Periodic and solitary wave solutions of generalized nonlinear Schrödinger equation using a Madelung fluid description

The hydrodynamic fluid description, proposed many years ago by E. Madelung (1927) for quantum mechanics, is used to discuss the class of nonlinear Schr̈odinger equations. In the case of stationary profile solutions the equation satisfied by the fluid density ρ = {pipe}Ψ{pipe}2 is integrated and periodic solutions expressed through Jacobi elliptic functions are derived for cubic and cubic + quintic nonlinearities. In the limit case k2 = 1 the solitary wave solution found for the cubic + quintic nonlinearity proves to be much steeper and narrower than the one-soliton solution of the cubic NLS equation.