In this Lecture I will discuss fundamental phenomena occurring in Cosmology, Quantum Field Theory and early Universe in the framework of models based on new achievements in Material Science.  In particular, I will focus on comparison of relativistic physics with nonlinear hyperbolic metamaterials designed by using condensate lattices in atomic systems or in semiconductor microstructures. We have demonstrated mapping of the initial Gross-Pitaevskii equation on nonlinear Klein-Gordon and/or Ginzburg-Landau-Higgs equations, which is inherent in matter fields within φ4-field theories. Formation of breather-like oscillating localized states—atomic oscillons—as well as kink-shaped states have been examined in details. The obtained results pave the way to simulating some analogues of fundamental cosmological processes occurring during our Universe’s evolution and to modeling nonlinear hyperbolic metamaterials with condensed matter (atomic) systems.