Analytic functions and integrable hierarchies--characterization of tau functions

TL;DR

This paper proves the dispersionless Hirota equations for several integrable hierarchies and shows these equations uniquely characterize their tau functions, linking the hierarchies through complex analysis techniques.

Contribution

It establishes the dispersionless Hirota equations as characterizations of tau functions for multiple integrable hierarchies using classical complex analysis.

Findings

Proved dispersionless Hirota equations for Toda, mKP, and KP hierarchies.

Showed Hirota equations uniquely characterize tau functions.

Linked different integrable hierarchies through these equations.

Abstract

We prove the dispersionless Hirota equations for the dispersionless Toda, dispersionless coupled modified KP and dispersionless KP hierarchies using an idea from classical complex analysis. We also prove that the Hirota equations characterize the tau functions for each of these hierarchies. As a result, we establish the links between the hierarchies.