Nonlocal general vector nonlinear Schroedinger equations:Integrability, PT symmetribility, and solutions

A family of new one-parameter (\epsilon_x=\pm 1) nonlinear wave models (called G_{\epsilon_x}^{(nm)} model) is presented, including both the local (\epsilon_x=1) and new integrable nonlocal $(\epsilon_x=-1)$ general vector nonlinear Schr\"odinger (VNLS) equations with the self-phase, cross-phase, and multi-wave mixing modulations. The nonlocal G_{-1}^{(nm)} model is shown to possess the Lax pair and infinite number of conservation laws for $m=1$. We also establish a connection between the G_{\epsilon_x}^{(nm)} model and some known models. Some symmetric reductions and exact solutions (e.g., bright, dark, and mixed bright-dark solitons) of the representative nonlocal systems are also found. Moreover, we find that the new general two-parameter (\epsilon_x, \epsilon_t) model (called G_{\epsilon_x, \epsilon_t}^{(nm)} model) including the G_{\epsilon_x}^{(nm)} model is invariant under the PT-symmetric transformation and the PT symmetribility of its self-induced potentials is discussed for the distinct two parameters (\epsilon_x, \epsilon_t)=(\pm 1, \pm 1).