Abelian solutions of the soliton equations and geometry of abelian varieties

TL;DR

This paper introduces abelian solutions to certain integrable equations and demonstrates that all such solutions are algebro-geometric, linking soliton equations with algebraic geometry.

Contribution

It defines abelian solutions for 2D Toda lattice and Hirota equations and proves their algebro-geometric nature, establishing a new connection between integrable systems and algebraic geometry.

Findings

All abelian solutions are algebro-geometric.

Established a link between soliton equations and algebraic geometry.

Provided a framework for classifying solutions using algebraic geometry.

Abstract

We introduce the notion of abelian solutions of the 2D Toda lattice equations and the bilinear discrete Hirota equation and show that all of them are algebro-geometric.