The general mixed nonlinear Schrödinger equation: Darboux transformation, rogue wave solutions, and modulation instability

In this paper, the Darboux transformation method has been successfully applied to a general mixed nonlinear Schrödinger equation and some rogue wave solutions are proposed. First of all, the determinant representation of an n-fold DT is given explicitly. Then starting with a periodic seed solution, we obtain some rogue wave solutions of the general mixed nonlinear Schrödinger equation through iteration of a generalized DT. Second, the three-dimensional images and density profiles of the rogue waves are plotted to show the structures of these rogue wave solutions. Finally, we give evidence for the connection between the occurrence of rogue wave solutions and the modulation instability.