Approximating tau-functions by theta-functions

TL;DR

This paper demonstrates that tau-functions of the KdV hierarchy can be approximated by hyperelliptic theta-functions, providing a new perspective on their structure and connections to algebraic geometry.

Contribution

It establishes a method to approximate tau-functions using theta-functions of finite genus, linking integrable systems with algebraic geometry.

Findings

Tau-functions can be approximated by hyperelliptic theta-functions up to quadratic terms.

The approximation is valid in the topology of graded formal series.

Application to the Witten--Kontsevich tau-function illustrates the method.

Abstract

We prove that the logarithm of an arbitrary tau-function of the KdV hierarchy can be approximated, in the topology of graded formal series by the logarithmic expansions of hyperelliptic theta-functions of finite genus, up to at most quadratic terms. As an example we consider theta-functional approximations of the Witten--Kontsevich tau-function.