Integrability and exact solutions for a nonlocal matrix nonlinear Schrödinger equation with self-induced $\mathcal{PT}$-symmetric potentials

A nonlocal matrix nonlinear Schrödinger equation with the self-induced parity-time $\mathcal{PT}$-symmetric potentials and its Darboux transformation have been studied. We obtain several types of solutions as taking different spectral parameter and seeds, include localized soliton solutions, dark-antidark soliton solutions, exact solutions which describe the onset and nonlinear development of the modulational instability of continuous wave states, parallel traveling wave solution, and rogue wave solutions. Besides, the Infinite conservation laws have been derived.