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

摘要

We use two families of parameters {(epsilon(xj), epsilon(tj)) vertical bar epsilon(xj), t(j) = +/- 1, j = 1, 2, ... , n} to first introduce a unified novel hierarchy of two-family-parameter equations (simply called Q(epsilon x (n) over bar,epsilon t (n) over bar)((n)) hierarchy), connecting integrable local, nonlocal, novel mixed-local-nonlocal, and other nonlocal vector nonlinear Schrodinger (VNLS) equations. The Q(epsilon x (n) over bar,epsilon t (n) over bar)((n)) system with (epsilon(xj), epsilon(tj)) = (+/- 1,1), j = 1, 2, ... , n is shown to possess Lax parrs 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 {(epsilon(xj), epsilon(tj))vertical bar epsilon(xj) = e(i theta xj) , epsilon(tj) = e(i theta tj) , theta(xj) , theta(tj) is an element of[0, 2 pi), 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. (C) 2017 Elsevier Ltd. All rights reserved.