Reality problems for the Algebro-Geometric Solutions of Fokas-Lenell hierarchy

TL;DR

This paper investigates the reality conditions of algebro-geometric solutions for the Fokas-Lenell hierarchy, focusing on physically relevant solutions where the potential functions satisfy specific conjugation relations.

Contribution

It introduces a new approach to construct physically relevant solutions of the FL hierarchy by analyzing reality conditions using Vinikov's homological basis.

Findings

Identified conditions for real-valued solutions in the FL hierarchy.

Constructed classes of solutions satisfying physical reality constraints.

Extended the understanding of solution structures for the FL hierarchy.

Abstract

In a previous study, we obtained the algebro-geometric solutions and $n$-dark solitons of Forkas-Lenells (FL) hierarchy using algebro-geometric method. In this paper, we construct physically relevant classes of solutions for FL hierarchy by studying the reality conditions for $q=\pm \bar{r}$ based on the idea of Vinikov's homological basis.