Abstract: In the last years, the study of relativistic degenerate dense magnetoplasmas has become one of the most important activities of quantum plasma physics because of their vital role in the investigation of the compact astrophysical objects. Therefore, in this thesis, the electrostatic travelling wave structures have been investigated in degenerate dense magnetoplasmas consisting of relativistic degenerate inertialess electronsg and positrons, as well as nondegenerate inertial cold ions. The well-known reductive perturbation method has been applied in obtaining the nonlinear Zakharov-Kuznetsov equation. The formation of the electrostatic solitary waves and periodic travelling waves is carried out by the bifurcation theory of planar dynamical systems. The results demonstrate that the nonlinear electrostatic travelling waves strongly depend on the concentration of positrons, the ion cyclotron frequency, and the direction cosines of the wave vector. The obtained results here may be useful for better understanding of the structure of nonlinear electrostatic travelling wave in dense plasmas, such as the interior of massive white dwarfs.