Negaton and positon solutions of the soliton equation with self-consistent sources

The Korteweg–de Vries (KdV) equation with self-consistent sources (KdVES) is used as a model to illustrate this method. We present a generalized binary Darboux transformation (GBDT) with an arbitrary time-dependent function for the KdVES as well as the formula for N-times repeated GBDT. This GBDT provides non-auto-Backlund transformation between two KdV equations with different degrees of sources and enables us to construct more general solutions with N arbitrary t-dependent functions. By taking the special t-function, we obtain multisoliton, multipositon, multinegaton, multisoliton–positon, multinegaton–positon and multisoliton–negaton solutions of the KdVES.