Abstract
We construct matrix integrable fourth-order nonlinear Schrodinger equations through reducing the Ablowitz-Kaup-Newell-Segur matrix eigenvalue problems. Based on properties of eigenvalue and adjoint eigenvalue problems, we solve the corresponding reflectionless Riemann-Hilbert problems, where eigenvalues could equal adjoint eigenvalues, and formulate their soliton solutions via those reflectionless Riemann-Hilbert problems. Soliton solutions are computed for three illustrative examples of scalar and two-component integrable fourth-order nonlinear Schrodinger equations.