Darboux transformation of the Heisenberg ferromagnet equation is constructed by the Darboux matrix method. In application, the rogue wave solutions of the Heisenberg ferromagnet equation are obtained. In particular, rogue waves are discussed and illustrated.

An explicit N-fold Darboux transformation with multi-parameters for coupled mKdV equation is constructed with the help of a gauge transformation of the Ablowitz-Kaup-Newell-Segur (AKNS) system spectral problem. By using the Darboux transformation and the reduction technique, some multi-soliton solutions for the complex mKdV equation are obtained.

We study the multi—peakon solutions for two new coupled Camassa—Holm equations, which include two-component and three-component Camassa—Holm equations. These multi—peakon solutions are shown in weak sense. In particular, the double peakon solutions of both equations are investigated in detail. At the same time, the dynamic behaviors of three types double peakon solutions are analyzed by some figures.

A Darboux transformation of the generalized derivative nonlinear Schrödinger equation is derived. As an application, some new periodic wave solutions of the generalized derivative nonlinear Schrödinger equation are explicitly given.