Construction of KdV flow I. Tau function via Weyl function

TL;DR

This paper presents a new representation of tau-functions for the KdV equation using Weyl functions associated with 1D Schrödinger operators, enabling broader initial data classes.

Contribution

It introduces a novel representation of tau-functions via Weyl functions, extending the scope of initial data for the KdV equation beyond previous frameworks.

Findings

New representation of tau-functions using Weyl functions

Extension of initial data classes for KdV solutions

Potential for broader applicability in integrable systems

Abstract

Sato introduced the tau-function to describe solutions to a wide class of completely integrable differential equations. Later Segal-Wilson represented it in terms of the relevant integral operators on Hardy space of the unit disc. This paper gives another representation of the tau-functions by the Weyl functions for 1d Schr\"odinger operators with real valued potentials, which will make it possible to extend the class of initial data for the KdV equation to more general one.