Integrable PTPT-symmetric local and nonlocal vector nonlinear Schrödinger equations: A unified two-parameter model

We introduce a new unified two-parameter {(ϵx,ϵt)|ϵx,t=±1} wave model (simply called Qϵx,ϵt(n) model), connecting integrable local and nonlocal vector nonlinear Schrödinger equations. The two-parameter (ϵx,ϵt)(ϵx,ϵt) family also brings insight into a one-to-one connection between four points (ϵx,ϵt)(ϵx,ϵt) (or complex numbers ϵx+iϵtϵx+iϵt) with {I,P,T,PT}{I,P,T,PT} symmetries for the first time. The Qϵx,ϵt(n) model is shown to possess a Lax pair and infinite number of conservation laws, and to be PTPT symmetric. Moreover, the Hamiltonians with self-induced potentials are shown to be PTPT symmetric only for Q−1,−1(n) model and to be TT symmetric only for Q+1,−1(n) model. The multi-linear form and some self-similar solutions are also given for the Qϵx,ϵt(n) model including bright and dark solitons, periodic wave solutions, and multi-rogue wave solutions.