Quasi-periodic solutions for an extension of AKNS hierarchy and their reductions

Quasi-periodic solutions of an extension of the AKNS hierarchy are derived. Based on finite-order expansion of the Lax matrix, the elliptic coordinates are introduced, from which the equations are separated into solvable ordinary differential equations. Then various flows are straightened out through the Abel–Jacobi coordinates. By the standard Jacobi inversion treatment, explicit quasi-periodic solutions of the evolution equations are constructed in terms of the Riemann theta functions. Furthermore, the solutions of a new generalized nonlinear Schrödinger equation, which are the reductions of the above system, are deduced.