  Plasma state of matter can be studied in various types of situations. These
studies are of great interest in Astrophysical objects like galaxies, accretion
disk, neutron stars, etc, and laboratory plasma as well. Different objects
demand different approaches to investigate the dynamics of the plasma. The
relativistic effects in the motion of electrons in Quantum Plasma highly affect
the characteristics of the solitary structure of the wave with two-temperature
electrons. In this paper, considering the quantum hydrodynamic (QHD) model a
dispersion relation is derived, and using standard perturbation technique, a
mathematical model (i.e. nonlinear Schr\"odinger Equation) is studied for a
wave with relativistic and quantum effects in it. We study the analysis for
different values of diffraction coefficient, streaming velocity, and other
plasma parameters as well. We analyze the stable rogue wave structure using
NLSE and run simulations of those solitary profiles and rogue waves.
