Integrable nonlocal nonlinear Schrödinger hierarchies of type (-λ*,λ) and soliton solutions

Two simultaneous nonlocal group constraints of the Ablowitz-Kaup-Newell-Segur matrix eigenvalue problems are discussed, of which one constraint changes the eigenvalue parameter into its negative of the complex conjugate and the other constraint does not change the eigenvalue parameter. Under those two constraints, mixed-type nonlocal integrable nonlinear Schrödinger hierarchies are generated. Further, based on specific distributions of eigenvalues and adjoint eigenvalues, a formulation of soliton solutions is established via the corresponding reflection-less generalized Riemann-Hilbert problems, where eigenvalues and adjoint eigenvalues could be equal.