Indeterminate equation and the soliton solution of KdV equation

Under the Cole-Hopf transformation, the KdV equation becomes the tau-equation. The tau-equation is solved using an algebraic method. The expressions of the two-parameter and three-parameter solution family for the tau-equation are given. Some singular solutions for the KdV equation are also presented. The comparison with the known results is discussed. The two-parameter and three-parameter solution family become the well-known two-soliton and three-soliton solutions of the KdV equation if the parameters k(i) (i=1,2; or i=1,2,3) are taken as some special values. Therefore, the results given here are more general than those obtained by means of the inverse scattering transformation (IST) approach. The key step of method presented here is to solve a system of indeterminate equations. So, the authors assert that the properties of two-parameter and three-parameter solution family of the KdV equation must be considered from the algebraic-geometry approach.