The recent ability of plasmonic nanostructures to probe subnanometer and even atomic scales demands theories that can account for the nonlocal dynamics of the electron gas. The hydrodynamic Drude model (HDM) captures much of the microscopic dynamics of the quantum mechanical effects when additional boundary conditions are considered. Here, we revisit the HDM under the Madelung formalism to reexpress its coupled system of equations as a single nonlinear Schrödinger equation in order to have a natural quantum mechanical description of plasmonics. Specifically, we study the response of two overlapping nanowires with this formalism. We ensure that an proposed frame concurs with classical electrodynamics when the local response approximation holds in the plasmonic system by finding the correction needed.