Uniformly constructing finite-band solutions for a family of derivative nonlinear Schrödinger equations

Based a spectral problem with an arbitrary parameter and Lenard operator pairs, we derive a generalized Kaup-Newell type hierarchy of nonlinear evolution equations, which is explicitly related to many important equations such as the Kundu equation, the Kaup-Newell (KN) equation, the Chen-Lee-Liu (CLL) equation, the Gerdjikov-Ivanov (GI) equation, the Burgers equation, the MKdV equation and the Sharma-Tasso-Olver equation. Furthermore, the separation of variables for x- and tm-constrained flows of the the generalized Kaup-Newell hierarchy is shown. Especially the Kundu, KN, CLL and GI equations are uniformly decomposed into systems of solvable ordinary differential equations. A hyperelliptic Riemann surface and Abel-Jacobi coordinates are introduced to straighten the associated flow, from which the algebro-geometric solutions of these equations are explicitly constructed in terms of the Riemann theta functions by standard Jacobi inversion technique.