Quasi-periodic solutions of mixed AKNS equations

A hierarchy of new nonlinear evolution equations, which are composed of the positive and negative AKNS flows, is proposed. On the basis of the theory of algebraic curves, the corresponding flows are straightened using the Abel–Jacobi coordinates. The meromorphic function ϕϕ, the Baker–Akhiezer vector ψ̄, and the hyperelliptic curve KnKn are introduced and, by using these, quasi-periodic solutions of the first three nonlinear evolution equations in the hierarchy are constructed according to the asymptotic properties and the algebro-geometric characters of ϕϕ, ψ̄ and KnKn.