Low-frequency ion-acoustic waves are analysed on the ion time-scale, in a three-component electron–ion space plasma. The solitary waves propagate in the positive $x$ direction relative to an ambient magnetic field $\overrightarrow{B}_{0}$ which forms static background for a configuration consisting of cool fluid ions and both warm and hot Boltzmann distributed electrons with temperatures $T_{ic}$, $T_{ew}$ and $T_{eh}$, respectively. We derive linear dispersion relation for the waves by introducing first-order density, pressure and velocity perturbations into the ion fluid equations. Additionally, the variation in the nonlinear structure of the waves are investigated by carrying out a full parametric analysis utilising our numerical code. Our results reveal that ion-acoustic waves exhibit well-defined nonlinear spikes at speeds of $M \geq 2.25$ and an electric field amplitude of $E_{0} = 0.85$. It is also shown that low wave speeds ($M \leq 2$), higher densities of the hot electrons, antiparallel drifting of the cool fluid ions, and increased ion temperatures all lead to significant dispersive effects. The ion-acoustic plasma waves featured in this paper have forms that are consistent with those classified as the type-A and type-B broadband electrostatic noise (BEN) observed in the data obtained from earlier satellite missions.