Computations Related to the Construction of Finite Genus Solutions to the Kadomtsev-Petviashvili Equation

Abstract

Krichever's method of integrating certain partial differential equations using algebro-geometric techniques provides an explicit approach to the construction of finite-genus solutions to the Kadomtsev-Petviashvili (KP) equation. The closed-form expression that results can be used as an ansatz provided that the parameters of the ansatz have meaning in the context of this construction. The mathematical framework that is the basis for the algcurves package, a Maple package that provided computational tools for working with Riemann surfaces, is used to produce two procedures: the Krichever Construction Method (KCM), which encapsulates the construction of finite-genus solutions to KP using Krichever's method, and the Extended Dubrovin Method (EDM), which computes the parameters of the closed-form expression using no more than a Riemann matrix and a few parameters while extending ideas in Dubrovin's 1981 survey. Moreover, an approach to computing the Riemann constant vector that modifies Patterson's work is presented.