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Along with these solutions, singular solutions also exist. The rational solutions of the KdV equation with decay -2/x2 as 1x1 00 are typical. The bilinear transformation 46 2 The Bilinear Transformation Method method was used to obtain rational solutions of the KdV equation [83, 841. Another class of solutions, similarity-type decay-mode (or ripplon) solutions, also exist in two-dimensional nonlinear systems; the Backlund transformation in the bilinear formalism was applied to obtain ripplon solutions for the two-dimensional KdV equation [85, 861, the two-dimensional nonlinear Schrodinger equation [87], and the two-dimensional Toda equation [88].

10) x,W17 = 1,2,. . 1 1) where x j (j= 1 , 2 , . . , N) are complex functions of time t whose imaginary parts are positive. 1l), we obtain i a f* Hu(x, t ) = - H - In 2 ax f = -i H Z N 2 j=1 -_ = (y--) 1 1 x-x* x-xi 1 i a 2 ax Wf *f1. 16) 2 f X + fxxf') = where an integration constant is assumed to be zero. f - - iD, f * - f = D: f*-f, which is the bilinearized BO equation. 18) which can be verified by direct substitution. 19) where cl, c, c 3 ,and b are unknown constants. 19) may be arbitrarily chosen.

The two-dimensional nonlinear evolution equations can be treated similarly. 266) with flyp', and 6 being real constants, are typical examples. Other important classes of nonlinear evolution equations to which the bilinear transformation method has been successfully applied are nonlinear differential-difference and nonlinear partial difference equations. 267) which was discussed in detail in Ref. [70]. 269) also belongs to a class of nonlinear differential-difference equations which can be bilinearized through the dependent variable transformation.

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