Integrable nonlocal vector nonlinear Schr\"odinger equation with self-induced parity-time-symmetric Potential

A two component nonlocal vector nonlinear Schr\"odinger equation (VNLSE) is considered with a self-induced ${\cal PT}$ symmetric potential. It is shown that the system possess a Lax pair and an infinite number of conserved quantities and hence integrable. Some of the conserved quantities like number operator, Hamiltonian etc. are found to be real-valued, in spite of these charges being non-hermitian. The soliton solution for the same equation is obtained through the method of inverse scattering transformation and the condition of reduction from nonlocal to local case is also mentioned. An inhomogeneous version of this VNLSE with space -time modulated nonlinear interaction term is also considered and a mapping of this Eq. with standard VNLSE through similarity transformation is used to generate its solutions.