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By Philip L. F. Liu

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2 at 4-7T J_00 J_00 (166c) By expanding Eq. (166b) in the derivatives, linear evolution equations to all orders can be obtained. Expansion up to fifth-order derivatives yields the linear part of Eq. (159). Trulsen et al. (2000) use the exact linear dispersive equation (166b) for the linear part of the higher-order NLS-type equation instead of some power-series approximation such as Eq. (159). Keeping the usual nonlinear term, the equation considered then is: 3A — + L(dx,dy)A 1 + -iuj0k20\A\2A = 0 .

100) 32 M. W. Dingemans & A. K. Otta Since we look for stationary solutions, we put, A($, T) = &(X)e i ^- S T ) where X (101) is a moving frame with respect to the frame (£, r) and v and £ are constants. Substituting Eq. (101) in the NLS equation (97) leads to an ordinarily differential equation in b(X), Ai d2b + ab- i/ib3 = 0 dX2 with (102) 4AT' and where has been substituted £ = v/{2\{) in order to obtain real solutions for the amplitude b. Imposing the condition that b(X) and db(X)/dx —> 0 for X —»• ^oo, the following solution is obtained (for details, see Dingemans, 1997, p.

40), now the following equation is obtained for the leadingorder mean flow (1,0), 2^(1,0) t3V (1,0) , 2 f ,, 2^ 2 ,. „ ,, 1 d\Bf (65) Introducing a quantity Q by b : 8*(1'°> , «. f„. 7 the set of Eqs. (48) is recovered now with the coefficients0 modified by the surface tension effect, see Djordjevic and Redekopp (1977), Ablowitz and Segur (1979, 1981, p. 320) or Dingemans (1997, p. 903). A few notable differences between the situation with and without surface tension are listed below, • With surface tension included, it becomes possible that cg > gh.

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