New Construction of Algebro-Geometric Solutions to the Modified Kadomtsev-Petviashvili Hierarchy

TL;DR

This paper develops a new unified method for constructing algebro-geometric solutions to the entire modified Kadomtsev-Petviashvili hierarchy in 2+1 dimensions, extending previous approaches.

Contribution

It introduces a novel extension of Gesztesy-Holden's method to the 2+1 dimensional case, linking solutions of GI and mKP hierarchies using advanced algebraic geometry tools.

Findings

Unified construction of solutions for the mKP hierarchy.

Relations established between GI and mKP solutions.

Introduction of a special function on algebraic curves for solution representation.

Abstract

We extend Gesztesy-Holden's method to 2+1 dimensional case to obtain a unified construction to the algebro-geometric solutions of the whole modified Kadomtsev-Petviashvili (mKP) hierarchy. Our tools include the relations between solutions of the Gerdjikov-Ivanov (GI) and mKP hierarchy, the Baker-Akhiezer functions in 2+1 dimensions, a special function $\psi_1(P)\psi_2(P^*)$ on $X\times \mathbb{R}^3$ and Dubrovin-type equations for auxiliary divisors.