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We present the construction of ground state equilibrium configurations of the Schrödinger–Poisson (SP) system in the Madelung frame and evolve such configuration using finite volume methods. We compare the behavior of these configurations when evolved within the SP and Madelung frames, in terms of conservation of mass and energy. We also discuss the issues of the equations in the Madelung frame and others inherent to the numerical methods used to solve them.

We introduce a tool that solves the Schrödinger-Euler-Poisson system of equations and allows the study of the interaction between ultralight bosonic dark matter, whose dynamics is described with the Schrödinger-Poisson system and luminous matter which, as a first approximation, is modeled with a single component compressible ideal fluid. The two matter fields are coupled through the Poisson equation, whose source is the addition of both, dark matter and gas densities. We describe the numerical methods used to solve the system of equations and present tests for each of the two components, that show the accuracy and convergence properties of the code. As simple possible applications we present some toy scenarios: i) the merger between a core of dark matter with a cloud of gas that could be the process of galaxy formation, ii) the merger of bosonic dark matter plus gas configurations emulating basic models of galaxy mergers, and iii) the post merger properties, including the dark-matter offset from gas and the correlation between oscillations of the bosonic core and those of the gas.

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