Loewner evolution and finite dimensional reductions of integrable systems

TL;DR

This paper explores how time splitting in the Loewner and Loewner-Kufarev equations can generate known integrable systems, linking these equations to broader integrable hierarchies.

Contribution

It introduces a reverse process demonstrating that time splitting of Loewner equations produces integrable systems, revealing new connections between these equations and integrable hierarchies.

Findings

Time splitting in Loewner equations yields known integrable systems.

The approach links Loewner evolution to integrable hierarchies.

Provides a new perspective on reductions of integrable systems.

Abstract

The Loewner equation is known as a one-dimensional reduction of the Benney chain as well as the dispersionless KP hierarchy. We propose a reverse process showing that time splitting in the Loewner or the Loewner-Kufarev equation leads to some known integrable systems.