  The Sakai-Sugimoto model is the preeminent example of a string theory
description of holographic QCD, in which baryons correspond to topological
solitons in the bulk. Here we investigate the validity of various
approximations of the Sakai-Sugimoto soliton that are used widely to study the
properties of holographic baryons. These approximations include the flat space
self-dual instanton, a linear expansion in terms of eigenfunctions in the
holographic direction and an asymptotic power series at large radius. These
different approaches have produced contradictory results in the literature
regarding properties of the baryon, such as relations for the electromagnetic
form factors. Here we determine the regions of validity of these various
approximations and show how to relate different approximations in contiguous
regions of applicability. This analysis clarifies the source of the
contradictory results in the literature and resolves some outstanding issues,
including the use of the flat space self-dual instanton, the detailed
properties of the asymptotic soliton tail, and the role of the UV cutoff
introduced in previous investigations. A consequence of our analysis is the
discovery of a new large scale, that grows logarithmically with the 't Hooft
coupling, at which the soliton fields enter a nonlinear regime. Finally, we
provide the first numerical computation of the Sakai-Sugimoto soliton and
demonstrate that the numerical results support our analysis.
