristic wave periods without losing energy.

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“…It is currently the simplest form for the 1D deep water waves equation (see [13]) derived from the compact Zakharov equation [14,15]. The breathers were obtained in [16] and the envelope of such a breather can be called the soliton. It is shown in [17] that special solutions of the SCZE called breathers can exchange energy during their pairwise interactions.…”

Section: Introductionmentioning

confidence: 99%

Multiple Soliton Interactions on the Surface of Deep Water

Kachulin

Dyachenko

Dremov

2020

Fluids

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The paper presents the long-time dynamics with multiple collisions of breathers in the super compact Zakharov equation for unidirectional deep water waves. Solutions in the form of breathers were found numerically by the Petviashvili method. In the terms of envelope and the assumption of the narrow spectral width the super compact equation turns into the well known exact integrable model—nonlinear Schrödinger equation, and the breather solution in this case turns into envelope soliton. The results of numerical simulations show that two main scenarios of long-time dynamics occur during numerous collisions of breathers. In the first case, one of the breathers regularly takes a number of particles from the other one at each collision and in the second one a structure resembling the bi-soliton solution of nonlinear Schrödinger equation arises during the collision. Despite these scenarios, it is shown that after numerous collisions the only one breather having initially a larger number of particles remains.

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Collisions of two breathers at the surface of deep water

Cited by 12 publications

References 20 publications

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

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

Bound Coherent Structures Propagating on the Free Surface of Deep Water

Multiple Soliton Interactions on the Surface of Deep Water

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