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The Theory of General Relativity is currently the most accepted model to describe gravity, and although many experiments and observations continue to validate it, recent astrophysical and cosmological observations require to include new forms of matter and energy (dark matter and dark energy), to be consistent. Modified theories of gravity with non-minimal coupling between curvature and matter are extensions of the Theory of General Relativity and have been proposed to address these shortcomings. Interestingly, matter at large scales behaves as a fluid and under certain approximations, the field equations can be approximated to a generalized Schr¨odingerNewton system of equations. This model is largely found in the nonlinear optical systems, in particular to describe light propagating in nonlinear and nonlocal optical materials and also as a base model for the development of many optical analogues. Due to this, there are a wide variety of numerical methods developed to tackle this type of mathematical models, and that can be used to study these alternative gravity models. In this work, we explore the application of these numerical techniques based on GPGPU supercomputing, initially developed to study light propagating in nonlinear optical systems, to explore a particular non-minimal coupled gravity model. This model, in the nonrelativistic limit, modifies the hydrodynamic equations with the introduction of an attractive Yukawa potential and a repulsive one proportional to the matter density. We used the Schr¨odinger-Newton formalism to numerically study this model and, through the imaginary-time propagation method, we found stationary solutions that were sustained by the repulsive potential introduced by the non-minimal coupled model, even in the absence of a pressure term. We developed an analytical study in the Thomas-Fermi approximation and compared the predictions with numerical solutions. Finally, we explored how this gravity model may be emulated in the laboratory as an optical analogue.
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