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

We introduce a new unified two-parameter {(is an element of(x), is an element of(t)) vertical bar is an element of(x,t) = +/- 1} wave model (simply called Q(is an element of x,is an element of t)((n)) model), connecting integrable local and nonlocal vector nonlinear Schrodinger equations. The two-parameter (is an element of(x), is an element of(t)) family also brings insight into a one-to-one connection between four points (is an element of(x), is an element of(t)) (or complex numbers is an element of(x) +i(is an element of t)) with {I, P, T,PT} symmetries for the first time. The Q(is an element of x,is an element of 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 model. The multi-linear form and some self-similar solutions are also given for the Q(is an element of x,is an element of t)((n)) model including bright and dark solitons, periodic wave solutions, and multi-rogue wave solutions. (C) 2015 Elsevier Ltd. All rights reserved.