Classical limit and asymptotic Bethe ansatz for integrable stringy sigma models

I will show how, starting from the Zamolodchikovs S-matrix for the physical particles of  SO(n) quantum sigma models, the full set of finite gap solutions can be reproduced, in terms of the underlying algebraic curve. In a slightly subtler approximation, the asymptotic Bethe ansatz of Arutynov-Frolov-Staudacher for string theory will be reproduced from the same setup. The problems with generalization to the full supersting on AdS5xS5 background will be discussed.