The coupled Schodinger hierarchy associated with third-order algebraic curves and algebro-geometric solutions

TL;DR

This paper develops a comprehensive algebro-geometric framework for solving the coupled nonlinear Schrödinger hierarchy, linking algebraic curves, Baker-Akhiezer functions, and explicit theta function solutions.

Contribution

It introduces a novel connection between the CNLS hierarchy and third-order algebraic curves, providing explicit algebro-geometric solutions and associated functions.

Findings

Derived the CNLS hierarchy using Lenard recursion equations.

Established algebraic curve and Baker-Akhiezer functions for solutions.

Obtained explicit theta function representations of solutions.

Abstract

By introducing Lenard recursion equations, we derive a general coupled nonlinear Sch$\mathrm{\ddot{o}}$dinger (CNLS) hierarchy associated with well-known Manakov system and Sasa-Satsuma system. Based on the characteristic polynomial of Lax matrix for CNLS hierarchy, we obtain a third order algebraic curve $\mathcal{K}_{m-2}$ of arithmetic genus $m-2$, from which we establish the associated Baker-Ahhiezer functions, meromorphic function and Dubrovin-type equations for analogs of Dirichlet and Neumann divisors. Using these results and the theory of algebraic curve, we obtain the explicit theta function representations of the Baker-Ahhiezer functions, the meromorphic function, and in particular, of the algebro-geometric solutions for the entire CNLS hierarchy.