Darboux transformation and soliton solutions in the parity-time-symmetric nonlocal vector nonlinear Schrödinger equation

In this Letter, we study a nonlocal vector nonlinear Schrödinger (NVNLS) equation with self-induced parity-time-symmetric potential. We construct the N-fold Darboux transformation in terms of compact determinant forms. Starting from the non-vanishing background, we give the general solution of spectral problem, which allows us to derive many different types of exact analytical solutions of the NVNLS equation, like the breathers, dark and anti-dark solitons. With three-component case as an example, we display three types of two-soliton elastic collision behaviors: breather and dark soliton, breather and anti-dark soliton, dark soliton and anti-dark soliton.