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

Abstract

By introducing Lenard recursion equations, we derive a general coupled nonlinear Sch$\ddot{\mathrm{o}}$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.

We have found the results of this manuscript have been obtained by others