Scaling self-similar formulation of the string equations of the hermitian one-matrix model

Abstract

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.