Whitham hierarchy in growth problems

TL;DR

This paper explores the connection between Laplacian growth and the Whitham hierarchy, enabling new exact solutions for complex growth problems through integrable systems techniques.

Contribution

It establishes a novel equivalence between Laplacian growth and the Whitham hierarchy, facilitating the derivation of exact solutions in multiply-connected cases.

Findings

Identifies a class of exact solutions for Laplacian growth

Links growth problems to integrable hierarchies

Provides a method to solve these solutions via hydrodynamic equations

Abstract

We discuss the recently established equivalence between the Laplacian growth in the limit of zero surface tension and the universal Whitham hierarchy known in soliton theory. This equivalence allows one to distinguish a class of exact solutions to the Laplacian growth problem in the multiply-connected case. These solutions corerespond to finite-dimensional reductions of the Whitham hierarchy representable as equations of hydrodynamic type which are solvable by means of the generalized hodograph method.