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Fz + (3/C)f0(Dxf0 . ~ I z ) , + (3/2c)fO(fo J; dXx + 3P: + P W X - +D;31fl * f Z - ( 6 / ~ ) f o . Tlz}. 224) =O, since fo fi # 0. 211) has been verified. 212, and therefore we have completed the proof of the commutability relation for the time part of the Backlund transformation. Finally, we shall consider an important structure of the Backlund transformation. We have seen that the Backlund transformation enables us to obtain new solutions starting with a known solution, such as a vacuum solution.
262) reduces to the BO equation in the deep-water limit d + 00 and to the KdV equation 44 2 The Bilinear Transformation Method in the shallow-water limit d -+ 0. 261) and the N-soliton solution were given in Refs. [27,65]. For the development of bilinearization of soliton equations the reader is referred to Ref. . These equations are concerned with the one-dimensional system. The two-dimensional nonlinear evolution equations can be treated similarly. 266) with flyp', and 6 being real constants, are typical examples.
6 Bibliography The bilinear transformation method has been illustrated by applying it to the KdV equation, a typical nonlinear evolution equation. From the bilinearized KdV equation various exact solutions, the Backlund transformations, an infinite number of conservation laws, and the inverse scattering transform have been derived in a systematic way; their interrelation is shown in Fig. 1. We believe that the essential part of the bilinear transformation method has been fully discussed. However, other references describe the applications of this method to other nonlinear evolution equations.
Bilinear Transformation Method by Matsuno