In this study, we have explored the existence of electrostatic periodic (cnoidal) and solitary wave structures in a weak relativistic quantum electron-positron-ion plasma within a spherical geometry. By employing the perturbation technique, we derived a spherical Kadomtsev-Petviashvili equation (SKP), and through a suitable variable transformation, we proposed an exact solution for this equation. Furthermore, we utilized bifurcation theory to demonstrate the possibility of compressive structures in the form of electrostatic solitary waves and periodic traveling waves in such plasmas. Analytical and numerical results were obtained using the Sagdeev potential approach. We investigated the effects of electron number density and the positron to electron density ratio on the nonlinear electrostatic traveling and solitary waves. The cnoidal wave structure always lies within the separatrix, according to the phase plane portraiture of these nonlinear wave structures. We discovered through numerical analysis that the amplitude and phase of the cnoidal wave both decrease as the positron to electron density ratio rises. Additionally, an increase in positron density resulted in a decrease in the amplitude of the soliton and an increase in its width. These findings are of interest in studying nonlinear electrostatic traveling waves in dense quantum e-p plasmas, which may exist in dense stellar objects such as massive white dwarfs.
