A fourth-order evolution equation for deep water surface gravity waves in the presence of wind blowing over water

Dhar, A. K.; Das, K. P.

doi:10.1063/1.857731

Cited by 27 publications

(16 citation statements)

References 17 publications

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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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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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“…Since the evolution equation from which we have started our analysis is valid for narrow spectral band, stability beyond the critical spectral width does not imply stability for large spectral band widths. For vanishing spectral band widths we recover the deterministic growth rate of instability obtained in my paper (Dhar and Das, 1990). From this growth rate of instability the maximum growth rate of instability has been obtained.…”

Section: The Limit Of Vanishing Bandwidthmentioning

confidence: 93%

“…An expression for the critical value of bandwidth has been obtained from the long wave length approximation of the dispersion relation whose order is found to be of the order of root mean square wave steepness 0 a . For vanishing spectral bandwidth and for perturbation direction 0   , we find the deterministic maximum growth rate of instability which was obtained in Dhar and Das (1990), when the value of the non dimensional surface tension s is zero. ' / , ,…”

Section: Introductionmentioning

confidence: 99%

“…The dominant new effect that comes in the fourth order is the influence of wave induced mean flow and this produces a significant deviation in the stability character. Fourth order nonlinear evolution equation for deep water surface gravity waves including different effects were derived and stability analysis was considered by several authors (Stiassnie, 1984;Hogan, 1985;Dhar and Das, 1990;1991;1994;Janssen, 1983).…”

Section: Introductionmentioning

confidence: 99%

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The Effect Of Randomness On The Stability Of Capillary Gravity Waves In The Presence Of Air Flowing Over Water

Dhar¹

2015

International Journal of Applied Mechanics and Engineering

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A nonlinear spectral transport equation for the narrow band Gaussian random surface wave trains is derived from a fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves. The effect of randomness on the stability of deep water capillary gravity waves in the presence of air flowing over water is investigated. The stability is then considered for an initial homogenous wave spectrum having a simple normal form to small oblique long wave length perturbations for a range of spectral widths. An expression for the growth rate of instability is obtained; in which a higher order contribution comes from the fourth order term in the evolution equation, which is responsible for wave induced mean flow. This higher order contribution produces a decrease in the growth rate. The growth rate of instability is found to decrease with the increase of spectral width and the instability disappears if the spectral width increases beyond a certain critical value, which is not influenced by the fourth order term in the evolution equation.

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“…12) In nonlin-ear optics high speed phenomena (femtosecond regime) also demand the introduction of additional nonlinearity, and (5.1) is a typical example. 13) The bright N-soliton solution (5.1) has been ed earlier the literature by the Hirota bilinear method.…”

Section: Coalescence Of Dark Solitonsmentioning

confidence: 99%

Coalescence of Ripplons, Breathers, Dromions and Dark Solitons

Lai

Chow

2001

J. Phys. Soc. Jpn.

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New solutions of several nonlinear evolution equations (NEEs) are obtained by a special limit corresponding to a coalescence or merger of wavenumbers. This technique will yield the multiple pole solutions of NEEs if ordinary solitons are involved. This limiting process will now be applied through the Hirota bilinear transform to other novel solutions of NEEs. For ripplons (self similar explode-decay solutions) such merger yields interacting self similar solitary waves. For breathers (pulsating waves) this coalescence gives rise to a pair of counterpropagating breathers. For dromions (exponentially decaying solutions in all spatial directions) this merger might generate additional localized structures. For dark solitons such coalescence can lead to a pair of anti-dark (localized elevation solitary waves on a continuous wave background) and dark solitons.KEYWORDS: nonlinear evolution equations, Hirota bilinear method, breathers, dromionsof (1.1) is obtained. This agrees with results obtained by the inverse scattering transform. 2) Computer plot of (1.2) (Fig. 1) shows a pair of counterpropagating soliton, anti-soliton in a frame moving to the right with speed m 2 . There is an important difference between (1.2) and the ordinary multi-soliton of mKdV. Distance of separation of ordinary solitons of mKdV after interaction will be 666 proportional to t (time). However, the distance between the soliton and anti-soliton in (1.2) goes like log t. The hnique of coalescence of wavenumbers will now be applied to ripplons, breathers, dromions and dark solitons in the following sections. tec §1. IntroductionNonlinear evolution equations (NEEs) and localized solitons have been studied intensively recently. Novel and sophisticated solutions for NEEs have been obtained by a variety of ingenious techniques. Examples treated in the present work include: (a) ripplons are self similar explode-decay type solutions, 1) (b) breathers are pulsating waves which can be generated from multi-soliton solutions by employing complex conjugate wavenumbers, 2) (c) dromions are localized solutions of (2 + 1) (2 spatial and 1 temporal) dimensional evolution equations decaying exponentially in all directions, 3) and (d) dark solitons are localized minima in intensity propagating on a continuous wave background. 4) For most soliton equations double or multiple poles solutions arise as special cases of multi-soliton expressions. In the language of the inverse scattering transform this occurs as a result of the merger of simple poles of the reflection coefficient. 2, 5) Alternatively the same result can be obtained by the 'coalescence' of wavenumbers in the Hirota bilinear solutions of the multi-soliton expressions. 6) The main goal of this paper is to apply this technique of 'coalescence' of wavenumbers in the Hirota formulation of ripplons, breathers, dromions, and dark solitons. New solutions of several famous and widely applicable evolution equations, as well as some less well known cases, are generated. The validity of these new solutions is estab...

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A fourth-order evolution equation for deep water surface gravity waves in the presence of wind blowing over water

Cited by 27 publications

References 17 publications

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

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

The Effect Of Randomness On The Stability Of Capillary Gravity Waves In The Presence Of Air Flowing Over Water

Coalescence of Ripplons, Breathers, Dromions and Dark Solitons

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