Whitham-Toda Hierarchy in the Laplacian Growth Problem

Abstract

The Laplacian growth problem in the limit of zero surface tension is proved to be equivalent to finding a particular solution to the dispersionless Toda lattice hierarchy. The hierarchical times are harmonic moments of the growing domain. The Laplacian growth equation itself is the quasiclassical version of the string equation that selects the solution to the hierarchy.Comment: 7 pages, no figures, Talk given at the Workshop NEEDS 99 (Crete, Greece, June 1999