An algebraic method for solving the KdV equation (III). N-parameter solution family

By direct solving Hirotaʹs τ-equation, we obtain the explicit expressions of N-parameter solution family which include known N-soliton solution. The transformation relations among the N-parameter solution family and the results obtained by using the IST method, Hirota method and Backlund transformation, respectively, are given. Thus the complete pattern of the soliton theory is revealed.