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

A family of new one-parameter (ϵx=±1)(ϵx=±1) nonlinear wave models (called Gϵx(nm) model) is presented, including both the local (ϵx=1)(ϵx=1) and new integrable nonlocal (ϵx=−1)(ϵx=−1) general vector nonlinear Schrödinger (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=1m=1. We also establish a connection between the Gϵx(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 (ϵx,ϵt)(ϵx,ϵt) model (called Gϵx,ϵt(nm) model) including the Gϵx(nm) model is invariant under the PTPT-symmetric transformation and the PTPT symmetribility of its self-induced potentials is discussed for the distinct two parameters (ϵx,ϵt)=(±1,±1)(ϵx,ϵt)=(±1,±1).