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

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A family of new one-parameter ( ϵ x = ± 1 ) nonlinear wave models (called G ϵ x ( n m ) model) is presented, including both the local ( ϵ x = 1 ) and new integrable nonlocal ( ϵ x = − 1 ) general vector nonlinear Schrödinger (VNLS) equations with the self-phase, cross-phase, and multi-wave mixing modulations. The nonlocal G − 1 ( n m ) 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 ϵ x ( n m ) 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 ( ϵ x , ϵ t ) model (called G ϵ x , ϵ t ( n m ) model) including the G ϵ x ( n m ) model is invariant under the P T -symmetric transformation and the P T symmetribility of its self-induced potentials is discussed for the distinct two parameters ( ϵ x , ϵ t ) = ( ± 1 , ± 1 ) .