Scaling Self-Similar Formulation of the String Equations of the Hermitian Matrix Model

The string equation appearing in the double scaling limit of the Hermitian one--matrix model, which corresponds to a Galilean self--similar condition for the KdV hierarchy, is reformulated as a scaling self--similar condition for the Ur--KdV hierarchy. A non--scaling limit analysis of the one--matrix model has led to the complexified NLS hierarchy and a string equation. We show that this corresponds to the Galilean self--similarity condition for the AKNS hierarchy and also its equivalence to a scaling self--similar condition for the Heisenberg ferromagnet hierarchy.