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 $\{({\epsilon }_{x_j},{\epsilon }_{t_j})|{\epsilon }_{x_j,t_j}=\pm 1,j=1,2,...,n\}$ to first introduce a unified novel two-family-parameter system (simply called ${\mathcal{Q}}_{{\epsilon }_{x_{\vec{n}}},{\epsilon }_{t_{\vec{n}}}}^{(n)}$ system), connecting integrable local, nonlocal, novel mixed-local-nonlocal, and other nonlocal vector nonlinear Schrödinger (VNLS) equations. The ${\mathcal{Q}}_{{\epsilon }_{x_{\vec{n}}},{\epsilon }_{t_{\vec{n}}}}^{(n)}$ system with $({\epsilon }_{x_j},{\epsilon }_{t_j})=(\pm 1,1),j=1,2,...,n$ is shown to possess Lax pairs and infinite number of conservation laws. Moreover, we also analyze the $\mathcal{P}T$ 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 $\{({\epsilon }_{x_j},{\epsilon }_{t_j})|{\epsilon }_{x_j}=e^{i{\theta }_{x_j}},{\epsilon }_{t_j}=e^{i{\theta }_{t_j}},{\theta }_{x_j},{\theta }_{t_j}\in [0,2\pi ),j=1,2,...,n\}$ 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.

8 pages

References in corpus (1)

• Spontaneous Parity--Time Symmetry Breaking and Stability of Solitons in Bose-Einstein Condensates