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

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