Explicit quasi-periodic solutions of the Kaup–Newell hierarchy

An algebraic curve of arithmetic genus n is introduced with the aid of the Lax matrix, from which we construct the meromorphic function on the algebraic curve and investigate its properties. Further, we straighten out all the flows associated with the Kaup–Newell hierarchy under the Abel–Jacobi coordinates. Finally, we achieve the explicit quasi-periodic solutions for the whole Kaup–Newell hierarchy, including the coupled derivative nonlinear Schrödinger equations.