Univalent Functions and Integrable Systems

Abstract

We study one-parameter expanding evolution families of simply connected domains in the complex plane described by infinite systems of evolution parameters. These evolution parameters in some cases admit Hamiltonian formulation and lead to integrable systems. One example of such parameters is complex moments for the Laplacian growth that form a Whitham-Toda integrable hierarchy. Another example we deal with is related to expanding coefficient bodies for conformal maps given by Löwner subordination chains. The coefficients bodies are proved to form a Liouville partially integrable Hamiltonian system for each fixed index and the first integrals are obtained. We also discuss the contact structure of this system.

Energy characteristics of subordination chains

“…Originally, these arguments have been made by Prokhorov and the author in [21]. We remark also that an analogous Hamiltonian H can also be found in [4].…”

“…In [21] we studied another Hamiltonian system for a finite number of the coefficients of the function w(z, t) generated by the equation (5). It turns out that the Hamiltonian system generated by the coefficients is Liouville partially integrable and the first integrals were obtained and were proved to form a finite-dimensional grading algebra generated by truncated Kirillov operators [1], [10] and [11] for a representation of the Virasoro algebra.…”

From the Hele-Shaw Experiment to Integrable Systems: A Historical Overview

“…An attempt to formulate the classical action was launched. As for the Löwner evolution in general, it was shown [66,86] that there are conservative quantities which represent the generators of the Virasoro algebra. Relations to partially integrable systems were revealed.…”

Loewner Theory in annulus II: Loewner chains

“…Finally, we would like to mention recently discovered relations between classical Loewner Theory, integrable systems and representation of the Virasoro algebra, which appears in a number of fundamental problems in Mathematical Physics, see [13,23,24,26].…”