  Based on the quantum hydrodynamic model, a new relationship between the
electrostatic-potential and the electron-density in the ultra-dense plasma is
derived. Propagation of arbitrary amplitude nonlinear ion waves is, then,
investigated in a completely degenerate dense dusty electron-ion plasma, using
this new energy relation, for the relativistic electrons in the ground of
quantum hydrodynamics model and the results are compared to the case of
semiclassical Thomas-Fermi dusty plasma. Using the standard pseudo-potential
approach it is remarked that the Fermi-Dirac plasma, in contrast to the
Thomas-Fermi counterpart, accommodates a wide variety of nonlinear excitations
such as positive-/negative-potential ion solitary and periodic waves,
double-layers and double-wells. It is also remarked that, the relativistic
degeneracy parameter which relates to the mass-density of plasma has
significant effects on the allowed matching-speed range in dusty Fermi-Dirac
plasmas.
