In this paper, the propagation of acoustic solitons in an electron-positron (e-p) degenerate plasma with the relativistically degenerate electrons and positrons is studied. Moreover, we considered stationary positive ions for neutralizing the plasma background. Using the reductive perturbation method, we obtained a modified Korteweg–de Vries equation. By applying a Jacobi elliptic function expansion method, we have proposed the exact analytical solitary, kink, and periodic solutions for the nonlinear wave structures. Also, using the phase plane analysis three equilibrium points have been perceived two of them should be centers and the last is a saddle for quantum acoustic waves. The numerical and computational results show that firstly some of them should be discarded by physical values and then these solutions have been influenced by the plasma parameters such as the electron degenerate density parameter () and the ion-to-electron density ratio () which peaked out in white dwarf stars. It is found that both the amplitude and the width of the nonlinear structures are affected by the mentioned parameters. This investigation is useful for the relativistically degenerate plasma media in ultra-dense astrophysical environments (such as neutron stars, white dwarfs, etc.), inertia fusion science, experiments involving the interaction of laser and solid density plasma, and inertial confinement fusion in which the relativistic effects play a major role.
