Integrable ${\mathcal PT}$-symmetric local and nonlocal vector nonlinear Schr\"odinger equations: a unified two-parameter model

We introduce a new unified two-parameter $\{(\epsilon_x, \epsilon_t)\,|\epsilon_{x,t}=\pm1\}$ wave model (simply called ${\mathcal Q}_{\epsilon_x,\epsilon_t}^{(n)}$ model), connecting integrable local and nonlocal vector nonlinear Schr\"odinger equations. The two-parameter $(\epsilon_x, \epsilon_t)$ family also brings insight into a one-to-one connection between four points $(\epsilon_x, \epsilon_t)$ (or complex numbers $\epsilon_x+i\epsilon_t$) with $\{{\mathcal I}, {\mathcal P}, {\mathcal T}, {\mathcal PT}\}$ symmetries for the first time. The ${\mathcal Q}_{\epsilon_x,\epsilon_t}^{(n)}$ model with $(\epsilon_x, \epsilon_t)=(\pm 1, 1)$ is shown to possess a Lax pair and infinite number of conservation laws, and to be ${\mathcal PT}$ symmetric. Moreover, the Hamiltonians with self-induced potentials are shown to be ${\mathcal PT}$ symmetric only for ${\mathcal Q}_{-1,-1}^{(n)}$ model and to be ${\mathcal T}$ symmetric only for ${\mathcal Q}_{+1,-1}^{(n)}$ model. The multi-linear form and some self-similar solutions are also given for the ${\mathcal Q}_{\epsilon_x,\epsilon_t}^{(n)}$ model including bright and dark solitons, periodic wave solutions, and multi-rogue wave solutions.