An alternative approach to solve the mixed AKNS equations

The algebraic–geometric solutions of the mixed AKNS equations are investigated through a finite-dimensional Lie–Poisson Hamiltonian system, which is generated by the nonlinearization of the adjoint equation related to the AKNS spectral problem. First, each mixed AKNS equation can be decomposed into two compatible Lie–Poisson Hamiltonian flows. Then the separated variables on the coadjoint orbit are introduced to study these Lie–Poisson Hamiltonian systems. Further, based on the Hamilton–Jacobi theory, the relationship between the action-angle coordinates and the Jacobi-inversion problem is established. In the end, using Riemann–Jacobi inversion, the algebraic–geometric solutions of the first three mixed AKNS equations are obtained.