Constrained Reductions of 2D dispersionless Toda Hierarchy, Hamiltonian Structure and Interface Dynamics

TL;DR

This paper investigates finite-dimensional reductions of the 2D dispersionless Toda hierarchy constrained by the string equation, exploring their Hamiltonian structure and applications to interface dynamics like Laplacian growth.

Contribution

It proves the consistency of these reductions and derives their Hamiltonian and Poisson structures, advancing understanding of integrable systems and interface evolution.

Findings

Reductions include polynomial, rational, and logarithmic solutions.

Hamiltonian structure of reduced dynamics is explicitly derived.

Poisson structure for rational reductions is established.

Abstract

Finite-dimensional reductions of the 2D 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'' 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