Tau functions associated to pseudodifferential operators of several variables

TL;DR

This paper extends the concept of tau functions to pseudodifferential operators in multiple variables, enabling the expression of Baker functions in terms of tau functions, thus generalizing integrable systems theory.

Contribution

It introduces a new framework for tau functions associated with multivariable pseudodifferential operators, expanding the classical single-variable theory.

Findings

Tau functions are defined for multivariable pseudodifferential operators.

Baker functions can be expressed via these tau functions.

The approach generalizes integrable systems to several variables.

Abstract

Pseudodifferential operators of several variables are formal Laurent series in the formal inverses of $\partial_1, ..., \partial_n$ with $\partial_i = d$ $1 \leq i \leq n$. As in the single variable case, Lax equations can be constructed using such pseudodifferential operators, whose solutions can be provided by Baker functions. We extend the usual notion of tau functions to the case of pseudodifferential operators of several variables so that each Baker function can be expressed in terms of the corresponding tau function.