Constrained reductions of two-dimensional dispersionless Toda hierarchy, Hamiltonian structure, and interface dynamics

Abstract

Finite-dimensional reductions of the two-dimensional dispersionless Toda hierarchy constrained by the “string equation” are studied. These include solutions determined by polynomial, rational, or logarithmic functions, which are of interest in relation to the “Laplacian growth” or Hele-Shaw problem governing interface dynamics. The consistency of such reductions is proved, and the Hamiltonian structure of the reduced dynamics is derived. The Poisson structure of the rationally reduced dispersionless Toda hierarchies is also derived.