Loewner equations, Hirota equations and reductions of universal Whitham hierarchy

TL;DR

This paper explores finite variable reductions of the universal Whitham hierarchy of genus zero using dispersionless Hirota equations, providing a unified framework for understanding and deriving reduction equations.

Contribution

It introduces a unified approach to derive and justify reduction equations of the universal Whitham hierarchy via dispersionless Hirota equations, including multi-variable cases.

Findings

Unified derivation of L"owner-type and hydrodynamic equations

Reconfirmation of previous reduction results

Framework applicable to multi-variable reductions

Abstract

This paper reconsiders finite variable reductions of the universal Whitham hierarchy of genus zero in the perspective of dispersionless Hirota equations. In the case of one-variable reduction, dispersionless Hirota equations turn out to be a powerful tool for understanding the mechanism of reduction. All relevant equations describing the reduction (L\"owner-type equations and diagonal hydrodynamic equations) can be thereby derived and justified in a unified manner. The case of multi-variable reductions is not so straightforward. Nevertheless, the reduction procedure can be formulated in a general form, and justified with the aid of dispersionless Hirota equations. As an application, previous results of Guil, Ma\~{n}as and Mart\'{\i}nez Alonso are reconfirmed in this formulation.