By Philip L. F. Liu

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**Sample text**

17)) is present: the third term between square brackets in the expression for v\ should have the numerical factor one and not four, see also Eqs. (48). Nonlinear Modulation of Water Waves 21 The coefficient v\ of the term \B\2B is singular when cg = y/gh and when 7 = cr 2 /(3 — <72). For cg = y/gh, one has long-wave/short-wave resonance in which the group velocity of the short (capillary) waves equals the phase velocity of the long (gravity) waves. Near cg2 = gh, another scaling of the independent variables has to be used and the asymptotic expansion of $ and ( is also different (Djordjevic and Redekopp, 1977), $ = 2 3 £ /

40) and (41), 2ik8B_{khfd^+8^ JghdT v ' d£2 drj2 = ^ ^B3J^^ 2gh3* ' yfgh (62a) dt, The validity of these equations has been checked by Freeman and Davey (1975) who started from the original equations of motion and introduced the appropriate long-wave scaling. They showed that the double limit e —>• 0, kh —> 0 in the asymptotic expansions for $ and £ was uniform when the condition e/{kh)2 <§; 1 was fulfilled. Because e = ka, this condition can be written as (a/h)/(kh). This condition does not necessarily imply that the Stokes parameter which is given by (a/h)/{kh)2 is a small quantity.

Dingemans & A. K. Otta Some examples of these solutions have been considered in Dingemans (1997, p. 925). We replot solutions for the dn, en and sn-solutions given there. In two dimensions, let us consider the deep-water case Eq. (61) because of its simplicity. dA 1 (d2A d2A\ . AI2A ,,„, We note that for NLS-type of equations for water waves, the coordinate along the propagation direction £ and the lateral one r/ are not interchangeable. Fig. 2. A dn-solution. Fig. 3. A cn-solution. Fig. 4. A sn-solution.