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

•Envelope soliton-like solutions are derived for a generalized derivative nonlinear Schrödinger equation within the framework of the Madelung fluid description.•Bright- and dark-type (including gray and black) solitary waves exist due to associated parametric constraints.

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∂Ψ∂t=∂2Ψ∂x2+ia∂∂x(|Ψ|2Ψ)+b|Ψ|2Ψ, 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.