A theoretical investigation has been made on the nonlinear propagation of electrostatic perturbation modes in an unmagnetized relativistic degenerate dense plasma without and with charged dust grains. The electron-ion plasma (plasma without dust) mainly contains relativistic degenerate electrons and ions, while the dusty plasma (plasma with dust) mainly consists of relativistic degenerate electrons and ions and negatively or positively charged static dust. The reductive perturbation method, which allows us to derive the Korteweg-de Vries (K-dV) and Burgers equations, has been employed. The nonlinear structures (viz. solitary and shock waves), which are formed due to balance between nonlinearity and dispersion (in case of solitary structures) or dissipation (in case of shock structures), have been theoretically analyzed for certain ranges of plasma parameters in case of both non-relativistic and ultra-relativistic limits. The results, which have been found from these investigations have been pin-pointed in the end of each chapter. The implications of the results in some astrophysical compact objects have been briefly discussed.