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

We introduce a new unified two-parameter ${(\epsilon_x, \epsilon_t),|\epsilon_{x,t}=\pm1}$wavemodel(simplycalled${\mathcal Q}{\epsilon_x,\epsilon_t}^{(n)}$ model), connecting integrable local and nonlocal vector nonlinear Schr"odinger 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 ${{\mathcal I}, {\mathcal P}, {\mathcal T}, {\mathcal PT}}$symmetriesforthefirsttime\mathrm{.}The${\mathcal Q}{\epsilon_x,\epsilon_t}^{(n)}$modelwith$(\epsilon_x, \epsilon_t)=(\pm 1, 1)$ is shown to possess a Lax pair and infinite number of conservation laws, and to be $\mathcal{P}T$ symmetric. Moreover, the Hamiltonians with self-induced potentials are shown to be $\mathcal{P}T$ 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 ${\mathcal{Q}}_{{ϵ}_x,{ϵ}_t}^{(n)}$ model including bright and dark solitons, periodic wave solutions, and multi-rogue wave solutions.