Loewner equations and reductions of dispersionless hierarchies

TL;DR

This paper explores the connection between Loewner equations from complex analysis and their application in reducing dispersionless integrable hierarchies, providing a comparative analysis of both approaches and their solutions.

Contribution

It compares the derivation of Loewner equations in complex analysis with their role in integrable systems reductions, including multi-variable cases and hydrodynamic systems.

Findings

Different Loewner equations describe various reductions of dispersionless hierarchies.

Multi-variable reductions involve Loewner equations coupled with hydrodynamic PDEs.

The generalized hodograph method proves the solvability of these systems.

Abstract

The equations of Loewner type can be derived in two very different contexts: one of them is complex analysis and the theory of parametric conformal maps and the other one is the theory of integrable systems. In this paper we compare the both approaches. After recalling the derivation of L\"owner equations based on complex analysis we review one- and multi-variable reductions of dispersionless integrable hierarhies (dKP, dBKP, dToda, and dDKP). The one-vaiable reductions are described by solutions of different versions of Loewner equation: chordal (rational) for dKP, quadrant for dBKP, radial (trigonometric) for dToda and elliptic for DKP. We also discuss multi-variable reductions which are given by a system of Loewner equations supplemented by a system of partial differential equations of hydrodynamic type. The solvability of the hydrodynamic type system can be proved by means of the generalized hodograph method.