Asymptotic Bethe Ansatz from String Sigma Model on S³ x R

We derive the asymptotic Bethe ansatz (AFS equations) for the string on S^3 x R sector of AdS_5 x S^5 from the integrable nonhomogeneous dynamical spin chain for the string sigma model proposed in GKSV. It is clear from the derivation that AFS equations can be viewed only as an effective model describing a certain regime of a more fundamental inhomogeneous spin chain.