The vector nonlinear Schrödinger hierarchy

An article published recently by one of the authors (JNE) presents closed form solutions for zero-curvature representations of the vector nonlinear Schrödinger hierarchy. Several of these results are confirmed computationally but left unproven. In this article we begin by providing strict algebraic proofs of these results. The forms of the hierarchy’s associated spectral curves are investigated and proven to have only odd genus without the introduction of integration constants. We admit specific non-zero constants of integration to the hierarchy and show that even genus curves may be introduced in a very straightforward manner, thereby accessing the full family of finite gap solutions to the vector nonlinear Schrödinger equation. Finally an original construction of the infinite set of conserved densities of the hierarchy is given.