Tau functions and KP solutions on families of algebraic curves

TL;DR

This paper investigates tau functions on algebraic curves, analyzing their asymptotics and applying findings to construct KP hierarchy solutions using nonarchimedean theta functions and mixed solutions on M-curves.

Contribution

It provides a universal expression for tau functions on degenerating algebraic curves and constructs novel KP solutions involving nonarchimedean theta functions.

Findings

Universal asymptotic behavior of tau functions on degenerating curves

Construction of KP solutions with nonarchimedean theta functions

Mixed quasi-periodic and soliton solutions on M-curves

Abstract

Using abelian differentials and periods of the universal Mumford curve, we study the universal expression and asymptotic behavior of tau functions defined for stably degenerating families of algebraic curves with additional data. Furthermore, we apply this study to constructing solutions to the KP hierarchy which are expressed by nonarchimedean theta functions and by mixtures of quasi-periodic solutions with solitons containing real solutions defined on families of M-curves.