Solitary waves with the Madelung fluid description: A generalized derivative nonlinear Schrödinger equation

Within the framework of the Madelung fluid description, we will derive bright and dark (including gray- and black-soliton) envelope solutions for a generalized derivative nonlinear Schrödinger model i, by virtue of the corresponding solitary wave solutions for the stationary Gardner equations. Note that we only consider the motion with stationary-profile current velocity case and exclude the motion with constant current velocity case for a ≠ 0; on the other hand, our results are derived under suitable assumptions for the current velocity associated with corresponding boundary conditions of the fluid density, and under corresponding parametric constraints.