Note on the modified nonlinear Schrödinger equation for deep water waves

Stiassnie, Michael

doi:10.1016/0165-2125(84)90043-x

Cited by 98 publications

(101 citation statements)

References 3 publications

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“…Hereafter, we will study a special case of (2.6), which is a symplectic version of the temporal Dysthe equation ( [11], see also [31,30,16,33]). Keeping terms to O(ε) in (2.6) yields…”

Section: A Hamiltonian Dysthe Equationmentioning

confidence: 99%

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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“…Hereafter, we will study a special case of (2.6), which is a symplectic version of the temporal Dysthe equation ( [11], see also [31,30,16,33]). Keeping terms to O(ε) in (2.6) yields…”

Section: A Hamiltonian Dysthe Equationmentioning

confidence: 99%

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

Fedele

Dutykh

2012

J. Fluid Mech.

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“…Considering the importance of the fourth order evolution equation, which was first pointed out by Dysthe [8] and later elaborated by Janssen [14] and considered by many authors ([2, [5][6][7]11,13,21]) in studying stability of water waves, two coupled nonlinear evolution equations correct to fourth order in wave steepness are obtained for a surface gravity wave packet in the presence of a thin thermocline. These two coupled equations are reduced to a single equation on the assumption that the space variation of the amplitudes takes place along a line making an arbitrary fixed angle with the direction of propagation of the wave packet.…”

Section: Introductionmentioning

confidence: 99%

Fourth-order nonlinear evolution equations for surface gravity waves in the presence of a thin thermocline

Bhattacharyya

Das

1997

J. Aust. Math. Soc. Series B, Appl. Math

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Two coupled nonlinear evolution equations correct to fourth order in wave steepness are derived for a three-dimensional wave packet in the presence of a thin thermocline. These two coupled equations are reduced to a single equation on the assumption that the space variation of the amplitudes takes place along a line making an arbitrary fixed angle with the direction of propagation of the wave. This single equation is used to study the stability of a uniform wave train. Expressions for maximum growth rate of instability and wave number at marginal stability are obtained. Some of the results are shown graphically. It is found that a thin thermocline has a stabilizing influence and the maximum growth rate of instability decreases with the increase of thermocline depth.

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“…(1) can be performed (see [23] for details) and the following set of equations is obtained in physical (configuration) space:…”

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confidence: 99%

“…Note that the expansion is performed on the nonlinear term, while leaving the linear part of the equation unchanged. As a result, the following particular kernel is used in eq.(1) (see [23] ):1 …”

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Extreme wave events in directional, random oceanic sea states

2002

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We discuss the effects of the directional spreading on the occurrence of extreme wave events. We numerically integrate the envelope equation recently proposed by Trulsen et al., Phys of Fluids 2000, as a weakly nonlinear model for realistic oceanic gravity waves. Initial conditions for numerical simulations are characterized by the spatial JONSWAP power spectrum for several values of the significant wave height, steepness and directional spreading. We show that by increasing the directionality of the initial spectrum the appearance of extreme events is notably reduced.We address the occurrence of extreme wave events in numerical models of random, directional, oceanic sea states. Extreme ocean waves of this type are of unusually large size with respect to the background, surrounding waves. They are often refered to as "freak" or "rogue" waves and have been given a rough definition in terms of an arbitrary threshold, H max > 2.2H s , for H s the significant wave height [1]. Because extreme waves are rare, and hence are not often measured , the physical mechanisms for their occurrence have not been unequivocally established. Possible causes of freak waves in the open sea have been investigated by many researchers [2] and basically three mechanisms have been proposed: the linear interaction of the waves with the currents (geometrical optic theory), see [3][4][5]; the simple linear superposition of Fourier phases (the resulting large waves are also known as transient waves [6]) and the modulation instability [7].The last mechanism and indeed the method addressed herein is based on the nonlinear phenomenon known as the Benjamin-Feir instability [8], which describes how a monochromatic wave train can be unstable to small, side-band perturbations. This physical phenomena has been widely studied in wave tank facilities (see for example [9], [10] and references therein) and from a theoretical and numerical point of view for the 1+1 Nonlinear Schroedinger equation (NLS equation in one space and one time dimensions) [11,12] and for the fully nonlinear Euler equations [13]. A major complication arises from the fact the envelope equations (for example NLS) are derived from the primitive equations of motion under the hypothesis of a narrow-banded spectrum. Higher order equations in the envelope hierarchy have then been proposed [15,16] in order to overcome the limitation of narrow bandwidth. Numerical simulations of these equations in 1+1 dimension [14,17] have shown that the probability of occurrence of freak waves is strictly related to the "enhancement" coefficient γ and the Phillips parameter α of the JONSWAP power spectrum [18]. In [14,17] it was found that increasing the values of γ and α has the effect of increasing the probability of finding freak wave events. In addition to the physical limitations associated with the narrow spectral width assumption, an even more severe limitation of the above results arises because these conclusions have been obtained from essentially one-dimensional numerical simulations.In this co...

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Note on the modified nonlinear Schrödinger equation for deep water waves

Cited by 98 publications

References 3 publications

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

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

Fourth-order nonlinear evolution equations for surface gravity waves in the presence of a thin thermocline

Extreme wave events in directional, random oceanic sea states

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