The string equation for non-univalent functions

TL;DR

This paper explores the applicability of the string equation, originally for conformal maps, to non-univalent functions, demonstrating its validity for polynomials and certain rational functions.

Contribution

It extends the concept of the string equation to non-univalent functions, specifically polynomials and a class of rational functions, broadening its theoretical scope.

Findings

String equation holds for polynomials.

String equation holds for a special class of rational functions.

The study connects the string equation to Laplacian growth models.

Abstract

For conformal maps defined in the unit disk one can define a certain Poisson bracket that involves the harmonic moments of the image domain. When this bracket is applied to the conformal map itself together with its conformally reflected map the result is identically one. This is called the string equation, and it is closely connected to the governing equation, the Polubarinova-Galin equation, for the evolution of a Hele-Shaw blob of a viscous fluid (or, by another name, Laplacian growth). In the present paper we investigate to what extent the string equation makes sense and holds for non-univalent analytic functions. We give positive answers in two cases: for polynomials and for a special class of rational functions.