Madelung fluid description of the generalized derivative nonlinear Schrödinger equation: Special solutions and their stability

A correspondence between the families of generalized nonlinear Schrodinger (NLS) equations and generalized KdV equations was recently found using a Madelung fluid description. We similarly consider a special derivative NLS equation. We find a number of solitary waves and periodic solutions (expressed in terms of elliptic Jacobi functions) for a motion with a stationary profile current velocity. We study the stability of a bright solitary wave (ground state) by conjecturing that the Vakhitov-Kolokolov criterion is applicable.