Two binary Darboux transformations for the KdV hierarchy with self-consistent sources

Two binary (integral type) Darboux transformations for the KdV hierarchy wi th self-consistent sources are proposed. In contrast with the Darboux trans formation for the KdV hierarchy, one of the two binary Darboux transformati ons provides non-auto-Backlund transformation between two nth KdV equations with self-consistent sources with different degrees. The formula for the m -times repeated binary Darboux transformations are presented. This enables us to construct the N-soliton solution for the KdV hierarchy with self-cons istent sources. (C) 2001 American Institute of Physics.