By Philip L. F. Liu

ISBN-10: 981024620X

ISBN-13: 9789810246204

ISBN-10: 9812794573

ISBN-13: 9789812794574

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**Sample text**

The coefficient YQ^2 3 n a s been given in Rasmussen (1998, Eq. 18)). , 1995): T< 2 )(fc,k u k 2 ,k 3 ) = T^(kuk,k2,fc3) = T^(kuk, fc3, k2) = T(-2\k2,k3,k,k1). 4. 1. Narrow-band approximation in both dispersion and nonlinearity Stiassnie (1984) showed that the NLS equation can be derived from the Zakharov equation by restricting the waves to have narrow spectra only. To this end, it is supposed that the energy is concentrated around the wave number k = ko = (fco,0)T which is in accordance with the usual assumption for NLS equations that the waves have one predominant direction, taken here as the x = X\ direction.

Nonlinear Modulation of Water Waves 39 We first sketch the steps along which the Zakharov equation can be obtained. Here, we follow Stiassnie and Shemer (1984), see also Shemer and Stiassnie (1991), where the bottom has been assumed to be horizontal. A detailed account can also be found in Rasmussen (1998). The kinematic and dynamic free-surface conditions are written in terms of the free-surface potential ip(x,t) = $>{x,z = £(x, £),£}, the vertical velocity ivs = (d$>/dz)\z=£ and have been given in Eqs.

68c) -«i^(l^l2). (69) In this case, the ID version becomes, dB dr , d2B 2 di 9F FB, where i,o) nt Si d£ and Si l + g(l-a2)(l+7) and Qi = -5ik2{l-a2) > 0. A major aid in discussing the general behaviour of the equation system including surface tension is the figure for the parameter space of the coefficients with respect to the surface tension parameter 7 = "~fk2/(pg) and kh (Fig. 1, taken from Ablowitz and Segur, 1981). A similar picture is also given by Djordjevic and Redekopp (1977). The corresponding set of equations is Eq.

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