Fourth-order nonlinear evolution equation for two Stokes wave trains in deep water

Abstract

Fourth-order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves as first pointed out by Dysthe [Proc. R. Soc. London Ser. A 369, 105 (1979)] and later elaborated by Janssen [J. Fluid Mech. 126, 1 (1983)], are derived for a deep-water surface gravity wave packet in the presence of a second wave packet. Here it is assumed that the space variation of the amplitude takes place only in a direction along which the group velocity projection of the two waves overlap. Stability analysis is made for a uniform Stokes wave train in the presence of a second wave train. Graphs are plotted for maximum growth rate of instability and for wave number at marginal stability against wave steepness. Significant deviations are noticed from the results obtained from the third-order evolution equations which consist of two coupled nonlinear Schrödinger equations.

Akhmediev breathers, Ma solitons, and general breathers from rogue waves: A case study in the Manakov system

“…The RW solution of the CNLS equation matches with the one presented in [21]. We note that the restriction τ 2 = 0 in (11) provides the RW solution of the scalar NLS equation. A typical evolution of the RW is shown in Fig.…”

“…Eq. (1) also appears in multi-component Bose-Einstein condensates [8], bio-physics [9], finance [10] and oceanographic studies [11]. Solitons in coupled NLSEs have been the subject of intense study over the past few years because of their interesting collision properties and their robustness against external perturbations.…”

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

“…These equations terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S033427000000881X [6] Fourth-order nonlinear evolution equations for surface gravity waves 219 (A5) -(A8) determine the quantities A o , B o , C o , £ 0 -Eliminating C o between (A7) and (A8) we get the equation (A9) given in the appendix. Thus we get three sets of equations.…”

“…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.…”

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

“…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).…”