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

TL;DR

This paper introduces a unified two-family-parameter framework for vector nonlinear Schrödinger equations, encompassing local, nonlocal, and mixed types, with analysis of integrability, symmetries, and potential extensions.

Contribution

The paper presents a novel two-family-parameter system unifying various local and nonlocal VNLS equations, including their Lax pairs and conservation laws, and explores symmetry and extension possibilities.

Findings

Existence of Lax pairs and infinite conservation laws for the system.

Analysis of PT symmetry in Hamiltonians with self-induced potentials.

Extension of parameters to generate more nonlinear equations.

Abstract

We use two families of parameters $\{(\epsilon_{x_j}, \epsilon_{t_j})\,|\,\epsilon_{x_j,t_j}=\pm1,\, j=1,2,...,n\}$ to first introduce a unified novel two-family-parameter system (simply called ${\mathcal Q}^{(n)}_{\epsilon_{x_{\vec{n}}},\epsilon_{t_{\vec{n}}}}$ system), connecting integrable local, nonlocal, novel mixed-local-nonlocal, and other nonlocal vector nonlinear Schr\"odinger (VNLS) equations. The ${\mathcal Q}^{(n)}_{\epsilon_{x_{\vec{n}}}, \epsilon_{t_{\vec{n}}}}$ 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 ${\mathcal PT}$ 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 $\{(\epsilon_{x_j}, \epsilon_{t_j}) | \epsilon_{x_j} = e^{i\theta_{x_j}}, \epsilon_{t_j} = e^{i\theta_{t_j}},\, \theta_{x_j}, \theta_{t_j}\in [0, 2\pi),\, j=1,2,...,n\}$ 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.