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.