The electrostatic pulses recorded by the Geotail spacecraft and labeled electrostatic solitary waves (ESW) are considered within the framework of Bernstein--Greene--Kruskal (BGK) solitons. The main goal of this paper is to develop sufficiently general and simple theoretical models of the nonlinear wave perturbations under study. Such models are necessary for understanding the physical structure and properties of the observed waves as well as for the treatment of the extensive experimental material, which has been accumulated on the ESW waveforms and for meaningful data processing. Examples of localized BGK modes described previously in literature are difficult to compare with the experimental data due to their particular nature. In contrast to these particular models, the general approach developed in the article applies to arbitrary particle distributions of the background plasma, velocities of the BGK solitons and wide variety of the recorded ESW waveforms. Relying on the theory of BGK solitons, with an emphasis on their general properties, the work contains a consistent analytical description and physical interpretation of ESW as effective spatial charges shielded by a collisionless plasma. The general analysis of localized BGK waves leads to a number of new results, such as unified physical models of ESW based on the additivity of the distribution function, as well as energetic relations and universal restrictions on the parameters and waveforms of the solitons. The new models and physical interrelations reveal universal features of the BGK soliton structure and allow a direct juxtaposition with the observations. The established interconnections between the physical characteristics of the waves agree well with the Geotail data on ESW waveforms. We also discuss less studied nonstationary phenomena, among which are wave emission by the BGK solitons, mechanisms of electron phase density hole interaction and certain questions of stability. Some of the most typical examples of the various BGK solutions described in the article are useful for a correlation between the theory and measurements as well as for an identification of the ESW waveforms observed in the magnetosphere.