Matrix Integrable Fourth-Order Nonlinear Schr?dinger Equations and Their Exact Soliton Solutions

We construct matrix integrable fourth-order nonlinear Schrödinger 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 Schrödinger equations.