The N-soliton solutions of some soliton equations with self-consistent sources

The hierarchy of the mKdV–sine-Gordon equation with self-consistent sources is derived. The N-soliton solutions of the mKdV–sine-Gordon equation with N self-consistent sources are obtained through Hirota method and Wronskian technique, respectively, from which we also reduce solutions for some soliton equations with self-consistent sources, such as one-dimensional atomic grid equation with self-consistent sources, the sine-Gordon equation with self-consistent sources, the mKdV equation with self-consistent sources and the KdV equation with self-consistent sources (KdVESCS). Finally, the mixed rational-soliton solutions in Wronskian form for the KdVESCS are discussed.