Periodic and stationary wave solutions of coupled NLS equations

A system of coupled NLS equations (integrable and non-integrable) is discussed using a Madelung fluid description. The problem is equivalent with a two component fluid of densities ρ1 and ρ2 and velocities υ1 and υ2, which satisfy equations of continuity and equations of motion. Provided that the nonlinear coupling coefficients are identical, several periodic solutions, expressed through Jacobi elliptic functions, and localized solutions in the form of bright, dark and grey solitons were obtained in different simplifying conditions (motion with constant but equal velocities, i.e. υ1 = υ2 = υ, and equal "energies", i.e. E1 = E2 = E; motion with stationary profile of the current velocity). For different"energies" (E1 ≠ E2) a direct method is used, which can be easily extended to more complex situations (different nonlinear coupling coefficients, i.e. β and γ).