Abstract
Starting from a specific example of 4 × 4 matrix spectral problems, an integrable coupled hierarchy, which includes a three-component coupled nonlinear Schrödinger system as the first nonlinear one, is generated, and an associated oscillatory Riemann–Hilbert problem is formulated. With the nonlinear steepest descent method, the leading long-time asymptotics for the Cauchy problem of the three-component coupled nonlinear Schrödinger system is computed, through deforming the oscillatory Riemann–Hilbert problem into a model one which is solvable.