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

We use two families of parameters {(ϵxj,ϵtj)|ϵxj,tj=±1,j=1,2,…,n} to first introduce a unified novel hierarchy of two-family-parameter equations (simply called Qϵxn→,ϵtn→(n) hierarchy), connecting integrable local, nonlocal, novel mixed-local-nonlocal, and other nonlocal vector nonlinear Schrödinger (VNLS) equations. The Qϵxn→,ϵtn→(n) system with (ϵxj,ϵtj)=(±1,1),j=1,2,…,n is shown to possess Lax pairs and infinite number of conservation laws. Moreover, we also analyze the 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 {(ϵxj,ϵtj)|ϵxj=eiθxj,ϵtj=eiθtj,θxj,θtj∈[0,2π),j=1,2,…,n} to generate more types of nonlinear equations. The novel two-family-parameter (or multi-family-parameter for higher-dimensional cases) idea can also be applied to other local nonlinear evolution equations to find novel integrable and non-integrable nonlocal and mixed-local-nonlocal systems.