We investigate the character of ion-acoustic (IA) solitary waves in plasma structures that experience rotational motions. The plasma properties associated with white dwarf stars that are controlled by the relativistic parameter, ratio of ion and electron Fermi temperatures, rotational frequency, and total plasma frequency besides the equilibrium conditions are highlighted. This requires modelling a confined plasma system with cold relativistically degenerated electrons and thermal ions embedded in a straight external magnetic field. The plasma is considered initially rotating with specific angular frequencies with respect to the axis. The magnetic field is aligned with the axis. The nonlinear dynamics of the low-frequency ion-acoustic plasma modes of various nature are shaped through two governing equations, namely the Zakharov-Kuznetsov (ZK) equation and the extended Zakharov-Kuznetsov (eZK) equation. Solutions of the governing equations provide insight on the interplay of the controlling actors on the energy packet and the instability growth rate of the IA waves. This is in a sense that higher relativistic parameters of the medium elevate the instability growth rate of IA waves. Although both amplitude and width of the wave are inversely proportional to the relativistic parameter, it is their ratio that justifies the behaviour of the instability growth rate of IA waves. This ratio decreases when the ratio of the ion temperature and electron Fermi temperature increases. In contrast, this ratio increases with the total frequency, while the instability growth rate is directly proportional to the ratio of the ion temperature and electron Fermi temperature. The instability growth rate experiences saturation at high relativistic parameters where the ratio of the ion temperature to the electron Fermi temperature expedites the instability saturation. As the energy packet of IA waves measures the amplitude to width of their profile, the growth rate and saturation provide insight
