An Algebraic Method for Solving the KdV Equation (II). Three-parameter and Four-parameter Solution Family

The three-parameter and four-parameter solution family of the KdV equation are given. The comparison of our expressions with the known results which were obtained by means of the inverse scattering transformation (IST) or Backlund transformation is discussed in detail. We find that the known result obtained by the IST approach is the special case of our expressions and under some proper transformations, the expressions given here become the result obtained by Backlund transformation. It is emphasized that the properties of these multiple-parameter solution families should be studied from algebraic-geometry approach.