/ 93 07 01 7 v 1 2 J ul 1 99 3 AKNS Hierarchy , Self – Similarity , String Equations and the Grassmannian ∗

In this paper the Galilean, scaling and translational self–similarity conditions for the AKNS hierarchy are analysed geometrically in terms of the infinite dimensional Grassmannian. The string equations found recently by non–scaling limit analysis of the one–matrix model are shown to correspond to the Galilean self–similarity condition for this hierarchy. We describe, in terms of the initial data for the zero–curvature 1–form of the AKNS hierarchy, the moduli space of these self–similar solutions in the Sato Grassmannian. As a byproduct we characterize the points in the Segal–Wilson Grassmannian corresponding to the Sachs rational solutions of the AKNS equation and to the Nakamura–Hirota rational solutions of the NLS equation. An explicit 1– parameter family of Galilean self–similar solutions of the AKNS equation and the associated solution to the NLS equation is determined. Partially supported by CICYT proyecto PB89-0133 Research supported by British Council’s Fleming award —postdoctoral MEC fellowship GB92 00411668, and postdoctoral EC Human Capital and Mobility individual fellowship ERB40001GT922134