A novel hierarchy of two-family-parameter equations: Local, nonlocal, and mixed-local-nonlocal vector nonlinear Schrodinger equations

Abstract

We use two families of parameters $\{(ε_{x_j}, ε_{t_j})\,|\,ε_{x_j,t_j}=\pm1,\, j=1,2,...,n\}$ to first introduce a unified novel two-family-parameter system (simply called ${\mathcal Q}^{(n)}_{ε_{x_{\vec{n}}},ε_{t_{\vec{n}}}}$ system), connecting integrable local, nonlocal, novel mixed-local-nonlocal, and other nonlocal vector nonlinear Schrödinger (VNLS) equations. The ${\mathcal Q}^{(n)}_{ε_{x_{\vec{n}}}, ε_{t_{\vec{n}}}}$ system with $(ε_{x_j}, ε_{t_j})=(\pm 1, 1),\, j=1,2,...,n$ is shown to possess Lax pairs and infinite number of conservation laws. Moreover, we also analyze the ${\mathcal PT}$ symmetry of the Hamiltonians with self-induced potentials. The multi-linear forms and some symmetry reductions are also studied. In fact, the used two families of parameters can also be extended to the general case $\{(ε_{x_j}, ε_{t_j}) | ε_{x_j} = e^{iθ_{x_j}}, ε_{t_j} = e^{iθ_{t_j}},\, θ_{x_j}, θ_{t_j}\in [0, 2π),\, j=1,2,...,n\}$ to generate more types of nonlinear equations. The two-family-parameter idea used in this paper can also be applied to other local nonlinear evolution equations such that novel integrable and non-integrable nonlocal and mixed-local-nonlocal systems can also be found.