Darboux transformation and explicit solutions for two integrable equations

A new N-fold Darboux transformation for two integrable equations is constructed with the help of a gauge transformation for the spectral problem proposed by Qiao [Z.J. Qiao, Phys. Lett. A 192 (1994) 316–322]. By the Darboux transformation, explicit soliton and multi-soliton solutions for the two equations are obtained. In particular, soliton and complexiton solutions are shown through some figures.