Abstract : Equations of the Schrödinger-Newton type appear naturally in different domains of research. In cosmology, they are used to model the formation of large scale structures in the universe, while, in astrophysics, they are employed as a semi-classical description for the evolution of elementary particles of dark matter. The Schrödinger-Newton equation is also used in optics, to describe the propagation of a beam of light through a medium with a thermo-optical nonlinearity. All of these systems, despite being very different from each other, are characterized by long range interactions. Long range interacting systems, relax generically to out-of-equilibrium quasi-stationary states, with significant astrophysical examples such as galaxies and globular clusters. These states cannot be described by standard statistical mechanics, but are formed through a process of very different nature, called violent relaxation. However, astrophysical time-scales are so large that it is not possible to directly observe the relaxation dynamics. In this thesis, we develop a table-top experimental model that captures the dynamics of long range interacting systems and allows to directly observe violent relaxation mechanism, leading to the formation of a table-top optical analogue of a galaxy. The experiment allows to control a range of parameters, including the nonlocal (gravitational) interaction strength and the strength of quantum gravity coupling ħ/m, thus providing an effective test-bed for gravitational models that cannot otherwise be directly studied in experimental settings. In addition, a central part of the thesis is devoted to the numerical study of the Schrödinger-Newton equation. Here, we present a new efficient numerical method which can be employed not only to solve the Schrödinger-Newton equation, but any Schrödinger-like equations, with different kind of interactions. The numerical method developed, which exploits the freedom provided by the gauge condition of the potential, is an improvement of the integrating factor technique. Optimal gauge conditions are derived considering the equation and the temporal numerical resolution with an adaptive embedded scheme of arbitrary order. We show that this optimization increases significantly the overall computational speed, sometimes by a factor five or more. Moreover, we make an extensive comparison of the new method developed, together with other popular integrators, commonly use to numerically solve this kind of equations. We focus in particular on splitting algorithms and methods belonging to the integrating factor family. Comparisons are done in one and two spatial dimensions, with different boundary conditions, both for the Schrödinger--Newton equation and the non-linear Schrödinger equation. We conclude that for the short range potential of the non-linear Schrödinger equation, integrating factor methods perform better than the Split-Step algorithm, while for the long range potential of the Schrödinger--Newton equation it depends on the particular system considered.
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