Integrable Systems in the Infinite Genus Limit

TL;DR

This paper explores the extension of integrable systems linked to hyperelliptic curves to the infinite genus case, analyzing classical theories and transformations within this limit, especially for the KdV hierarchy.

Contribution

It introduces an elementary approach to infinite genus hyperelliptic curves and generalizes classical integrable systems theory to this new setting.

Findings

Generalization of Burchnall-Chaundy theory to infinite genus

Analysis of Darboux transformations on infinite genus curves

Identification of the Riemann surface with a limit of Burchnall-Chaundy curves

Abstract

We provide an elementary approach to integrable systems associated with hyperelliptic curves of infinite genus. In particular, we explore the extent to which the classical Burchnall-Chaundy theory generalizes in the infinite genus limit, and systematically study the effect of Darboux transformations for the KdV hierarchy on such infinite genus curves. Our approach applies to complex-valued periodic solutions of the KdV hierarchy and naturally identifies the Riemann surface familiar from standard Floquet theoretic considerations with a limit of Burchnall-Chaundy curves.