Hyperelliptic sigma functions and the Kadomtsev-Petviashvili equation

TL;DR

This paper develops a hyperelliptic function theory using multidimensional sigma functions to explicitly solve the KP equations and analyze how solutions depend on hyperelliptic curve coefficients.

Contribution

It introduces a new hyperelliptic function framework and provides explicit formulas for KP solutions, addressing the dependence on curve coefficients.

Findings

Explicit formulas for hyperelliptic solutions to KP equations

Solution to the problem of coefficient dependence in hyperelliptic solutions

Development of a multidimensional sigma function theory

Abstract

In this paper, a theory of hyperelliptic functions based on multidimensional sigma functions is developed and explicit formulas for hyperelliptic solutions to the Kadomtsev-Petviashvili equations KP-I and KP-II are obtained. The long-standing problem of describing the dependence of these solutions on the variation of the coefficients of the defining equation of a hyperelliptic curve, which are integrals of the equations, is solved.