A Characterization of All Elliptic Solutions of the AKNS Hierarchy

Abstract

An explicit characterization of all elliptic algebro-geometric solutions of the AKNS hierarchy is presented. Our approach is based on (an extension of) a classical theorem of Picard, which guarantees the existence of solutions which are elliptic of the second kind for n-th order ordinary differential equations with elliptic coefficients associated with a common period lattice. As by-products we offer a detailed Floquet analysis of Dirac-type differential expressions with periodic coefficients, specifically emphasizing algebro-geometric coefficients, and a constructive reduction of singular hyperelliptic curves and their Baker-Akhiezer functions to the nonsingular case.