Degenerate Addition Formulas of the KP hierarchy and Applications

TL;DR

This paper explores degenerate addition formulas of the KP hierarchy, revealing new relations between tau functions, wave functions, and theta functions, with applications to vertex operators, Darboux transformations, and Riemann surfaces.

Contribution

It introduces a new addition formula for the KP hierarchy derived from determinant limits, linking various solution methods and Riemann theta functions.

Findings

New addition formula for tau functions expressed via Wronskian determinants.

Clarification of the relation between vertex operator and Darboux transformation solutions.

Derivation of a novel addition formula for Riemann's theta functions.

Abstract

It is well known that tau functions of the KP hierarchy satisfy addition formulas. We consider the general addition formula in the determinant form and take a certain limit of it. It expresses certain shifts of a tau function in terms of the Wronskian determinants of wave functions at various values of the spectral parameter. As an application the relation between solutions created by vertex operators and those created by Darboux transformations is clarified. As another application the new addition formula for Riemann's theta functions of Riemann surfaces is obtained by considering theta function solutions of the KP hierarchy. This addition formula is different from any of formulas in Fay's book.