Degenerate Addition Formulas of the KP hierarchy and Applications

TL;DR

This work derives degenerate addition formulas for KP hierarchy tau functions by taking limits of a determinant-form addition formula, expressing shifts $\tau(t-\sum_i[\alpha_i])$ as Wronskians of wave functions $\Psi(t,\alpha_i)$ multiplied by exponential factors. It then proves the equivalence of tau functions generated by vertex operators and by Darboux transformations, linking the two solution-generating mechanisms through explicit determinant (Wronskian) representations and Grassmannian parametrizations. The theta-function extension translates the tau-formulas into new addition identities for Riemann theta functions, with genus-$g$ general forms and genus-1 reductions in terms of Weierstrass data, distinct from Fay’s classical limits. Overall, the paper unifies algebraic-geometric KP solutions, soliton constructions, and theta-function identities via universal addition formula structures, providing practical tools for relating backgrounds, singular curves, and quasi-periodic solutions.

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.

Theorems & Definitions (20)

• Theorem 2.1
• Theorem 2.2
• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 3.1
• Proposition 3.1
• Definition 4.1
• Theorem 4.1