Algebro-geometric solutions and their reductions for the Fokas-Lenells hierarchy

TL;DR

This paper develops theta function representations for algebro-geometric solutions of the Fokas-Lenells hierarchy and derives N-dark solutions via Riemann surface degeneration.

Contribution

It introduces explicit theta function formulas for solutions and presents a reduction method to obtain dark solutions from algebro-geometric data.

Findings

Theta function representations for solutions are constructed.

N-dark solutions are obtained through surface degeneration.

The approach links algebro-geometric methods with solution reduction techniques.

Abstract

This paper is dedicated to provide theta function representations of algebro-geometric solutions for the Fokas-Lenells (FL) hierarchy through studying an algebro-geometric initial value problem. Further, we reduce these solutions into $N$-dark solutions through the degeneration of associated Riemann surfaces.